Universidade Estadual Paulista “Júlio de Mesquita Filho” Faculdade de Engenharia de Ilha Solteira Campus de Ilha Solteira Julio Cesar Eduardo de Souza Optimization in ultrasonic imaging systems: Use of metaheuristics to design sparse arrays and a heuristic to create an acquisiton system Ilha Solteira 2022 Julio Cesar Eduardo de Souza Optimization in ultrasonic imaging systems: Use of metaheuristics to design sparse arrays and a heuristic to create an acquisiton system Tese apresentada à Faculdade de Engenharia de Ilha Solteira – UNESP como parte dos requisi- tos para obtenção do título de Doutor na área de Engenharia Elétrica Área: Automação. Dr. Ricardo Tokio Higuti UNESP-FEIS Supervisor: Dr. Óscar Fernando Martínez-Graullera CSIC-ITEFI Co-supervisor: Ilha Solteira 2022 de Souza Optimization in ultrasonic imaging systems: Use of metaheuristics to design sparse arrays and a heuristic to create an acquisiton systemIlha Solteira2022 134 Sim Tese (doutorado)Engenharia ElétricaAutomaçãoNão . FICHA CATALOGRÁFICA Desenvolvido pelo Serviço Técnico de Biblioteca e Documentação Souza, Julio Cesar Eduardo de. Optimization in ultrasonic imaging systems: use of metaheuristics to design sparse arrays and a heuristic to create an acquisiton system / Julio Cesar Eduardo de Souza. -- Ilha Solteira: [s.n.], 2022 133 f. : il. Tese (doutorado) - Universidade Estadual Paulista. Faculdade de Engenharia de Ilha Solteira. Área de conhecimento: Automação, 2022 Orientador: Ricardo Tokio Higuti Coorientador: Óscar Fernando Martínez Graullera Inclui bibliografia 1. Algoritmos de otimização estocástico. 2. Arrays 2D esparsos. 3. Array lineares esparsos. 4. Metaheurísticas. S729o Dedico este trabalho a Sônia Souza que com muito carinho e apoio não mediu esforços para que eu chegasse até esta etapa na minha vida. Ao meu pai, Paulo Souza (in memoriam), que me inspira a buscar por meus objetivos de vida. Quando criança, o senhor desejava que eu me tornasse doutor, faltou esclarecer qual. ACKNOWLEDGMENT À minha amada Jéssica, por muitas vezes foi coagida a debater sobre a minha tese, me segurou quando desejei cair e juntos estamos enfrentando os desafios. À minha cachorrinha Luna, companheira durante o desenvolvimento e escrita da tese. Os pas- seios diários me ajudaram a ter ideias e pensar nas soluções dos problemas encontrados. Ao meu orientador Prof. Dr. Ricardo Higuti, pela confiança no meu trabalho, conselhos e mentoria na minha carreira. Ao meu coorientador Dr. Óscar Martínez Graullera, pelos ensinamentos repassados e ajuda no desenvolvimento da tese. À minha avó Maria Takahashi (in memoriam), que esteve presente na minha criação e ajudou a formar a pessoa que sou hoje. Aos meus companheiros do LUS, em específico Marcelo Tiago e Vander Prado, pela colabo- ração direta e indireta para que este trabalho fosse construído. Aos professores que me ajudaram e motivaram a chegar até este estágio da minha vida. Espero conseguir motivar as pessoas assim como os senhores me motivaram. A Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) e o Conselho Na- cional de Desenvolvimento Científico e Tecnológico (CNPq), pelo auxílio financeiro. À minha família, pois acredito que sem o apoio deles seria muito difícil vencer esse desafio. E a todos os que direta ou indiretamente fizeram parte da minha formação, o meu obrigado. "I have not failed. I have found 10,000 ways that it will not work" Thomas Edison RESUMO Imagens ultrassônicas possuem importante papel no diagnóstico médico e ensaio não destrutivo. Uma das alternativas para geração de imagens consiste em utilizar um array, um transdutor composto por um conjunto de elementos piezelétricos, para gerar diversas frentes de ondas e amostrar suas reflexões. Uma característica física importante para construção dos arrays é a sua dimensão. Quanto maior a extensão do array, melhor será a resolução lateral da imagem gerada. Além disso, uma recomendação de fabricação dos arrays é que o centro do seus elementos precisam estar espaçados por uma distância (pitch) menor ou igual a 0, 5λ, em que λ é o comprimento da onda gerada pelo transdutor. Desta forma, as imagens geradas por esses arrays não apresentam artefatos causados pelos lóbulos de espaçamento. A recomen- dação para o uso de arrays é que seja o maior array possível, respeitando a restrição de pitch. No entanto, o volume de dados, recursos e custo de fabricação aumentam proporcionalmente à medida que o número de elementos no array aumenta, o que pode torná-lo impraticável de- pendendo da aplicação. Esta tese propõe técnicas para reduzir o uso de recursos em sistemas ultrassônicos visando obter imagens com alta resolução lateral e mitigar eventuais desvanta- gens. No primeiro estágio, arrays esparsos lineares cujos pitches são superiores que 0, 5λ são estudados. Propõe-se uma nova estratégia para projetar esses arrays, na qual é apresentada uma nova codificação matemática para os arrays esparsos e uma função de aptidão baseada na equação de energia e entropia das PSFs (Point Spread Function). Posteriormente, algorit- mos de otimização estocástico são utilizados para desenhar as configurações esparsas lineares. A função aptidão proposta foi comparada com a função aptidão mais utilizada na literatura, baseada no diagrama de radiação. Identificou-se que a função proposta valoriza configurações de arrays esparsos que geram imagens com melhor equilíbrio entre contraste e resolução lateral. Além disso, foi identificado que a função aptidão proposta na literatura apresenta inconsistên- cias ao avaliar os arrays esparsos, o que não ocorre na função proposta. Em seguida, uma nova estratégia de aquisição de dados para arrays bidimensionais que não estão em uma malha é proposta. A estratégia se baseia em analisar as projeções dos elementos do coarray e manter somente as combinações de elementos emissores e receptores mais importantes para a geração de imagens. Assim, o número de aquisições e volume de dados de aberturas esparsas bidi- mensionais, cujos elementos não estão posicionados em uma malha, é reduzido, bem como o tempo de geração de imagens. As análises dos resultados indicaram a viabilidade em reduzir os sinais adquiridos sem comprometer a qualidade das imagens geradas. Adicionalmente, foram desenvolvidas duas figuras de mérito baseadas na análise da disposição espacial dos elementos, que por sua vez foram utilizadas para avaliar arrays 2D esparsos. A relação entre as figuras de mérito desenvolvidas e a energia irradiada pelos arrays foi estudada e, a partir desta análise, uma função custo desenvolvida. Em seguida, é apresentada uma estratégia para projetar arrays 2D esparsos utilizando o algoritmo simulated annealing. As análises do diagrama de radiação das aberturas bidimensionais esparsas obtidas pelo algoritmo de busca possuem característi- cas desejáveis com alta resolução lateral e baixa intensidade nos artefatos. A tese possui três contribuições para os sistemas de geração de imagens por ultrassom que reduzem os custos de manufatura e computacional. Palavras-chave: Algoritmos de Otimização Estocástico. Arrays 2D Esparsos. Array Lineares Esparsos. Metaheurísticas. ABSTRACT Ultrasonic images have an important contribution to medical diagnosis and non-destru- ctive testing. One strategy to generate an image is to use an array, which is a transducer com- posed of a set of piezoelectric elements, that emits several wavefronts and samples the reflected waves. An important physical characteristic of arrays is their dimension. The wider the array extension, the better the lateral resolution of the generated image will be. Additionally, a con- struction recommendation for arrays is that the centre of their elements must be spaced by a distance (pitch) less or equal to 0.5λ, where λ is the generated wavelength by the transducer. Thus, the images generated by these arrays do not present artefacts caused by the grating lobes. The recommendation for using arrays is that it has to be the wider array possible, respecting the pitch recommendation. However, the data volume, resource, and manufacturing cost pro- portionally increase as the number of elements in the array rises, which might be impractical to use this array depending on the application. This thesis investigates techniques to reduce the use of resources in ultrasonic systems aiming to achieve images with high lateral resolution and mitigate any disadvantages. In the first part of this thesis, the linear sparse arrays, which are arrays that pitch higher than 0.5λ are studied. A new strategy to design these arrays is proposed, where a new mathematical codification for sparse arrays and fitness function based on the equa- tion of energy and entropy of the PSFs (Point Spread Function) are presented. Subsequently, stochastic optimization algorithms are used to design sparse configurations. The proposed fit- ness function was compared with the most used fitness function in the literature based on the radiation pattern. The sparse arrays found using the proposed fitness function generated images with a balance between contrast and lateral resolution. Moreover, it was noticed that the fitness function proposed in the literature has inconsistencies when evaluating sparse array configu- rations which do not happen with the proposed fitness function. Next, a new data acquisition strategy for synthetic aperture for two-dimensional arrays that are not in a grid is proposed. This strategy is based on analysing the projections of the elements of the coarray and keeping only the combinations of emitter and receiver elements that are most important for image gen- eration. Consequently, the number of acquisitions and data volume of sparse two-dimensional apertures, whose elements are not positioned in a grid, is reduced, as well as the image gen- eration time. The results indicate that it is possible to reduce the number of acquiring signals without compromising the quality of the ultrasonic image generated. In addition, two figures of merit based on the spatial distribution of the elements were used to evaluate sparse 2D arrays. A study of these parameters and how they influence the energy irradiated by arrays is done, and a fitness function is created. Then, a strategy to design a sparse 2D array is proposed using the simulated annealing algorithm. The radiation pattern analysis of the sparse arrays obtained from the search algorithm shown that the aperture generated images with high lateral resolution and low artefact intensities. The radiation pattern analysis of the sparse arrays obtained from the search algorithm showed that the aperture generated images with high lateral resolution and low artefact intensities. This thesis has three main contributions to ultrasonic systems that reduce manufacturing and computational costs. Keywords: 2D Sparse Arrays. Linear Sparse Arrays. Metaheuristics. Stochastic Optimization Algorithms. LIST OF FIGURES Figure 1 – (a) Linear and (b) annular segmented array. . . . . . . . . . . . . . . . . . . 30 Figure 2 – (a) Matrix, (b) Fermat spiral, (c) segmented annular and (d) mills cross arrays. 31 Figure 3 – Pressure sum of a linear array at a point P(R, θ) considering the elements are point sources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Figure 4 – (a) Radiation Pattern of 4 (solid blue line), 8 (dash-dot red line) and 16 (dashed green line) elements array. (b) Apodization of a 16 linear array with Rectangular (solid blue line), Hanning (dash-dot red line) and Hamming (dashed green line) apodization. . . . . . . . . . . . . . . . . . . . . . . . 35 Figure 5 – Radiation Pattern of 256 elements matrix array with 0.5λ pitch. . . . . . . . 35 Figure 6 – Wideband response of (a) linear array with different numbers of elements and (b) 256 elements matrix array with 0.5λpitch. . . . . . . . . . . . . . . 37 Figure 7 – Pressure of an array element with width a. . . . . . . . . . . . . . . . . . . 37 Figure 8 – (a) Energy irradiated by an element with different a/λ values: (b) Radiation Pattern narrowband response modulated by the directivity of the element with a a = 0.5λ width. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Figure 9 – 32 elements radiation pattern considering a pitch equal to: (a) 0.50λ, (b) 0.75λ, (c) 1λ and (d) 2λ. . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Figure 10 – (a) Fully populated array (blue dot), sparse aperiodic array (red dot), sparse periodic array (green dot) and the missing elements to complete a fully pop- ulated array. (b) the radiation pattern narrow band response of the presented arrays with the MLW and PS L highlighted. . . . . . . . . . . . . . . . . . 41 Figure 11 – Illustration of Crt. (a) the array elements are periodically spaced (in a grid). (b) One reception element is shifted by 0.3. . . . . . . . . . . . . . . . . . . 45 Figure 12 – (a) Transmit and Receiver Linear FPA with 32 elements. (b) the coarray formed from the two arrays with triangular shape with all elements respect- ing a pitch 0.5λ. (c) the radiation pattern of the transmit array and the coarray (two-way radiation pattern). . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Figure 13 – (a) Sparse Transmit and Receiver Linear arrays created based on Vernier dis- tribution, (b) the coarray created where it has the triangular shape, respects a pitch 0.5λ but it has adjacents elements that alternate the amplitudes and (c) the two-way radiation pattern of the coarray created. . . . . . . . . . . 47 Figure 14 – (a) Matrix array with 256 elements (16 × 16) where the adjacent elements in the x-axis and y-axis have a pitch equals to 0.5λ (λ = 1[mm]) and the elements diagonally adjacents have a pitch equals to 0.70λ. (b) The coarray is created from the matrix array which has a pyramidal shape. . . . . . . . . 48 Figure 15 – An example of the minimum redundancy array (KARAMAN et al., 2009) where (a) 30 elements are disposed of two vertical lines and (b) 30 elements are disposed of two horizontal lines where the elements in the lines have a pitch equal to 0.5λ (λ = 1[mm]). (c) the coarray is generated from these two apertures where the majority of the elements have amplitude one. . . . . . 48 Figure 16 – Simple imaging system using a single element transducer. (a) representation of the imaging system where the transducer is shifted only in the x axis and the resulting image (b) created in the x and z-axis. (c) An imaging strategy where the transducer moves in x and y-axis. (d) the segmented image generated in a different depth. . . . . . . . . . . . . . . . . . . . . . 50 Figure 17 – Phased Array inspection modalities. (a) B-scan. (b) Focused B-scan. (c) Sector Scan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Figure 18 – Synthetic Aperture Radar. . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Figure 19 – Synthetic Aperture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Figure 20 – Total Focusing Method (TFM). Representation in (x, z) grid (a), and in (r, θ) grid (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Figure 21 – Data set simulation. (a) electrical input response. (b) shifted signal to the time of flight (9µ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Figure 22 – PSF of an FPA with 32 elements. (a) Image generated from −90◦ to 90◦. (b) as the array is symmetric it is only necessary to generate half of the image (from 0◦ to 90◦). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Figure 23 – PSFs of (a) FPA with 32 elements, (b) SPA with 32 elements a pitch λ and (c) SPA with 32 elements and pitch 2λ. (d) normalized sum of the amplitudes. 55 Figure 24 – (a) a sparse array created by selecting 16 elements in a grid corresponding to a 32 FPA. (b) the PSF of the sparse array where the energy besides the point reflector is not concentrated in a specific area. . . . . . . . . . . . . . 56 Figure 25 – Eight elements FPA (red dot), where four elements were selected (blue dot) to create a sparse array. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Figure 26 – PSF of (a) 96 elements FPA, (b) 128 elements FPA and (c) amplitude are summed and then normalized at each angle of the PSF, where the blue con- tinuous line corresponds to the 96 elements FPA and the red dashed line corresponds to 128 elements FPA. . . . . . . . . . . . . . . . . . . . . . . 63 Figure 27 – PSFs created using the 66 elements sparse array configurations related to Table 2, where (a) is the WOA Run 1 configuration and (b) the AOA Run 1 configuration. In (c), the normalized means of the two PSF are illustrated where the PSF in (a) is illustrated in a blue continuous line and (b) is illus- trated in a red dashed line. . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Figure 28 – PSFs created using the 88 elements sparse array configurations related to Table 2, where (a) is the SA Run 1 configuration and (b) the SA Run 3 configuration. In (c), the normalized means of the two PSF are illustrated where the PSF in (a) is illustrated in a blue continuous line and in (b) is illustrated in a red dashed line. . . . . . . . . . . . . . . . . . . . . . . . . 65 Figure 29 – PSFs created using the 102 elements sparse array configurations related to Table 3, where (a) is the WOA Run 1 configuration and (b) the SA Run 3 configuration. In (c), the normalized mean of the two PSF are illustrated where the PSF in (a) is illustrated in a blue continuous line and (b) is illus- trated in a red dashed line. . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Figure 30 – (a) Distance where the defect is placed based on the array size (D). (b) PSF of a FPA with 64 elements is used as an example. (c) normalized mean extracted from the PSF showing the first valley and (d), the segmented PSF used to calculate the Energy. . . . . . . . . . . . . . . . . . . . . . . . . . 68 Figure 31 – PSFs created using the 48 elements sparse arrays in a search space of 96 FPA, where the FF of each configuration is illustrated in Table 4. (a) is the WOA Run 1 configuration and (b) the WOA Run 3 configuration. In (c), the normalized means of the two PSF are illustrated where the PSF in (a) is illustrated in a blue continuous line and (b) is illustrated in a red dashed line. 70 Figure 32 – PSFs created using the 88 elements sparse arrays in a search space of 128 FPA, where the FF of each configuration is illustrated in Table 4. (a) is the WOA Run 1 configuration and (b) the AOA Run 2 configuration. In (c), the normalized means of the two PSF are illustrated where the PSF in (a) is illustrated in a blue continuous line and (b) is illustrated in a red dashed line. 71 Figure 33 – PSFs created using the 110 elements sparse arrays in a search space of 128 FPA, where the FF of each configuration is illustrated in Table 4. (a) is the WOA Run 3 configuration and (b) the WOA Run 3 configuration. In (c), the normalized means of the two PSF are illustrated where the PSF in (a) is illustrated in a blue continuous line, in (b) is illustrated in a red dashed line and the. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Figure 34 – Dansk Phantom Model 525. . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Figure 35 – Ultrasonic images of (a) Section 3, (b) Section 6 and (c) Section F of the Dansk Phantom Model 525. . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Figure 36 – Phantom Section 3 Image using: (a) 128 Elements FPA, (b) best 96-configuration found using the Radiation Pattern FF and (c) best 96-configuration found us- ing the FF proposed in this work. . . . . . . . . . . . . . . . . . . . . . . . 75 Figure 37 – Section F of the phantom generated using the (a) 96 FPA, (b) the best 48 elements sparse array found using the FF based on radiation pattern and (c) the best 48 elements sparse array found using the proposed FF. Section 6 of the phantom generated using the (d) 96 FPA, (e) the sparse array found using the FF based on radiation pattern and (f) the 48 elements sparse array found using the proposed FF. . . . . . . . . . . . . . . . . . . . . . . . . . 79 Figure 38 – Section 3 of the phantom generated using the (a) 96 FPA, (b) the 80 elements best sparse array found using the FF based on radiation pattern and (c) the 80 elements best sparse array found using the proposed FF. . . . . . . . . . 79 Figure 39 – (a) Section F and (d) Section 6 of the phantom generated using the 128 FPA, (b) and (e) the phantom images generated by the 96 elements best sparse array found using the FF based on radiation pattern and (c) and (f) the phantom images generated by the 96 elements best sparse array found using the proposed FF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Figure 40 – For matrix arrays: (a) array structure, (b) coarray footprint, (c) equivalente array projection at φ = 0◦, and (d) radiation pattern at φ = 0◦ (wideband re- sponse, BW=60%). For spiral array aparture: (e) array structure,(f) coarray footprint, (g) equivalente array projection at φ = 0◦, and (h) radiation pattern at φ = 0◦ (wideband response, BW=60%) . . . . . . . . . . . . . . . . . . . 84 Figure 41 – Matrix presentation of acquisition strategy (ACQ) for a generic array com- posed of 64 elements. White cells are the active channels. (a) ACQ1 - the reciprocity principle has been applied to reduce the number of signals. (b) ACQ2 - the reciprocity principle also has been used to optimise the elec- tronic resources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Figure 42 – Compact Representation of ACQ2 (a) and an ultrasonic acquisition system (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Figure 43 – Example of Coarray Projection Grid (CPG) where each [p, g] position con- tains a set of coarray elements ~Cer that meet the condition (42) . . . . . . . 88 Figure 44 – The 64 element spiral. (a) array footprint, (b) coarray footprint, (c) acoustic pressure in the semi-sphere [θ = 0◦ : 90◦, φ = 0◦ : 360◦] and (d) lateral profile showing the distribution of the sidelobes in elevation. The dashed line represents the spiral beampattern, and the light grey area within that line shows side lobe distribution at each elevation angle. The solid line is the corresponding beampattern of a 64 element matrix array. . . . . . . . . 92 Figure 45 – (a) strategy (64 : 64) for redundancy reduction. (b) sidelobe distribution at each elevation angle for strategy (64 : 64) (light grey area within the dashed line) and the FMC (dark grey area within the solid line). . . . . . . . . . . 93 Figure 46 – Acquisition matrix obtained from (a) ACQ(64 : 32), (b) ACQ(64 : 16) and (c) ACQ(64 : 8). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Figure 47 – Sidelobes distribution at each each elevation angle for strategies (a) ACQ(64 : 32), (b) ACQ(64 : 16) and (c) ACQ(64 : 16) (light grey area within the da- shed line) and the FMC (dark grey area within the solid line). . . . . . . . . 94 Figure 48 – ACQ obtained from (a) the setting (32 : 64) and (b) the setting (16 : 64) . . . 95 Figure 49 – Sidelobes distribution at each each elevation angle for strategies (a) (32 : 64) and (b) (16 : 64) (light grey area within the dashed line) and the FMC (dark grey area within the solid line). . . . . . . . . . . . . . . . . . . . . . . . . 95 Figure 50 – Segmented annular array. (a) Array prototype; (b) experimental setup in wa- ter, array faced downwards and a 3 mm-diameter spherical reflector placed 40 mm from the array; (c) element distribution, (d) coarray footprint. Image of a (e) point reflector (simulated) and (f) 3 mm-diameter metallic sphere (experimental) both placed at [X=−25 : 25 mm, Y=−25 : 25 mm, Z=40 mm]. 97 Figure 51 – Results for the segmented annular array obtained from ACQ (a) (64 : 64), (b) (64 : 24), (c) (64 : 16) and (d) (64 : 8), where the acquisition strategy, the simulated point reflector, the experimental metallic sphere image (cylin- drical coordinates: Z = 40 mm R = [0 : 25] mm, θ = [0o : 360o]), and the image reflectivity (maximum at each elevation angle) are illustrated, respec- tively. For the image reflectivity, the simulated response is illustrated in the dashed line and the experimental result in the solid line. . . . . . . . . . . . 