Bachelor's Thesis: Machine learning techniques for enhance amplitude patterns from phased-arrays augmented with an holographic plate

ESCUELA T´ ECNICA SUPERIOR DE INGENIER´ IA AGRON ´ OMICA Y BIOCIENCIAS NEKAZARITZAKO INGENIARITZAKO ETA BIOZIENTZIETAKO GOI MAILAKO ESKOLA TEKNIKOA Machine Learning Techniques for Enhance Amplitude Patterns from Phased-arrays Augmented with an Holographic Plate presentado por Mikel Aldea Esnaola aurkeztua GRADO EN CIENCIA DE DATOS GRADUA DATUEN ZIENTZIETAN Septiembre, 2023 / 2023, Iraila Abstract Phased-arrays are an array of transducers, in which the relative amplitudes and phases of each emitter are controllably varied in order to design the acoustic pressure pattern emitted by the array. The control of these transducers enables to generate a desired amplitude pattern at a certain distance. The quality of these patterns can be enhanced by adding an holographic plate modulator at an intermediate distance between the emitters and the target. Here, we propose new techniques using a Neural Network architecture to compute the parameters of the phased-array and the holographic plate needed to generate the desired pattern. Key Words Acoustic holograms, Ultrasound, Holographic techniques, Deep Learning, Neural Networks, Auto-Encoder Resumen Las antenas en fase son un conjunto de transductores en los que las amplitudes y las fases de cada emisor se var´ıan controladamente para alterar el patr´on de presi´on ac´ustica dise˜nado por los transductores. El control de estos transductores permite generar un patr´on de amplitud deseado a una distancia determinada. La calidad de estos patrones se puede mejorar agregando una placa hologr´afica a una distancia intermedia entre los emisores y el objetivo. Aqu´ı, se proponen nuevas t´ecnicas que utilizan una arquitectura de red neuronal para calcular los par´ametros de las antenas en fase y la placa hologr´afica necesarios para generar el patr´on deseado. Palabras Clave Hologramas ac´usticos, Ultrasonido, T´ecnicas Hologr´aficas, Aprendizaje Profundo, Redes Neuronales, Auto-Encoder 1 Agradecimientos En primer lugar, quiero agradecer a mi director de TFG Asier Marzo P´erez por todo el apoyo, ayuda e implicaci´on mostrada durante estos meses de trabajo. Gracias a ´el he podido aprender mucho acerca de ac´ustica, la cual era un tema que no dominaba en un principio y gracias a su apoyo me ha ayudado a intentar aportar, aunque sea un granito de arena al tema. Tambi´en me gustar´ıa agradecer a todo el equipo de UpnaLab, en especial a Manuel L´opezAmo Oc´on y a Jon Goikoetxea por su colaboraci´on y ayuda en el proyecto. Gracias a todos me he sentido muy bien acogido en su equipo y contagiado por sus ganas de aprender y de adquirir conocimiento. Por ´ultimo, me gustar´ıa agradecer a mis padres Javier Aldea Subiran y Mari Mar Esnaola Bermejo, a mi hermano Asier Aldea Esnaola y a toda mi familia y amigos, por todo el apoyo mostrado a lo largo de mi vida. De todo coraz´on, muchas gracias. 2 Contents 1 Introduction 4 2 State of Art 4 3 Experiments 6 3.1 ModelingoftheProblem............................... 6 3.2 Forward Step of the Physical Part of the Neural Network . . . . . . . . . . . . 8 3.3 Architectures Propose for the Virtual Part . . . . . . . . . . . . . . . . . . . . . 11 3.3.1 Diffracted Acoustic Network . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.3.2 One-Hot Layer Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3.3 Encoder Decoder Diffractive Network . . . . . . . . . . . . . . . . . . . 14 3.4 SimulationSetUp .................................. 15 4 Results 18 4.1 Pure Array Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2 Diffracted Acoustic Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.3 One-HotLayerModel ................................ 20 4.4 Encoder Decoder Diffractive Network . . . . . . . . . . . . . . . . . . . . . . . . 21 4.5 Comparison of all models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5 Conclusions and Future Work 25 6 References 26 7 Supplemental Information 27 3 1 Introduction Holographic techniques has diverse applications, such as in fields like medicine [1], biology [2] and engineering [3]. All, this sort of applications include different aspects such as volumetric displays [4] or particle manipulation [5]. In all of these applications, acoustic holograms must generate an acoustic pressure control pattern at a specified position. There multiple ways to generate an acoustic field, here we are use phased array transducers (PATs). A PATs is composed of many elements arranged that emit acoustic energy at different times in order to direct the sound wave in a specific direction. The amplitudes and phases needed to create that pattern can be computed with pure array optimization, but recently some deep learning techniques has been used to achieve this goal. Usually, with a high number of transducer a good resolution pattern can be achieved, but this is generally expensive. In order, to try to reduce this amount, new techniques that incorporate the use of holographic plates have recently been gaining popularity. 