99 Figure 52 – Results obtained from ACQ (a) (42 : 42), and (b) (32 : 32), where the ac- quisition strategy is presented on the left, the sidelobe distribution at each elevation angle for strategies is illustrated at the centre, and the configura- tion of the multiplexer net showing, at each shot, the distribution of reception transducer in the reception channels is illustrated in the right. For the side- lobe distribution, the current ACQ is represented in a light grey area within the dashed line, and the FMC is represented as a dark grey area within the solid line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Figure 53 – (a) Asymmetrical matrix sparse array, (b) Radiation Pattern narrowband re- sponse, (c) Line extracted at 0◦ of the narrowband response and (d) Radia- tion Pattern wideband Response with the amplitudes extracted at 0◦. . . . . 104 Figure 54 – (a) 360 Elements Fermat Spiral Array with 40 λ diameter ( λ = 1 mm) and α = 160.875◦. (b) the radiation pattern of the spiral array. (c) the radiation pattern of the spiral array with the same setting, but elements placed in a 60 λ diameter. (d) the lateral profile of the two radiation patterns. . . . . . . . . 106 Figure 55 – Radiation pattern wideband response of the three sparse arrays with 64 ele- ments emitting and 64 elements receiving with 40λ diameter, where the Mo and Mr information are given in Table (10). (a), (b) and (c) are generated using configurations 1, 2 and 3, respectively. . . . . . . . . . . . . . . . . . 107 Figure 56 – (a) lateral profile of configuration 1 and 2 of 128/40λ and (b) the radiation pattern comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Figure 57 – Radiation Pattern comparison, only the main lobe analysis. (a) Combination 1-3 (b) Combination 2-3. . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Figure 58 – Radiation pattern wideband response of the three sparse arrays with 96 ele- ments emitting and 96 elements receiving with 40λ diameter, where the Mo and Mr information are given in Table (10). (a), (b) and (c) are generated using configurations 1, 2 and 3, respectively. . . . . . . . . . . . . . . . . . 111 Figure 59 – Radiation pattern wideband response of the three sparse arrays with 128 elements emitting and 128 elements receiving with 60λ diameter, where the Mo and Mr information are given in Table (10). (a), (b) and (c) are generated using configurations 1, 2 and 3, respectively. . . . . . . . . . . . . . . . . . 112 Figure 60 – Optimization convergence. (a) 128/40λ, (b) 192/40λ and (c) 256/60λ. . . . 115 Figure 61 – Radiation pattern wideband response of the spiral optimized arrays. (a) 128, (b) 192 and (c) 256 elements array. . . . . . . . . . . . . . . . . . . . . . . 116 Figure 62 – Radiation pattern comparison between the optimized 128/40λ aperture and configuration-1 of 128/40λ . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Figure 63 – Segmented annular array example with 128 elements distributed in 8 radii. (a) radii in phase. (b) radii randomly rotated. . . . . . . . . . . . . . . . . 118 Figure 64 – Radiation pattern wideband response of the segmented annular optimized arrays. (a) 128, (b) 192 and (c) 256 elements array. . . . . . . . . . . . . . . 119 Figure 65 – Search Space. (a) Image of a second-order equation. (b) Image of a different FF where there are minimums that the search algorithm can be stuck. . . . . 125 Figure 66 – Search Space. (a) illustration of intensification and diversification process in metaheuristics. (b) real representation of a searching problem. . . . . . . 126 LIST OF TABLES Table 1 – Comparison between search algorithm using the fitness function proposed by Trucco (1999). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Table 2 – Comparison between search algorithms using the radiation pattern based fit- ness function given in (30). Search space corresponding to a 96 elements FPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Table 3 – Comparison between search algorithms using the radiation pattern based fit- ness function given in (30). Search space corresponding to a 128 elements FPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Table 4 – Comparison between search algorithms using the radiation pattern based fit- ness function given in (33). Search space corresponding to a 96 and 128 elements FPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Table 5 – MSSIM calculated between an image generated with a sparse array and a reference image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Table 6 – CNR of section 3, 6 and F of the phantom image generated by the sparse arrays and the 96 and 128 FPA . . . . . . . . . . . . . . . . . . . . . . . . . 78 Table 7 – Fermat spiral array: performance for the FMC, RCP and all six strategies considered (T x : Rx). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Table 8 – Segmented Annular Array. Performance for the (64 : 64), (64 : 24), (64 : 16) and (64 : 8) settings. Simulated and experimental data included for ∆θ−6dB, DR (dB) and ∆θDR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Table 9 – Spiral array: FMC, RCP, ACQ(42 : 24) and ACQ(32 : 32) acquisition stra- tegies results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Table 10 – Mo and Mr results of different configurations . . . . . . . . . . . . . . . . . 106 Table 11 – Main Lobe and Side Lobe distribution analysis for the spiral array with 40 and 60 λ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Table 12 – Mo, Mr and MoMr of different configurations with different settings and their optimized aperture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Table 13 – Main lobe and side lobe distribution analysis. Optimized aperture vs confi- gurations 1,2 and 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Table 14 – Mo, Mr and MoMr of spiral and segmented optimized array. . . . . . . . . . 118 Table 15 – Main lobe and side lobe distribution analysis. Spiral array vs segmented annular array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 LIST OF ABBREVIATIONS AND ACRONYMS ACQ Acquisition strategy AOA Arithmetic optimization algorithm bPSO binary particle swarm algorithm CP Crossing Point between lateral profiles CNR Contrast to noise ratio CPG Coarray Projection Grid DTFT Discrete-Time Fourier transform DR Dynamic range FF Fitness function FMC Full matrix capture FPA Fully populated array GFP Global Free Positions GNFP Global Non-Gree Positions GA Genetic algorithm GPGPU General-Purpose Graphics Processing Units ICPG Inverse Coarray Projection Grid MLW Main-lobe width MRC Minimum redundancy coarray MSSIM Mean structural similarity image measurement PSF Point spread function PS L Peak side-lobe PSO Particle swarm optimisation ROI Region of interest SA Simulated annealing SAA Sparse aperiodic array SNR Signal to noise ratio SPA Sparse periodic array SSIM Structural Similarity Image Measurement TFM Total focusing method LIST OF SYMBOLS ARe Set of emitters array elements ARr Set of receiver array elements a Array element width α Divergence Angle BW Relative bandwidth β decreasing temperature rate in the Simulated Annealing c Speed of sound CAN Signal sets Crt Coarray set position ~Cer Coarray element of emitter e and receiver r Ce1r1 Coarray element of emitter e1 and receiver r1 Ce2r2 Coarray element of emitter e2 and receiver r2 Cbu f Buffer containing the position of the coarray element projected. D Diameter of the aperture or Array Size d Pitch dp Infinitesimal pressure contribution dx Infinitesimal distance of the array element δ Kronecker delta function δ(t − τer) Time shifted impulse ∆τ Time delay between two elements ∆θ−6dB Main lobe width when the lateral profile decrease to -6dB ∆θDR Main lobe width is defined at the level of the dynamic range e Euler’s constant E Energy used to create the new FF e Index used to indicate emitter elements ~ee Vector with the position of the emitter element ~ei Vector with the position (xi, yi) of the element ~er Vector with the position of the receiver element η η = 1.5/(BW fc). fc Center frequency of the array fs Sample Frequency FP counter of the filled positions in the MAP matrix Gc Candidate to the acquisition set GFP Global free positions counter GNFP Global non-free positions counter g(t) Ultrasonic pressure pulse g Column position of the CPG matrix g(t − τer) Ultrasonic pressure pulse delayed h(θ, θs, t) Array impulse response in relation to θ and θs h(u, t) Array impulse response in relation to u h(~r, ~r f , t) Impulse response of 2D array H Entropy used to create the new FF H1(θ) Radiation Pattern of an array in a given angle θ H2(θ) Radiation Pattern of an element in a given angle θ H2(θ, φ) Radiation Pattern of an element in a given angle θ and φ i Variable used as an index i0 Pair of emitter and receiver elements current selected I(x, z) Image amplitude at pixel x and z I(R, θ) Image amplitude at pixel radius R and angle θ I1 First ultrasonic image used to calculate the MS S IM I1i Area of the image I1i I2 Second ultrasonic image used to calculate the MS S IM I2i Area of the image I2i j Imaginary unit ( j2 = −1) L Array element length λ wavelength k Wavenumber k1 Weights used in the fitness function k2 Weights used in the fitness function N Number of elements in the array Nc Maximum number of parallel channels Ng Number of angles where the coarray elements are projected Nk Exchange control in the Simulated Annealing algorithm Np Number of coarray grid positions Npix Number of pixels in the PSF image Ns Number of signals used to create an ultrasonic image NFP Counter for the redundancy introduced in the MAP matrix MAP Matrix that counts the number of coarray elements that fill a position (p, g) MPD Maximum number of Parallel Degree Mr Redundancy level Mo Occupancy rate µ Mean θ Azimuth angle θi Azimuth angle for the array element i θs Azimuth angle where the beam is steering P Location were all acoustic pressures sum PD Parallelism Degree per element counter P(R, θ, t) Sum of the acoustic pressures p Row position of the CPG matrix p(θ, θs, t) Acoustic pressure in wide band response p(R, t) Acoustic pressure in a radius R for a given time t pi(R, θ, t) Acoustic pressure in a radius Ri for a given azimuthal angle θ and time t p(~r, ~r f , t) Acoustic pressure wide band response for 2D arrays p0 Function of the wave number k PS Fre f Reference point spread function φ Projection angle φp Angular discretization ϕ Phase in the Fermat spiral array ρi Values of the probability distribution ρ Probability distribution used to calculate the Entropy r Index used to indicate receiver elements R Radius distance Ri Distance from the array element i to the point where the acoustic pressure is analysed R′ Distance from a point in the array element to the point where the acoustic pressure is analysed RCP Acquisition Strategy based on reciprocity R_x Number of parallel channels ser(t) Ultrasonic signal related to the emitter e and receiver r ŝer Hilbert transform of the ultrasonic signal related to the emitter e and recei- ver r σ Standard Deviation t Time Tk Current temperature used in the Simulated Annealing algorithm T_x Number of elements emitting τ Time of flight τer Delay time of the ultrasonic wave that propagates from the emitter e, is reflected and reaches the receiver r τer(x, z) Time of flight from the emitter element to the point (x, z) and back to the receiver element u u = sin(θ) − sin(θs) u0 Constant related to the ultrasound pulse transmitted by a single element x variable x used to represent any value x0 start the solution of a metaheuristic xv best solution of a metaheuristic y variable y used to represent any value wi Amplitude Modulation wer Amplitude Modulation related to the emitter e and receiver r ω Angular frequency ωθ ωθ = −kd sin(θ) + ω∆τ CONTENTS 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.1 OBJECTIVE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.1.1 Specifc objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.2 OUTLINE OF THE WORK . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2 FUNDAMENTALS AND LITERATURE REVIEW . . . . . . . . . . . . 30 2.1 ARRAYS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.1.1 The radiation pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.1.2 Sparse array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.1.3 Coarray . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.2 ULTRASONIC IMAGING . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.2.1 Phased array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.2.2 Synthetic aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.2.3 Point Spread Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.3 COMMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3 NEW STRATEGY FOR LINEAR SPARSE ARRAY DESIGN . . . . . . 57 3.1 FITNESS FUNCTION BASED ON THE RADIATION PATTERN AND SEARCH ALGORITHM ANALYSIS . . . . . . . . . . . . . . . . . . . . . 57 3.1.1 Countinous codification . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.1.2 Comparing algorithms with radiation pattern fitness function . . . . . . 59 3.2 NEW FITNESS FUNCTION . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.3 RESULTS AND DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . 73 3.4 COMMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4 ACQUISITION STRATEGY FOR NON-GRID APERTURES . . . . . . 82 4.1 THE COARRAY ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.1.1 Identification of redundancy in the coarray . . . . . . . . . . . . . . . . 86 4.2 ACQUISITION STRATEGIES ALGORITHM . . . . . . . . . . . . . . . . 87 4.2.1 Step one: Generation of the CPG database . . . . . . . . . . . . . . . . . 88 4.2.2 Step two: Acquisition Strategy Design . . . . . . . . . . . . . . . . . . . 89 4.3 EVALUATION OF THE PROCEDURE . . . . . . . . . . . . . . . . . . . 90 4.3.1 Fermat spiral array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.3.2 Segmented annular array - simulated and experimental results . . . . . 96 4.3.2.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.3.2.2 Synthetic aperture strategies . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.4 DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.5 COMMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5 2D SPARSE ARRAY EVALUATION . . . . . . . . . . . . . . . . . . . . 103 5.1 FITNESS FUNCTION BASED ON CPG . . . . . . . . . . . . . . . . . . . 103 5.1.1 Mo and Mr analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.2 PROBLEM FORMULATION AND SEARCH ALGORITHM . . . . . . . . 113 5.3 FERMAT SPIRAL OPTIMIZATION . . . . . . . . . . . . . . . . . . . . . 114 5.4 SEGMENTED ANNULAR ARRAY OPTIMIZATION . . . . . . . . . . . 117 5.5 COMMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6 FINAL COMMENTS AND FUTURE WORK . . . . . . . . . . . . . . . 121 6.1 RECAPITULATION AND APPLICATIONS . . . . . . . . . . . . . . . . . 121 6.2 CONTRIBUTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 6.3 PUBLICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 6.4 FUTURE WORKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 APPENDIX A – SEARCH PROBLEMS AND METAHEURISTICS . . 125 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 26 1 INTRODUCTION Ultrasonography is a relevant medical procedure used in several areas such as cardi- ology, orthopedics, and gynaecology. Using ultrasonic images, pathologies are detected in a non-ionizing, non-invasive, and non-traumatic way. It also allows the visualization of abnor- malities that are not detected using other techniques, such as the ventricular septal defect, which cannot be diagnosed using the electrocardiogram (DAKKAK; OLIVER, 2020). Ultrasonic systems are cheaper than other equipaments, such as tomography or mag- netic resonators. It allows fast diagnosis and can be used to combat pandemics by earlier iden- tifying a person suffering from an illness, such as the use of lung ultrasound to diagnose the coronavirus (COVID- 19) (DUGGAN et al., 2020; QIAN et al., 2020; YU et al., 2020; BE- VAN et al., 2020). It is also used for non-destructive testing to monitor parts and prevent accidents. There are several applications for ultrasound imaging in non-destructive testing to detect defects in parts and structures, such as cracks, delaminations and corrosions, and prevent accidents (LAROCHE et al., 2020; JOLLY et al., 2015; KHALILI; CAWLEY, 2018). The simplest way to generate an ultrasonic image is to use a single-element trans- ducer, for example, a piezoelectric element that converts electrical energy into acoustic pres- sure, and vice-versa. The transducer is excited, generating an acoustic wave that interacts with the medium (reflections, refractions, attenuation, etc.), and the same transducer can receive the echoes. By moving the transducer along a line, for example, an image can be created by plot- ting the reflected amplitudes for each transducer position. An array is a group of piezoelectric elements which can be linearly placed to image a section of an object, or bidimensionally (2D), to create volumetric images. One significant advantage of using arrays is that the individual elements can be independently excited, creating the possibility of beam steering and focusing without the need to move the array (or demanding small movements, in some cases). Although electronic complexity increases, lateral resolution and contrast can be improved compared to single-element operation (DRINKWATER; WILCOX, 2006). When using an array, a recommendation for the distance between the centre of adjacent elements is at most 0.5λ, where λ is the wavelength (LOCKWOOD et al., 1996a). This distance, called pitch, is relevant because, when the elements have a pitch higher than 0.5λ, the resulting irradiated energy sums in different areas resulting in image deterioration. Another feature that contributes to image quality is the size of an array: the broader an array, the better the lateral resolution of an image (TRUCCO, 1999). In this sense, the construction recommendation is to use larger arrays with 0.5λ pitch. Nevertheless, extending the size of an array increases the number of elements, and, in conse- quence, increases the electronic complexity, data volume, and time to generate an image, which might be undesirable for a specific application or even make the imaging system impractical. The need to find a solution for this problem is clear for 2D arrays, which are used to 27 create volumetric images. Considering an array with 32λ × 32λ dimension, if the elements are placed in a matrix pattern, 16384 elements would be required. Despite this type of array being physically fabricated using CMOS technology (ORALKAN et al., 2003), the resource to control all these elements turns its use impractical. For example, using the synthetic aperture technique (JENSEN et al., 2006), which is an imaging strategy where the emitter elements of the array are sequentially excited and the sampled ecos stored and post-processed, 268,435,456 mathematical operations would be necessary to create an image pixel, which is time-consuming and impractical for real-time applications. There are different strategies to enable the use of this array. For example, Karaman et al. (2009) created a method that defines the minimum number of elements in a matrix array, decreasing the time to generate an image. Combining the synthetic aperture technique and their strategy would be required 64,516 mathematical operations to create an image pixel, consider- ably reducing the number of operations. However, this strategy only works for matrix arrays. For arrays where the elements are not placed in a matrix grid (non-grid arrays), it is neces- sary to create a different method that selects the emitter and receiver elements of the aperture, which creates a routine that reduces the number of acquisitions required, enabling real-time applications and decreasing the computation cost to generate an image for non-grid arrays. Another strategy consists in creating a sparse array, in which elements’ pitches are higher than 0.5λ. With fewer elements, the area covered is still high and the lateral resolu- tion remains elevated. However, because of the energy summing in different areas, the sparse array has the problem of decreasing the image’s contrast compared to an array that respects the 0.5λ recommendation. Although linear arrays uses fewer elements (64 to 256) and is commer- cially spread, the use of sparse linear arrays might be still beneficial. For example, new imaging systems migth use the power supply and computation power of cellphones to generate images. Sparse arrays could be handy for this application, as fewer elements demand fewer resources (battery, processing). Moreover, if a certain number of elements in a probe burns, a sparse lin- ear array could be designed to work without this array. The burned and some specific elements from the array are excluded from the imaging process, and the resulting image still would have good lateral resolution and contrast. This strategy allows to recycle and extend the array’s life. The need for sparse 2D arrays is straightforward justified when compared to linear ar- rays because of the higher number of elements and signals involved. At this date, matrix ar- rays with a reasonable size, as mentioned previously, have an elevated number of elements, becoming impossible to control all of them simultaneously (MARMONIER et al., 2022). Con- sequently, 2D sparse arrays are practically mandatory to achieve high lateral resolution using a manageable number of elements. One strategy is to distribute the elements using a nature-based equation, such as the Fermat spiral equation, where the displacement is in a non-grid area. With fewer elements, it is possible to cover a wider area and distribute the elements with reduced periodicity, which is 28 a problem that contributes to increased artefact intensities in ultrasonic images (MARTÍNEZ– GRAULLERA et al., 2010). Another strategy is to use search engines to design a sparse configuration, where the algorithm determines the elements’ position (ROUX et al., 2017). These search algorithms, called metaheuristics, are part of stochastic optimization methods that can intelligently test solutions and return the best-found, according to a predefined condition (NESMACHNOW, 2014). In general, for ultrasonic imaging systems, the search mechanism creates different sparse configurations and finds one that attends to defined circumstances, such as image quality and system resources. Moreover, these two strategies can be merged to create sparse 2D arrays where, at first, the elements are placed in a non-grid distribution. Then, metaheuristics eliminate elements and create a more sparse aperture. 1.1 OBJECTIVE The main goal of this work is to find strategies to decrease the time, data volume and resources of an ultrasonic imaging system. 1.1.1 Specifc objectives • Understand how the ultrasonic images are generated and investigate the modelling meth- ods used to evaluate linear and 2D arrays. • Analyse the different metaheuristics to design sparse linear arrays. • Propose a technique to design linear sparse arrays. • Identify and remove redundant information for imaging systems based on synthetic aper- ture for 2D arrays. • Define a strategy to design 2D sparse arrays for non-grid apertures. 1.2 OUTLINE OF THE WORK Chapter 2 gives the theoretical background and the literature review regarding funda- mental and state-of-the-art techniques used in this work, where the mathematical functions used to evaluate, types and the strategies used to create sparse arrays are presented. Then, different metaheuristics working to design linear sparse array configurations are analysed, where a new codification and a fitness function based on the simulation of a point reflector are presented in chapter 3. In chapter 4, an acquisition strategy for synthetic aperture is presented where the aim is to decrease the acquisition for non-grid 2D arrays and enable real-time applications. In chapter 5, spatial parameters created during the previous chapter are analysed and used to define 29 a fitness function that is used with the simulated annealing algorithm to design sparse non-grid 2D arrays. Final considerations are made where the contributions of this work are highlighted and future ideas given. 30 2 FUNDAMENTALS AND LITERATURE REVIEW In this chapter, the theoretical background used in this work is given and the recent literature in sparse array design are reviewed. 2.1 ARRAYS One strategy to generate an ultrasonic image is to use an array, a group of piezoelectric elements, that converts electrical energy into vibration and vice-versa. The arrays are divided into two groups: the arrays used to generate 2D images, such as the linear and the annular array, and the arrays used to create 3D images, like the matrix array, the annular segmented array and the Fermat spiral array (DRINKWATER; WILCOX, 2006). The linear and annular arrays are illustrated in Figure 1, where d is the array pitch, which is the distance between the centre of adjacent elements, a and L are the element width and length of the linear array element, respectively. Both arrays generate 2D images, but the difference between them is that the linear array allows beam steerring in the imaging plane, and the annular array allows only the focal depth (DRINKWATER; WILCOX, 2006). In this work, linear arrays are divided into three groups: the fully populated array (FPA), which is an array where the pitch d has the maximum value of 0.5λ. The sparse periodic array (SPA), where the pitch is higher than 0.5λ but equal to all elements, and the sparse aperiodic array (SAA), where the pitches are not the same, and some of them are higher than 0.5λ. Figure 1 – (a) Linear and (b) annular segmented array. (a) Linear Array (b) Annular Array Source: Adpated from (DRINKWATER; WILCOX, 2006) Arrays used for 3D imaging have more design space, and different configurations are proposed. The most intuitive way to place the elements in a 2D array is in a matrix (Figure 2 (a)), where the array elements are placed into a matrix shape that creates a grid. However, different types of configurations result in images with considerable good contrast using fewer 31 elements, such as the Fermat spiral (Figure 2 (b))(MARTÍNEZ-GRAULLERA et al., 2010), segmented annular (Figure 2 (c))(MARTÍNEZ et al., 2003), and cross mills arrays (Figure 2 (d)) (MONDAL et al., 2005). Figure 2 – (a) Matrix, (b) Fermat spiral, (c) segmented annular and (d) mills cross arrays. (a) Matrix Array (b) Fermat Spiral Array (c) Segmented Annular Array (d) Mills Cross Array Source: Author Both linear and 2D arrays, when increased in size and number of elements, require a considerable amount of electronic and computational resources to operate. Depending on the application and available resources, it is necessary to find alternatives to reduce the time to acquire and generate an image or the manufacturing cost to produce an ultrasonic system. To achieve this goal, it is necessary to understand different methods used to analyse ultrasonic imaging systems, where one of these is the radiation pattern. 2.1.1 The radiation pattern The Radiation Pattern is the energy irradiated by an element source or array (JENSEN, 2002). It can be simulated considering the far field and elements continuously excited with a si- nus signal, referred to as the narrowband response, or considering different electrical excitation, ideally the pulse excitation, which is the wideband response (CARDONE et al., 2001). Although several ultrasonic applications in imaging occur in the near field and the array elements are pulsed, the radiation pattern narrowband response is a modelling tool used to anal- yse array response and its ability to generate images with high lateral resolution and contrast. Moreover, this modelling tool aids in understanding the concepts of ultrasonic imaging. Con- 32 sidering that a point source generates an omnidirectional acoustic wave, the acoustic pressure p as a function of time t and radius R is described as (WOOH; SHI, 1999a): p(R, t) = ( p0 R ) 1 2 e j(ωt−kR), (1) where ω is the angular frequency, and p0 is a function of the wavenumber k, derived from the solution of the acoustic wave. Therefore, the acoustic pressure of an array with N sources generators is given as the sum of pressures described in (1). Figure 3 illustrates a linear periodic array with N elements and pitch (d) 0.5λ. Element i has a distance Ri and angle θi to the point P, where the acoustic pressure (pi(Ri, θi, t)) is analysed and calculated as: Figure 3 – Pressure sum of a linear array at a point P(R, θ) considering the elements are point sources. Source: Author pi(Ri, θi, t) = ( p0 Ri ) 1 2 e j(ω(t−(i−1)∆τ)−kRi ), (2) where 1 ≤ i ≤ N, ∆τ is the difference in time-of-flight of the wave propagating from element i to the point P and its neighbour. Considering that a triangle with vertices at element 1, element i and point P, the follo- wing relationship between R and Ri is built: Ri = √ R2 + [(i − 1)d]2 − 2R(i − 1)d cos(π/2 − θ). (3) 33 However, if the distance where the pressure is analysed is sufficiently farther than the pitch d (far-field) and the lines Ri are considered parallel, (3) can be approximated as (WOOH; SHI, 1999a): Ri ≈ R − (i − 1)d sin(θ), (4) and using (4) in (2), the following relationship is obtained: pi(R, θ, t) = ( p0 R ) 1 2 e j(ωt−kR)e− j(ω(i−1)∆τ−k(i−1)d sin(θ)), (5) where the pressure of any element i is in function of R, θ and t. Therefore, the total acoustic pressure is the sum of all pressures of the N elements, given by: P(R, θ, t) = N ∑ i=1 wi pi(R, θ, t), (6) where wi is an amplitude modulation used to change the radiation pattern response to achieve the desired response, also known as apodization (SZABO, 2014). Substituting (5) in (6), the following relationship is achieved: P(R, θ, t) = ( p0 R ) 1 2 e j(ωt−kR) N ∑ i=1 wie − ji(−kd sin(θ)+ω∆τ), (7) defining ωθ = −kd sin(θ) + ω∆τ, (7) can be rewritten as: P(R, θ, t) = ( p0 R ) 1 2 e j(ωt−kR) N ∑ i=1 wie − jiωθ , (8) and considering w[i] = 0 for i < [1,N], (8) can be written as : P(R, θ, t) = ( p0 R ) 1 2 e j(ωt−kR) ∞ ∑ i=−∞ wie − jiωθ , (9) which the right part of the equation is a Discrete-Time Fourier transform (DTFT) of wi. Thus, the DTFT of the apodization coefficients is equal to the radiation pattern of the linear array. Solving (9), the energy irradiated by a linear array with uniform apodization (wi = 1, 1 ≤ i ≤ N ) can be written in the harmonic form as: P(R, θ, t) = ( p0 R ) 1 2 sin[((ω∆τ − kd sin(θ))/2)N] sin((ω∆τ − kd sin(θ))/2) e− j ( ω∆τ−kd sin(θ) 2 ) (N−1)e j(ωt−kr), (10) 34 and gives information about the energy irradiated from a set of emitters. Moreover, a normali- zation at any arbitrary steering angle θs is given by: H1(θ) = ∣ ∣ ∣ ∣ ∣ P(R, θ, t) P(R, θs, t) ∣ ∣ ∣ ∣ ∣ . (11) Substituting (8) in (11), the folowing relationship is obtained: H1(θ) = ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∞ ∑ i=−∞ wie− ji(ωθ−ωθs ) ∣ ∣ ∣ ∣ ∣ ∣ ∣ , (12) and using the relationship ∆τ = d sin(θs)/c, (12) has the following solution: H1(θ) = ∣ ∣ ∣ ∣ ∣ sin((πd(sin(θS ) − sin(θ))/λ)N] N sin(πd(sin(θs) − sin(θ)/λ) ∣ ∣ ∣ ∣ ∣ . (13) Using linear arrays with 4, 8 and 16 elements, where the elements are equally spaced with a 0.5λ pitch (FPA), the radiation pattern using (13) is illustrated in Figure 4(a). In sparse array design, two features are used to evaluate the array performance in gen- erating ultrasonic images, the main-lobe width (MLW), which is the angular length where the main lobe is -6 dB and related to the lateral resolution of the images, and the peak side-lobe (PS L), which is the maximum side-lobe peak related to artefacts and contrast of the ultrasonic images (YANG et al., 2006; HU et al., 2018; HU et al., 2017; ZHANG et al., 2020). In both features, it is desirable to have a low value. For MLW, increasing the array’s size will reduce its value. However, using an FPA, the number of elements will increase, requiring more resources. Sparse arrays are handy for this problem because a wider length size array can be achieved using fewer elements. For PS L, one strategy to reduce its value is using a sparse aperiodic array or apodization, as will be seen in this chapter. In Figure 4(b), the radiation pattern of a linear array with 16 elements is illustrated where three apodizations were used: Rectangular, Hamming and Hanning. As it can be seen, the Hamming and Hanning apodizations decrease side lobes levels, with a trade-off with increasing main lobe width (SZABO, 2014). The radiation pattern can be also expanded to 2D arrays, where the elements are po- sitioned on the x and y-axis. In this sense, instead of only considering the azimuthal angle, the radiation pattern of a 2D array is obtained in both azimuthal (θ) and elevation angle (φ), described as: H1(θ, φ) = ∣ ∣ ∣ ∣ ∣ ∣ ∣ N ∑ i=1 wie − jk(xi (cos(φ)−yi sin(φ)) sin(θ)−(xi cos(φs)+yi sin(φs)) sin(θs) ∣ ∣ ∣ ∣ ∣ ∣ ∣ . (14) 35 Figure 4 – (a) Radiation Pattern of 4 (solid blue line), 8 (dash-dot red line) and 16 (dashed green line) elements array. (b) Apodization of a 16 linear array with Rectangular (solid blue line), Hanning (dash-dot red line) and Hamming (dashed green line) apodization. (a) Radiation Pattern (b) Apodization Source: Author Figure 5 – Radiation Pattern of 256 elements matrix array with 0.5λ pitch. Source: Author Figure 5 illustrates the radiation pattern of a matrix array with 256 elements (16x16) with a 0.5λ pitch that separates each element. In this figure, it is possible to see the main lobe at the centre and the side lobes. The radiation pattern presented, so far, is created considering the narrowband response. 36 In order to obtain the wideband radiation pattern, which is a response close to ultrasonic imag- ing systems, it is necessary to consider the ultrasonic pressure pulse transmitted by a single element. Using the array impulse response, the acoustic pressure in wideband response is given by (CARDONE et al., 2001): p(θ, θs, t) = g(t) ∗ h(θ, θs, t), (15) where the asterisk denotes convolution, g(t) the ultrasonic pressure pulse and h(θ, θs, t) is the array impulse response given by (CARDONE et al., 2001): h(θ, θs, t) = N ∑ i=1 wiδ ( t − id(sin(θ) − sin(θs)) c ) , (16) and δ() is the Kronecker Delta function. The number of parameters in the equation can be reduced by considering the sinusoidal relationship as a real value u = sin(θ) − sin(θs), therefore (16) becomes: h(u, t) = N ∑ i=1 wiδ ( t − iu 2 fc ) , (17) where fc is the centre frequency of the transducer. The ultrasound pulse transmitted by a single array element g(t) can be simulated as a Gaussian-envelope sinusoidal pulse, which can be given by (CARDONE et al., 2001): g(t) = u0e π2 BW2 f 2 c (t−η)2 1.2 ln 10 sin(2π fct), (18) where u0 is a constant, BW the -6 dB relative fractional bandwidth, and η = 1.5/(BW fc). For 2D apertures, the radiation pattern wideband response is calculated using the follo- wing equation (TURNBULL; FOSTER, 1991): p(~r, ~r f , t) = g(t) ∗ h(~r, ~r f , t), (19) where h(~r, ~r f , t) is the impulse response of the 2D array, ~r it a vector with the azimutal angle and radious (θ, φ and R) where the pressure is sampled and ~r f is the focus position of the radiation pattern. Figure 6 (a) illustrates the FPA wideband radiation pattern with 4, 8 and 16 elements linear array, respectively. Figure 6 (b) illustrates the radiation pattern for the 256 matrix array. As it can be seen, the representation of the side lobes is attenuated in both images, compared to linear and 2D FPA. 37 Figure 6 – Wideband response of (a) linear array with different numbers of elements and (b) 256 elements matrix array with 0.5λpitch. (a) Linear Array (b) 2D array Source: Author All simulations presented consider the array elements as point sources, where the ir- radiated energy is omnidirectional. Real elements do not behave like this as they have finite dimensions. The energy irradiated by an array element can be calculated and included in the ra- diation pattern to give a better approximation of the practical performance of the array (WOOH; SHI, 1999b). In Figure 7, the dimensional array element is illustrated, where it is possible to see from a different perspective the dimension a illustrated in Figure 1 (a). x is the distance between the origin and a point of the element. Figure 7 – Pressure of an array element with width a. Source: Author Using the Huygens’ principle, which says that every point on the radiation surface is the origin of an outgoing spherical wave (JENSEN, 2002), the energy of a rectangular array element with width a and length L is calculated as: 38 p(R, θ, t) = ˆ a 0 dp, (20) where dp is the infinitesimal pressure contribution of the element with width a, written as: dp = ( p0 R )1/2 e j(ωt−kR)dx, (21) and the dx in the integral means an infinitesimal distance of the array width illustrated in Fi- gure 7. Considering that the element’s width a is smaller than the distance R′ (Ri >> a), Ri can be aproximated using a similar relationship used in (3) as: Ri = √ R2 + x2 − 2Rx cos(θ) ≈ R − x sin(θ), (22) and (20) can be written as: p(R, θ, t) = ˆ a 0 dp = ( p0 R )1/2 sin(ka sin(θ/2)) k sin(θ/2) e− jka sin(θ) 2 e j(ωt−kR) = ( p0 R )1/2 sinc(ka sin(θ/2))e− jka sin(θ) 2 e j(ωt−kR), (23) where normalizing the pressure at a max angle (θ → 0) gives the directivity for a single element as (WOOH; SHI, 1999a): H2(θ) = ∣ ∣ ∣ ∣ ∣ sin(πa sin(θ)/λ) πa sin(θ)/λ ∣ ∣ ∣ ∣ ∣ , (24) and considering the length L of the element, the directivity is given as (WOOH; SHI, 1999b): H2(θ, φ) = ∣ ∣ ∣ ∣ ∣ ∣ sinc ( πa sin(θ) sin(φ) λ ) sinc ( πL sin(θ) sin(φ) λ ) ∣ ∣ ∣ ∣ ∣ ∣ . (25) Figure 8(a) illustrates the irradiated energy by an element for different widths. As it can be seen, punctual elements (a/λ = 0) irradiate the energy in all directions. In contrast, as the element size increase, the irradiated energy is more concentrated at smaller angles, in front of the element. This directly affects the radiation pattern narrowband response of the array, where the energy irradiated by an element modulates the energy irradiated by the array. In Figure 8(b) this behaviour can be seen comparing the radiation pattern considering an array with 32 punc- tual elements (illustrated in blue) and the radiation pattern considering the same amount of elements but with elements with width a = 0.5λ. It is important to highlight that, in ultrasonic applications, elements are pulsed excited, and the wideband response would be a more reliable 39 Figure 8 – (a) Energy irradiated by an element with different a/λ values: (b) Radiation Pattern narrowband response modulated by the directivity of the element with a a = 0.5λ width. (a) H2(θ) (b) Radiation Pattern Source: Author simulation. However, the narrowband response illustrates the worst-case scenario, highlighting the arrays’ characteristics and helping to explain the characteristics of the arrays’ responses. Although the energy irradiated by a finite dimension element is normally not taken into account when designing sparse arrays, it is good to know how the elements behave experimen- tally. In this way, all radiation pattern simulations presented in this work consider punctual elements. 2.1.2 Sparse array Sparse arrays are an alternative to increasing the lateral resolution without increasing the electronic complexity. To understand the consequences of using sparse arrays, the narrow- band radiation pattern can be analysed considering punctual elements. Figure 9 illustrates four radiation patterns of a 32 elements arrays where the pitch are different for each case. In (a), the radiation pattern generated from an array with a pitch equal to 0.5λ (32 elements FPA) has the biggest MLW, and the side lobes’ intensities decrease as it gets away from the main lobe. When the pitch increases to 0.75λ (b), the MLW decreases from 2.2◦ to 1.5◦, but the side lobes intensities stop decreasing as it gets far from the main lobe and increase at the extreme angles. These lobes are called grating lobes, which are caused by the waves of the array summing in unwanted directions. In Figure 9(c), the pitch is increased to 1λ, and it is possible to see that the grating 40 lobes at ±90◦ have the same intensity as the main lobe. In (d), the pitch is 2λ, and the radiation pattern has additional grating lobes at ±30◦. In contrast, the MLW in these cases reduces to 1.1◦ and 0.6◦, respectively. As it can be seen, increasing the size of the array by just increasing the pitch will reduce the size of the MLW leading to better lateral resolution. However, the energy irradiated by the element will sum in different regions, which might result in images with high-intensity artefacts that can lead to misinterpretations of the images. Figure 9 – 32 elements radiation pattern considering a pitch equal to: (a) 0.50λ, (b) 0.75λ, (c) 1λ and (d) 2λ. (a) Radiation Pattern d = 0.5λ (b) Radiation Pattern d = 0.75λ (c) Radiation Pattern d = 1λ (d) Radiation Pattern d = 2λ Source: Author One way to decrease the intensity of the grating lobes is by breaking the periodicity of the elements by using sparse aperiodic arrays, where the pitch of the elements will be different from each other. Figure 10 (a) illustrates two sparse array configuration. In red dots, a sparse aperiodic array where 16 elements were randomly removed from a 32 FPA. In green dots, 16 elements sparse periodic array of 1λ pitch, which has the same length as the 32 FPA. Figure 10 (b), illustrates the radiation pattern created from these sparse configurations where the MLW 41 and PS L are highlighted. It is possible to see a reduction in the grating lobes’ intensity with an exchange of the rise of side lobes. Moreover, the MLW is slightly lower for the sparse periodic array as it has the same length as the 32 FPA. Figure 10 – (a) Fully populated array (blue dot), sparse aperiodic array (red dot), sparse periodic array (green dot) and the missing elements to complete a fully populated array. (b) the radiation pattern narrow band response of the presented arrays with the MLW and PS L highlighted. (a) Arrays (b) Radiation Pattern Source: Author Works such as Goss et al. (1996) propose to randomly select 64 in 108 elements in a 2D sparse array to obtain a sparse array configuration. Although the focus of the work is to use arrays in focal heating, the study of the radiation patterns presented shows a reduction in the intensity of the grating lobes, but these random configurations produce sparse arrays with unpredictable performance as the side lobes summed in different regions. Search mechanisms can be a better strategy to find sparse array configurations. They are composed of three parts: the codification process, where the problem is encoded into a mathematical representation. For example, the FPA illustrated in Figure 10 (a) can be encoded as a binary vector with 32 positions. In this case, it will be a vector with 32 ones as the ones indicate that the elements are used in the imaging process. In the sparse array illustrated in Figure 10 (a), the 32 binary vectors will have zeros indicating the elements not used and ones indicating that the elements are used. The second part is the fitness function (FF), a mathematical function that translates the encoded configuration into a number. The FF translates the sparse configuration to how well it can be to image. In this way, two configurations that have different FF can be compared, and with this value, it is possible to say which one is better to image. Creating this FF is a 42 Table 1 – Comparison between search algorithm using the fitness function proposed by Trucco (1999). Metaheuristic Fitness of the best Simulated Annealing 34,714 Genetic Algorithm 65,378 Particle Swarm Optimization 29,152 Harmony Search 34,823 Whale Optimization Algorithm 70,455 Bat Optimization Algorithm 31,505 Arithmetic Optimization Algorithm 28,976 difficult task, as most of the time it is necessary to quantify an image or a signal with different amplitudes into a number. The codification strategy and the FF are used in a search engine that will search different configurations aiming to minimize or maximize the FF. The search engine is called metaheuristics, and more information is provided in Appendix A. One of the classic algorithms used for this problem is simulated annealing (KIRKPA- TRICK et al., 1983). Murino et al. (1996) proposed using the energy of the narrowband radia- tion pattern response as a fitness function and use simulated annealing to find configurations of sparse linear arrays and also the apodization of the elements. Trucco (1999) proposed a similar strategy to optimize both the elements in 2D sparse arrays and the weight coefficients (apodiza- tion). In that work, the FF proposed by the author uses the mean squared error of the radiation pattern, where one signal is the radiation pattern of the sparse array, and the other is the desired radiation pattern, which is a matrix array where the elements has 0.5 λ pitch. In both strategies, the positions of the elements are predefined, which creates a grid, and the search engines select which elements are going to work. The search algorithm does not guarantee that the best result is found, and may vary among different algorithms. For example, the Genetic Algorithm (HOLLAND, 1992), Parti- cle Swarm (KENNEDY; EBERHART, 1995), Harmony Search (ZONG et al., 2001), Whale Optimization (MIRJALILI; LEWIS, 2016), Bat Optimization (YANG, 2010) and Arithmetic Optimization Algorithm (AOA) (ABUALIGAH et al., 2021) were implemented to work with the fitness function proposed by Trucco (1999). Table (1) gives the FF of the best sparse array found using the respective algorithm. As it can be seen, the results found by the algorithms vary considerably, with the AOA finding the best result. Trucco (2002) used this same optimization logic involving simulated annealing to find sparse linear array configurations. However, the author created a fitness function based on the radiation pattern wideband response. The author claimed the wideband response is closer to real ultrasound applications and, therefore, better layouts could be achieved. Nevertheless, the author observed that processing the wideband response is time-consuming and getting valuable information from the wideband response is more challenging. 43 In these works, a common problem is to control the number of elements selected in the array. The way that the search problem is constructed, it is required to add a penalization function in the FF to limit the number of active elements selected in the array during the search problem construction. This penalization function is a conditional statement inserted in the al- gorithm that if the current configuration analysed has more active elements than the predefined at the beginning of the search, an arbitrary constant is added to the fitness value. Although the penalization strategy can limit the number of elements selected in the sparse array, it is difficult to tune the penalization factor. A different optimization algorithm was used by Haupt (1994), where the author used the genetic algorithm to find sparse configurations for linear and matrix arrays. Although the work is for antennas, the fitness function is based on the radiation pattern narrowband response, which shares the same theory as ultrasonic arrays, and it can be used as an example. The author proposes to use the peak sidelobe (PSL) as a fitness function. The genetic algorithm has the advantage in comparison to simulated annealing because the algorithm works with bi- nary representation, so 0 is used to illustrate that the element in a certain position is turned off and 1 when the element is on. For example, the binary representation of the sparse array illustrated in Figure 10 (a) would be [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0 , 0, 1, 1, 0, 1, 1, 1, 1]. In the work presented by Haupt (1994), a penalization strategy was not used, and for a 200 element FPA and a 10x20 matrix array, the author achieves a 25% reduction in the number of elements for the linear array and 46% element reduction in the 2D array. The genetic algorithm was also used by Yang et al. (2006) to optimize the linear array. The FF proposed by the authors is based on the radiation pattern narrowband response where the two pieces of information are extracted from it: the main lobe width (MLW) and the peak sidelobe (PSL). The two features are combined using a Scalarization Method, which is a multi- objective strategy (MURATA et al., 1996). Equation (26) illustrates the fitness function used by the authors. FF = k1MLW + k2PS L, (26) where k1 and k2 are the weights used to focus the search. If the operator wants to focus on better lateral resolution, a higher value in k1, with respect to k2, would be defined. The equation (26) has been used in different works to find sparse linear array configura- tions. Hu et al. (2017) proposed an imaging algorithm that corrects the divergence of the sound beam and uses the same methodology proposed by Yang et al. (2006). The difference from the previous study is that the authors only find sparse emitters configurations and used a 32 FPA as a receiver. In the article published by Hu et al. (2018), the authors propose the use of a different variation of the genetic algorithm called Almost Different Sets Genetic Algorithm with (26) to 44 find emitters and receivers sparse linear arrays configurations and also proposed an algorithm to enhance the ultrasonic image by post-processing it interpolating its pixels Different algorithms were also used, Zhang et al. (2020) proposed to adapt the particle swarm optimisation algorithm (PSO) into a binary search mechanism and used the same FF described in (26) to find linear sparse configurations. For 2D arrays, metaheuristics have been used to design the sparse arrays. Trucco (1999) uses simulated annealing algorithm to select elements from 3 pre-defined matrix arrays with 112, 200 and 3228 elements. The simulation method used by the author is the radiation pattern narrowband response, where the FF created is the error between the radiation pattern generated by a sparse array and the pre-defined matrix array. Moreover, a penalization function is added to control the elements selected from the matrix array. Chen et al. (2011) refined this FF by changing the penalization function and expanding the analysis for multiple foci to find arrays with the same quality as Trucco (1999) but with fewer selected elements. In the article published by Diarra et al. (2013), this fitness function is used with a sim- ulated annealing-based algorithm to design 2D sparse array for non-grid arrays, where the ele- ments are placed anywhere in the space. The authors achieved the same lateral resolution with reduced side lobes intensities compared to the proposed works. One drawback of this strategy is the manufacturing process. Although CMUT technology enables the array manufacturing (ORALKAN et al., 2003), the cost of doing it is high which is necessary to increase produc- tion to compensate for the cost. Later in the same research group, Roux et al. (2017) propose a new fitness function based on the multi-depth focal points of the radiation pattern wideband response, which needs to define the element impulse response and might limit the sparse array to work under a specific setting. The optimization algorithms are used to find sparse array configurations using an FF that evaluates the configurations. Another strategy used in array design consists in evaluating the coarray shape to create different emitter and receiver sparse arrays. 2.1.3 Coarray The Coarray or effective aperture is a representation of a receiver aperture that would produce an identical image if the transmit aperture was considered a point source (omnidirec- tional radiation) (GEHLBACH; ALVAREZ, 1981). It is a mathematical tool that merges the array used for emission and reception and can be used to get information about the ability of the array imaging. By calculating the DTFT of the coarray, the two-way or pulse-echo radiation pattern narrowband response is calculated, which is the product of the radiation pattern of the emitting array with the receiving array (LOCKWOOD et al., 1996a). The coarray is defined as the convolution of the emitter and receiver apertures (GEHL- BACH; ALVAREZ, 1981; LOCKWOOD et al., 1996a; LOCKWOOD et al., 1996b; KARA- 45 MAN et al., 2009). Considering that the array elements are impulses in the space, each coarray element is the product of an emitting and receiving element of the array, which is also an im- pulse, and the coarray position is the sum of the emitting position with the receiving position. In this way, the coarray can be calculated as a set of its positions, where each position is calculated as: Crt = { ~Cer = ~ee + ~er} ∀ ~ee ∈ ARe, ~er ∈ ARr, (27) where ~Cer is a coarray element generated by the respective emitter e and the receiver r, ARe the position set of the emitter elements, ARr the position set of the receiver elements, and ei = (xi, yi), which is the XY position in the plane. Figure 11 illustrates two cases of Crt. In both cases, as the linear array is analysed, y is zero. In (a), the convolution of the apertures ARe and ARr generates a coarray with a triangular shape. The elements in the linear arrays are in one grid and the positions on the coarray overlap. In (b), one element of ARr is displaced by 0.3λ. In this example, the coarray elements do not overlap, where the first row of Crt gives the positions of the coarray elements considering the ~e1 of ARr and the second row gives the positions of the ~e2 of ARr. The weight of each coarray element is the number of emitter/receiver pairs that are coincident at the same position, and the analysis of its shape can be used to estimate the dy- namic range and lateral resolution of the imaging system (DRINKWATER; WILCOX, 2006; HOCTOR; KASSAM, 1990). For 2D non-grid arrays, such as the Fermat spiral and annular segmented arrays, this superposition is very rare to occur, turning the analysis of its shape diffi- cult. In this way, this definition of coarray is important as it will be further used to analyse the coarray elements’ distance and identify elements that are closer enough to be considered in the same position. Figure 11 – Illustration of Crt. (a) the array elements are periodically spaced (in a grid). (b) One reception element is shifted by 0.3. (a) Same position (b) Shifted Element Source: Author Considering a linear FPA with 32 elements, where a rectangular apodization is used, and all elements are used for transmission and reception (Figure 12(a)), the coarray Crt ge- 46 nerated has 1024 elements where summing the elements that overlaps has a triangular shape (Figure 12(b)). The radiation pattern narrowband of the transmit array (ARe) and the Coarray (Crt) are illustrated in Figure 12(c), where the two-way radiation pattern has a lower MLW, 3.20◦ in comparison to the radiation pattern of the emitter FPA (4.39◦), and lower side lobes levels. Figure 12 – (a) Transmit and Receiver Linear FPA with 32 elements. (b) the coarray formed from the two arrays with triangular shape with all elements respecting a pitch 0.5λ. (c) the radiation pattern of the transmit array and the coarray (two-way radiation pattern). (a) (b) (c) −75 −50 −25 0 25 50 75 Angle[◦] −100 −80 −60 −40 −20 0 A m p li tu d e [d B ] Radiation pattern Radiation pattern of Are Two-way radiation pattern Source: Author The analysis of the coarray is important and was used to design sparse array configura- tions. Lockwood et al. (1996a) used two linear sparse arrays, one working in emission and the other working in reception to create a coarray where the elements have a 0.5λ pitch, although the shape of the coarray created is uneven. The spacing between the elements is based on the Vernier scale, the same scale existing in rulers and pachymeters. Figure 13 (a) illustrates two sparse periodic arrays created based on the Vernier scale, one used for emission and the other for reception. The transmitting array has a 1.5 λ, and the receiver has a 1λ pitch. These two configurations are periodic and have grating lobes with the same intensity as the main lobe in the radiation pattern narrowband response. However, the resultant coarray illustrated in Figure 13 (b) resembles a triangular shape and the elements 47 respect the pitch of 0.5λ. The two-way radiation pattern is illustrated in Figure 13 (c), where it is possible to see the reduction in the grating lobes intensities. Figure 13 – (a) Sparse Transmit and Receiver Linear arrays created based on Vernier distri- bution, (b) the coarray created where it has the triangular shape, respects a pitch 0.5λ but it has adjacents elements that alternate the amplitudes and (c) the two-way radiation pattern of the coarray created. (a) (b) (c) Source: Author Another example of using sparse periodic arrays to create a desired coarray is given by Mitra et al. (2010), where the authors used the concept of factorization of polynomials to place the emitter and receiver elements of arrays. The problem in the arrays created using this methodology is the concentration of emitters on one side of the array, which unbalances the energy irradiated and might generate non-linear artefacts in the images. For bidimensional arrays, a 2D matrix array with 16 × 16 elements is illustrated in Figure 14(a) as an array example. Figure 14(b) illustrates the coarray formed from the matrix array. As it can be seen by analysing the colorbar, the coarray has a square-based pyramidal shape, similar to the example presented using linear arrays (Figure 12(b)). Following the strategy to use periodic sparse arrays and create desired coarrays, Lock- wood et al. (1996b) also proposed the use of the Vernier scale to design sparse planar arrays aiming at a desired coarray shape. Karaman et al. (2009) proposed four strategies using differ- ent emitter and receiver arrays to create a minimum redundancy coarray, where the elements of the coarray avoid overlapping. Figure 15 illustrates one example presented in that work. In (a), the elements of the emitter array are placed at the vertical at the extreme boundaries, with 0.5λ pitch. In (b), the elements of the receiver array are placed horizontally at the extreme bound- aries, with 0.5λ pitch. The coarray generated from these two arrays is illustrated in Figure 15 (c), where it is possible to see the majority of the elements have amplitude one (in white colour), which indicates the minimum redundancy. Although this strategy almost creates a coarray in which elements are unitary, this anal- ysis is restricted to matrix arrays, where the elements overlap and can be summed. In arrays where the elements are in a non-grid, for example, Fermat spiral arrays (Figure 2(b)), the coar- 48 Figure 14 – (a) Matrix array with 256 elements (16 × 16) where the adjacent elements in the x- axis and y-axis have a pitch equals to 0.5λ (λ = 1[mm]) and the elements diagonally adjacents have a pitch equals to 0.70λ. (b) The coarray is created from the matrix array which has a pyramidal shape. (a) (b) Source: Author ray elements do not overlap and the redundancy cannot be analysed in the same way. Figure 15 – An example of the minimum redundancy array (KARAMAN et al., 2009) where (a) 30 elements are disposed of two vertical lines and (b) 30 elements are disposed of two horizontal lines where the elements in the lines have a pitch equal to 0.5λ (λ = 1[mm]). (c) the coarray is generated from these two apertures where the majority of the elements have amplitude one. (a) Emitter Array (b) Receiver Array (c) Coarray Generated Source: Author The coarray is a mathematical tool that can be used to evaluate sparse configurations. It is considered a spatial apodization as the two-way radiation pattern is the Fourier transform of the amplitudes of the coarray (LOCKWOOD et al., 1996b). Another alternative to evaluate sparse configurations is the point spread function (PSF), which is the simulation of a reflector point in the medium and gives information about how a sparse array can image. Before ex- 49 plaining how the PSF is generated, which will be done in section (2.2.3), first, it is necessary to know how an ultrasonic image is generated. 