2 State of Art In terms of generating acoustic patterns, many methods involving the used of phased array have been proposed. Some remarkable methods use pure phased array optimization via iteration methods like proposed by Weibai Li et al. [6] on November 2022, where they used this iterative method to compute an optimized phase distribution for generating different figures such as an “S” or a kangaroo shape pattern. Another methods use automatic differentiation such as the one proposed by Fushimi et al [7] in June 2021. Fushimi et al. proposed a phase-only gradient-descent algorithm with automatic differentiation as a new platform for optimizing acoustic holograms. On their paper, they use a large variety of phased array like 14x14 or 32x32, adding flexibility to the problem and achieving better results that the previous works. Other methods involve the use of deep learning techniques such as Neural Network like the method proposed by Boyi Li et al. [8] in June 2022 where they come up with the design of an unsupervised Neural Network base on a U-Net architecture where they combined the use of CNN’s and Transformers to learn the local and the global features of the acoustic hologram. Similarly, Lin et al. [9] on July 2023 design a physics-enhanced deep neural network also base on a U-Net for generating multi-frequency acoustic holograms. All of these techniques are based on the use of phased arrays, which means that in order to achieve a good quality acoustic pressure pattern, these phased arrays need to have a sufficient number of transducers to achieve it. This means that many transducers are needed (on the order of 256 or 512 or up to 1024), which are expensive. In order, to try to reduce this amount, new techniques that incorporate the use of holographic plates have recently been 4 gaining popularity. The first appearance of the use of an holographic plate was proposed by Melde et al. in this paper [10] on 2016. On this paper they create a ripple on the surface of the water by using ultrasonic waves emanating from a speaker underneath. But crucially, the waves coming from the speaker pass through a hologram plate. On this holographic plate some parts are thicker than others. Where it’s thicker, the waves take longer to pass through and this creates a distortion pattern. As the waves reach the surface, they form the shape of the ripple. Using this technique, researchers can create all kinds of patterns, like a dove, this patterns will have much more resolution and accuracy than without using the holographic plate. The milestone of this paper was that they were able to generate an acoustic pattern with an incredible resolutions just by using one transducer and an hologram plate. The limitations were that the holographic plate modifies the wavefront of just a single ultrasonic transducer, which means that, for each pattern they want to do, they need to built a new holographic plate designed specifically to make that and only that pattern. Then on May 2023, Athanassiadis et al. [11] proposed a new method called Multiplane Diffractive Acoustic Networks (DAN) in which they were able to apply this technique to more than one transducer and thus they create more than one pattern using the same holographic plate. Of course, this holographic plate performs worse on a pattern than a plate specifically created for that pattern, but it was a huge milestone because it was the first time that an holographic plate was used to generate more than a single pattern. In order to achieve this, they transform the problem into a neural network including the holographic plate as customized layer of the model. The limitation of this method has to do with the way in which they modeled the inputs necessary to train the neural network. What they did was use the phased array itself as input, when the phased array transducers must be trained to know what to emit in order to obtain one pattern or another. This then leads to the question of what should the input be in order to generate a desired pattern (that is, what does the transducers in the phased array have to emit). So, in this way, the inputs also need to be trained along with the rest of the parameters of the neural network. But generally in a neural network the input are fixed and are not trained, which is why with that architecture that they proposed they were not able to train the part of the phased array. To solve this problem they decided to assign each emitter to each pattern, so they pass as input a one-hot vector (a vector of all zeros except a one) in which the one corresponded to the emitter on the phased array that was turned on to recreate that pattern and the rest emitters remained off. In this way they already knew the inputs to train the neural network. The limitation that this entails is that if you have N emitters you can only generate N patterns. They later tested configurations of pairs of emitters turned on at the same time being able of generate N 2=N·(N−1) 2patterns, but the results were experimentally worse than when they turned on a single emitters and they still have some amount of emitters that where not contributing in generating certain patterns. 5 Here, we propose different neural network architectures that are able to use all the transducer at the same time for generating all patterns, solving the issue of only just being able to use a single emitter for each pattern. 3 Experiments 3.1 Modeling of the Problem To compute the amplitude and phase of the phased array and the phase shift of the holographic plate needed to achieve the desired patterns we are going to use artificial intelligence techniques, precisely neural networks, but as the problem is based on a physical problem, the neural network must be subject it to some constraints in order to modulate how the real world works. A conventional neural network (as shown in Fig.1) is made of layers, in the figure there is one input layer, one hidden layer an one output layer, in reality there can be more than one hidden layer. Each layer can have an arbitrary amount of nodes. In Fig. 1 each layer has nnodes, but it doesn’t necessarily have to be like that. Input Layer Hidden Layer Output Layer Figure 1: A Conventional Neural Network with one hidden layer In a dense neural network (as shown in the figure) or also called fully connected neural networks, all nodes in the previous layer (N nodes) are connected to all the nodes in the next layer (M nodes) by a a weight wnm, also for each node mthere is a bias bm. Both parameters need to be training. For each individual node ja predicted value hjis computing by using the formula bellow: hj=f bj+ n X i=1 wi·xi!∀j∈[1, m] (1) However, for our problem we need to make some changes in the architecture and the parameters that need to be trained, because we are subject to suit the real life problem. What we 6 want to do is to model the problem in such a way that it resembles the real problem as closely as possible. For that, our input layer will represent our phased array, composed by N×N transducers, the hidden layer will be the holographic plate, formed by M×Mpixels and the output layer will be the chosen height at which the patterns will be constructed (discretized into S×Scontrol points). A 3D representation of the model is shown at Fig. 2. Figure 2: 3D Representation of the problem But the transformation of the problem into a neural network is tricky because here the input (which will be what should the emitters emit in order to create an specific pattern) needs to be trained, and usually in a neural network does not the case. But also, here the weights will represent the propagation of the acoustic wave from each emitter in the phased array to each pixel of the holographic plate, so they are fixed and they will not be trained. What will indeed be trained, is going to be the phase shift that each pixel on the holographic plate will applied to the wave that reaches that point. Furthermore, as we are working with waves we will work with complex numbers. So in summary, each emitter in the phased array is going to have two parameters, the amplitude and the phase of the wave it creates, this parameters will be trained. Then all the created waves will travel from the phased array to the height at which the holographic plate is located, this will represent the weights w1in a conventional neural network but with the particularity that this weights will not be trained and will be compute using physics laws. We will also have no activation function. So, with