2.2 ULTRASONIC IMAGING The basic principle of ultrasonic images is to use a transducer with a single element to emit an ultrasonic wave in the medium, and, in the case of reflection (change of density), the reflected mechanical wave is converted to an electrical signal by the same transducer, amplified, and then sampled. By doing this operation in different areas, creating a sweep, the acquired signals can be used to create an image. Figure 16 illustrates two simple imaging strategies. In (a), where the B-scan is repre- sented, the transducer is shifted and activated in the x-axis. The irradiated wave reflects at the end of the object or the hole. With the acquired data, a brightness image can be created in the x and z axis, as illustrated in Figure 16 (b). The C-scan is illustrated in (Figure 16(c)) where the transducer shifts in the x and y region. In this case, a section of the image in a certain depth is generated (Figure 16(d)) . The use of a single transducer to generate images is a simple example to understand the principle of ultrasonic images. In practice, images generated by a single element would have low lateral resolution and contrast. Moreover, the element needs to be mechanically moved, which decreases the frame rate and creates a margin for positioning errors. Arrays are used to overcome these problemss, and two main strategies are used for data acquisition and imaging: the Phased Array and the Synthetic Aperture. 2.2.1 Phased array In Phased Array systems, the elements in the array can be pulsed independently, and depending on the relative phases of element excitations, the ultrasonic beam can be steered or focused. The most common inspection modality is the B-scan (Figure 17(a)), where a sub- set of elements is used to inspect the object, and the reflected signals are plotted as intensity (DRINKWATER; WILCOX, 2006). This inspection modality is similar to the monolithic il- lustrated in Figure 16(a). However, the difference that the scan is done electronically and the mechanical movement is reduced. A different inspection modality is created when delays in the activation of the elements are applied. Figure 17(b) illustrates the focused B-scan, where the elements are fired with delays between each other and the beam focused in a region. The delays can also be used to create a sector scan (Figure 17(c)), where the beam is steered at an angle and objects can be scanned at positions not directly in front of the array. One drawback of Phased Arrays is that the elements need to work in parallel, increasing 50 Figure 16 – Simple imaging system using a single element transducer. (a) representation of the imaging system where the transducer is shifted only in the x axis and the resulting image (b) created in the x and z-axis. (c) An imaging strategy where the transducer moves in x and y-axis. (d) the segmented image generated in a different depth. (a) Imaging system (b) Image generated in x and z axis (c) Imaging system (d) Image generated in x and y axis in a specific depth Source: Author electronic complexity. An alternative to overcome this problem is the use of the Synthetic Aperture technique. 2.2.2 Synthetic aperture Synthetic aperture is an imaging technique based on the synthetic aperture radar, where the reflected electromagnetic wave of a region is sampled in different positions, and an image of a region is created (JENSEN et al., 2006). Figure 18 illustrates the working principle of this technique, an antenna is attached to the aircraft that moves above an area. The antenna emits and receives the waves in different positions and the data acquired is processed to create an image. In synthetic aperture, the elements of the array are sequentially emitted, which copies how the antenna is used in Figure 18. An illustration of the SA technique is given in Figure 19. At the first stage, element 1 emits, and all array elements receive. Afterwards, element 2 emits, and all array elements receive. This process is repeated for all N elements, which creates a 51 Figure 17 – Phased Array inspection modalities. (a) B-scan. (b) Focused B-scan. (c) Sector Scan. (a) B-Scan (b) Focused B-Scan (c) Sector Scan Source: Author Figure 18 – Synthetic Aperture Radar. Source: Author dataset with the echoes of all combinations of emitters and receivers, namely full matrix capture (FMC) (HOLMES et al., 2005). Figure 19 – Synthetic Aperture. Source: Adapted from Jensen et al. (2006) After the process of signal acquisition, an algorithm needs to be used to create an ul- 52 trasonic image, where the gold standard algorithm is called Total Focusing Method (TFM) (BANNOUF et al., 2013). This algorithm consists of applying delays and summing signals samples. Equation (28) gives the mathematical operation to calculate the TFM: I(x, z) = 1 N2                 N ∑ e=1 N ∑ r=1 wer ser(τer(x, z))        2 +        N ∑ e=1 N ∑ r=1 wer ŝer(τer(x, z))        2          1/2 , (28) where I(x, z) is the envelope of the amplitude image at a given (x, z) position, wer the apodiza- tion, ser(t) the ultrasonic signal related to the emitter e and receiver r, ŝer the Hilbert transform of ser(t), and τer(x, z) the time of flight from the emiter e to the pixel (x, z) back to the receiver r. Figure 20(a) illustrates how the TFM algorithm is calculated. The image area is discre- tised in a grid in the x, z plane, and the amplitude of the image is calculated at all points. At each point, the Euclidian distance from the emitter element e to a point in the grid (x, z) is calculated and summed to the distance from the point (x, z) to the receiver element r. This value is then divided by the velocity of the sound in the medium (c) to find the time of flight τer. Afterwards, the respective amplitudes at the time τer are summed. Figure 20 – Total Focusing Method (TFM). Representation in (x, z) grid (a), and in (r, θ) grid (b). (a) TFM (x, z) (b) TFM (R, θ) Source: Author Equation (28) gives the TFM calculate in respect to cartesian coordinates, which can be changed to polar using the cosine rule (Figure 20(b)). In this case, instead of using the notation I(x, z), I(R, θ) is used, where R is the radius and θ the angle that the image is calculated. The synthetic aperture technique reduces the electronic complexity and system cost as the emission and reception can be multiplexed. At the minimum, it is only necessary one Ana- 53 log to Digital (A/D) converter, but, in exchange, the acquisition time increases as N2 emissions are required. This long-duration acquisition time might turn impractical for the use of this technique; therefore, the solution is to use more A/D converters and add multiplexers to en- able several receivers to work in parallel. However, if the number of array elements increases, followed by the number of A/D converters and multiplexers, the system cost and data volume escalate at an unfeasible level. The sparse array is one solution for this problem, where some elements are placed in a large area. However, there is another alternative. Some emitters/receivers combinations can be removed from the acquisition process, which creates an acquisition strategy different from FMC. With fewer combinations of emitters and receivers, the number of elements in the array can increase, covering a larger area, and the number of receiver channels working in parallel can be managed, which helps to remain the system cost and data volume at an acceptable level. 2.2.3 Point Spread Function A different strategy to analyse the arrays’ ability to generate an image is the Point Spread Function (PSF). In this method, a point reflector is placed in space and, using simulated data, the ultrasonic image is generated (DRINKWATER; WILCOX, 2006). The data set of the combina- tions of emitters and receivers can be simulated by time-shifting the electrical input response of the transmitting element to the time of flight of the combination of emitter and receiver element. In Figure 21(a), an electrical input response defined using equation (18) is illustrated. In this example BW = 0.5, u0 = 1, fc = 3.5 MHz, and fs = 35 MHz. where fs is the sample frequency. The shift operation can be done in different ways, for example, the electrical signal g(t) can be convolved with a time-shifted impulse δ(t − τer), or multiplied by an exponential in the frequency domain. Another alternative, used in this work, is to time-shift (18), which will be written as: g(t − τer) = u0e π2BW2 f 2 c (t−η−τer)2 1.2 ln 10 sin(2π fc(t − τer)), (29) where the electrical input signal shifted at the time τer = 9µs is illustrated in Figure 21(b). The shifted signals are calculated for all combinations of emitters and receivers that are going to be used for imaging, and the data set is created. Afterwards, an algorithm is used to generate the PSF. Figure 22 illustrates the PSF of a FPA with 32 elements, where the point reflector is placed at x = 0 and z = 32λ. In (a) the image angle vary from −90◦ to 90◦. As the array is symmetric, it is only necessary to generate half of the angles (from 0◦ to 90◦). Figure 22 (b) illustrates only the half of the image that carries the same information as the full image. This strategy can be done only in symmetric arrays and is useful to reduce the simulation time. Figure 23 illustrates three cases of PSF created using only half of the image. (a) is the 54 Figure 21 – Data set simulation. (a) electrical input response. (b) shifted signal to the time of flight (9µ). (a) Electrical input response (b) Shifted signal Source: Author Figure 22 – PSF of an FPA with 32 elements. (a) Image generated from −90◦ to 90◦. (b) as the array is symmetric it is only necessary to generate half of the image (from 0◦ to 90◦). (a) Full Image (b) Half Image Source: Author PSF of a FPA with 32 elements; (b) is the PSF of a 16-element sparse periodic array (SPA) with λ pitch, and (c) is the PSF of an 8-element SPA with 2λ pitch. The point reflector remained the same size for all PSF, as the arrays have the same length. The number of signals processed and mathematical operations reduced from 1024 to 256 and 64, respectively, which decreases the time to generate images and resources. However, there is an increase in the intensity of the artefacts caused by the grating lobes. Figure 23(d) illustrates the normalized mean in the axial direction of the PSFs. To create this graph, the PSF is stored in a matrix where the lines correspond to the radii R of the image and the columns of the angles θ. At each θ, the amplitudes of the radii are summed, resulting in a vector that is then normalized. It is possible to see an increase in the artefacts’ intensity as the lateral resolution remained the same. The energy concentrated in a region outside the point reflector is distributed by breaking 55 Figure 23 – PSFs of (a) FPA with 32 elements, (b) SPA with 32 elements a pitch λ and (c) SPA with 32 elements and pitch 2λ. (d) normalized sum of the amplitudes. (a) FPA 32 elements (b) SPA 32 elements pitch λ (c) SPA 32 elements pitch 2λ (d) normalized sum of the amplitu- des Source: Author the periodicity of the elements. Figure 24 (a) illustrates a sparse array where 16 elements are selected from a grid with 32 positions that correspond to a FPA. The 32 elements sparse aperiodic array is created by mirroring the selected elements and, in Figure 24(b), the PSF of the sparse array is illustrated. The energy is not concentrated in an area as in Figure 23(b) and (d), but it spreads in the area beside the point reflector. This energy interferes with image quality by increasing the artefact’s intensity. In this way, it is interesting to find a sparse array configuration that has improved lateral resolution with low energy besides the point reflector. 2.3 COMMENTS In this chapter, the background to understand how arrays are used to create an ultra- sonic image is given. The differences between FPA, sparse periodic and aperiodic arrays are presented, and the benefits of using sparse arrays to increase the speed to generate an image in exchange for image quality are discussed. As it could be verified, different combinations of sparse arrays can be obtained, making it impossible to test all the combinations. In this sense, there is a need to define strategies that find sparse array configurations while reducing 56 Figure 24 – (a) a sparse array created by selecting 16 elements in a grid corresponding to a 32 FPA. (b) the PSF of the sparse array where the energy besides the point reflector