this computation we will know the acoustic pressure that reaches the holographic plate. An holographic plate is made up of many holes (M×M), of different widths and depending on the width each hole has, a different phase shift will be applied to the wave, our task will also be to compute which phase shift should each hole have to redirected the wave and create the desire patterns, this could be consider similar to a conventional bias b, because it is individual for each node. Then at the end, following the 7 law of physics, we will compute the propagation of the wave to the height where we want to produces the desired patterns, this will be analogous to w2in a conventional NN. The amplitude of the acoustic pressure produced at that height will be our pattern. A representation of the problem model as a neural network can be seen at Fig. 3. Phased Array )Flatten ( Holographic Plate )Flatten ( Target Pattern )Flatten ( Target Amplitude Pattern )Reshaped( Figure 3: Neural Network Architecture propose to model the problem The propagators (denoted as T1and T2in Fig. 3) to determine the acoustic pressure that a transducer nproduces at a point mare usually called transfer matrices. To compute this transfer matrices we will use the matrix method for acoustic levitation proposed in this paper [12] which is equivalent to the Rayleigh integral. So the transfer matrices can be computed as: Tmn =sn eikrnm rnm (2) Where i=√−1 and k=ω cis the wavenumber, usually called repetency and is the spatial frequency of a wave, it is made up of the angular frequency w= 2πf and in our case f= 40000Hz,cis the velocity of the sound in the medium, in our case air, c= 340(m/s) and rnm is the distance between the transducer n and the point in space m. 3.2 Forward Step of the Physical Part of the Neural Network Once all the elements in the physical model of the neural network have been explained, then it is timed to explain how the forward step is done. As we are using TensorFlow library [13] it is only needed to define the forward step of the NN and the software itself of TensorFlow will take care of the backward step. So, first of all, we are going to initialize randomly the amplitudes and phases of the Phased Array. Amplitudes (A) will be initialized with values between 0 and 1 and phases (θ) will be initialized with values between −πand π. With these two values we will be able to create the complex number that will represent the acoustic wave created for each transducer, that is to 8 interesting idea. One aspect to take into account when building the non real part is that the real part can not be totally invertible, the issue appears when we try to applied the the inverse of function f. f:C→Rf−1:R→C z=Aeiθ →f(z) = A f(z) = A→f(z)−1=Aeiθ (9) As in the function fwe go from being in the complex space that is a space of two dimensions (real part Re(z) and imaginary part Im(z)) into a space of just one dimension, there is no issue because we have amplitudes and phases and we just care about the amplitudes, so we discard the phases, but when we want to do it on the other way, we know the amplitudes, but we don’t know what the phases should be. To solve that, we can add arbitrary phases like all 0s or random phases or even been train it as a parameter of the model. One of the problems of this model is that it requires a huge amount of parameters apart from the ones in the real part (that are by itself a lot too). In fact, for the non real part in order to connect the target distance with the holographic plate T′ 2is a matrix S2×M2, then the holographic plate has M2phase shifts and at the end to connect the holographic plate with the phased array a transfer matrix T′ 1of M2×N2is needed, so this add S2·M2+M2+M2·N2 which is equal to M2·(S2+N2+ 1) more parameters just to compute the virtual part. On contrary, the one-hot layer model (previous model) just adds P·N2new parameters, which is a considerable difference. One way to reduce the number of parameters is, instead of having the transfer matrices T′ 1 and T′ 2in the virtual part be trained, we will make them fixed and compute it using the same physical formula that we used for the transfer matrices in the physical part, but this will reduce a lot the power of the model. Even so, we will do it on this way, to being able to run the model due to computational power problems. 3.4 Simulation Set Up In order to make a fair comparison between models, they will all be executed under the conditions, this means, same number of transducer in the phase array, same number of pixels in the holographic plate, same distance between the phased array, holographic plate and target distance, etc. Both the phased array, the holographic plate as well as the surface where the pattern is located, will be squares of 16x16 cm each. As the whole idea of using holographic plates is for trying to achieve good quality patterns without the need of using many transducers, the number of transducer in the phased array will be small, this is 16 (a squared of 4x4), so N= 4. The holographic plate will have 64x64 individual points for which the wave travel and changes his phase, so M= 64 and finally the image pattern is discretized into 128x128 pixels or control points (S= 128). 15 The dataset of patterns for training the neural network will be extracted from the MNIST dataset [14]. This dataset consist of 60.000 training images and 10.000 test images of handwritten numbers from 0 to 9. This images are all 28 ×28 pixels each. For obtaining the patterns just one random image of each digit will be taken and there will be resize into 128 ×128 using the bilinear interpolation method [15]. The images selected can be shown at Fig. 9. Figure 9: Dataset of the patterns So the dataset will consist of just 10 images, when specified, we will use a larger dataset that also contains letters. This dataset will be extracted from the EMNIST dataset [16]. This, when needed, will add 26 more patterns, so the complete dataset will have 36 total patterns. This new images can be seen at Fig. 10. Figure 10: Dataset of the patterns including Letters The framework used to create and modify the neural networks will be TensorFlow [13]. 16 The next discussion for the experimental set up must be: What should be the distances between the phased array and holographic plate? and to the target distance? In order to select the perfect distances we are going to do a grid search of distances between 5 and 30 cm, taking steps of 5 cm for both distances (from Emitters (Phased Array) to the Holographic Plate and from the Holographic Plate to the Target). To measure the quality of the results we are going to report the negative value of the logarithmic of the mean squared error loss: −log(MSE) (10) This decision was taken because the loss was very close to 0, so there were a lot of zeros and it was difficult for the eye to see which values where smaller. Now, using equation 10 results are in logarithmic form and are more intuitive to see the differences (now, unlike for the mse, the more the better). The grid search would be done for two models: •DAN •EDDN The results can be seen at fig 11: DAN EDDN Figure 11: Grid Search of the log loss for different trying distances As it can be see in the figure, the best distance for the Holographic Plate for both models is when the plate is very close to the Phased Array (5 cm). On the contrary, for the target distance it seems like around 20-30 cm from the Holographic Plate is the best distance. In this test the best distances are 5 cm and 20 cm for the DAN model and 5 cm and 25 cm for the EDDN method. As the EDDN method gives a higher loss, we will use 5 and 25 cm for 17 the final distance test for all the experiments. A 3d representation of the final simulation set up be seen at Fig. 12. 16cm 16cm 5cm 25cm Holographic Plate (64x64) Phased Array (4x4) Target (128x128) Figure 12: 3d representation of the simulation set up 4 Results 4.1 Pure Array Optimization The base method to compare against results, is usually pure array optimization. Pure array optimization consist of an optimization of just the phased array according to a target distance and just a pattern, which means to have no holographic plate and just optimize the phased array for an specific pattern. With just the phased array we can achieve good image resolution when the number of transducer is big enough [7]. For instance, by using 16 ×16 (512) transducers the simulations results have a very good resolution as it can be seen in Fig. 13. 18 Figure 13: Mean Log Loss for the MNIST dataset using Pure Array Optimization with 16x16 transducers Although, if instead of using 512 transducers that will involve a high cost, we used 16 transducers (4 ×4). The results are much worse. As it can be seen in Fig. 14. Figure 14: Mean Log Loss for the MNIST dataset using Pure Array Optimization with 4x4 transducers As it can be seen in the images, no pattern can be distinguished, so in order to achieve a good resolution with a very few amount of transducers an holographic plate is needed. We will use both the model with 4 ×4 and the one with 16 ×16 results to compare with the different models discuss in this paper. The idea will be to try to outperform the classic pure array optimization with the same number of transducers 4 ×4 by adding the holographic plate and see if it can compete with the phased array that has 240 more emitters. 19 4.2 Diffracted Acoustic Network Diffracted Acoustic Network (DAN) is the current best method for acoustic pattern generation using both a phased array and an holographic plate. Although in their paper they used more than one holographic plate, obtaining better results, here for our problem we will only use one holographic plate as the idea is to try to bring all this simulations to some real life experiments and we are forced to just use one holographic plate. The simulations results obtain replicating the DAN method with 4 ×4 emitters are shown at Fig. 15. Figure 15: Mean Log Loss for the MNIST dataset using DAN method with 4x4 transducers As it can be seen in the image the DAN method is able to almost double the results of pure array optimization 4 ×4, but still is 1 unit bellow this method when we use 240 more transducers. Even so, the patterns can be perfectly distinguish between them. One disadvantage of DAN is that we are forced to use just one transducer per pattern, so we loose a lot of power. Now, it is turn to see how it performs the models that use all transducers at the same time. 4.3 One-Hot Layer Model The One-Hot Layer model, incorporates a one hot input vector of (1 ×N.Patterns) dimension before the phased array layer, this makes that each pattern has its own pair of amplitudes and phases trained for make that specific pattern, resulting in a use of all the transducer at the same time. The acoustic pressure patterns generated with this models are shown at Fig. 22. 20 Figure 16: Mean Log Loss for the MNIST dataset using One Hot Layer Model with 4x4 transducers As it can be seen, the model achieves about 0.5 more log loss units than the DAN model. This proves that the use of all the emitters at once performs better than just using one emitter per pattern. An example of how the one hot layer model use all the transducers at the same time can be seen at the Supplemental Information in Fig. 27. 4.4 Encoder Decoder Diffractive Network Figure 17: Mean Log Loss for the MNIST dataset using the Encoder-Decoder Model with 4x4 transducers As it can be appreciated on Fig. 17, the EDDN model has better loss that the previous models, but also has more parameters so the training is more expensive. Still they are not able to obtain better resolution than pure array optimization when uses 240 more emitters. 4.5 Comparison of all models In order to make a comparison between all the models we will train the models with different number of patterns in the datasets. For this we will use both the MNIST and EMNIST 21 dataset explained in the experimental set up. We will compute the results for all the models explained and using both 4x4 and 16x16 transducers in the phased array. Results are shown in Table 1. N. Patterns ArrayAmp4 ×4 DAN4 ×4 OneHot4 ×4 EDDN4 ×4 ArrayAmp16 ×16 DAN16 ×16 OneHot16 ×16 EDDN16 ×16 1 2.575016 6.832472 5.647378 6.710172 5.498843 6.959398 5.962122 6.003410 2 2.869333 5.283944 6.367932 7.141631 5.761842 4.133021 6.590206 6.266028 3 2.796834 5.195768 6.377517 6.837344 5.622975 3.682206 6.825670 6.557240 4 2.742588 4.875667 6.338354 6.701730 5.604670 3.418805 6.452779 6.644028 5 2.831655 4.745792 6.331753 5.851975 5.685099 3.313510 6.611486 6.660971 6 2.823563 4.663406 5.999474 6.124025 5.652447 3.279502 6.448646 6.628277 7 2.790624 4.570724 5.831988 5.816370 5.695535 3.201134 6.281012 6.221437 8 2.783131 4.52635 5.506751 5.591366 5.469140 3.190218 6.288487 6.430957 9 2.770532 4.499083 5.320284 5.088617 5.405443 3.162776 6.502002 6.249623 10 2.755957 4.466246 5.067284 5.031923 5.434548 3.124454 6.128260 6.108799 11 2.745096 4.469185 4.908253 4.958215 5.402232 3.091897 6.195076 5.693188 12 2.729853 4.432047 4.855785 4.961229 5.377839 3.074676 6.088498 5.861040 13 2.709344 4.416377 4.546329 4.755369 5.395706 3.046923 6.257993 5.839122 14 2.717903 4.328341 4.461514 4.655101 5.367787 3.037177 6.132006 5.151098 15 2.728405 4.369721 4.447314 4.481391 5.350582 3.035635 5.965340 5.487315 16 2.724201 4.273734 4.364834 4.573129 5.393351 3.022809 6.064346 5.241032 17 2.717885 - 4.287466 4.426954 5.420611 3.006384 6.144506 5.191124 18 2.728515 - 4.231863 4.382349 5.406060 2.977415 5.875263 5.072109 19 2.740791 - 4.236287 4.392656 5.482078 2.972647 5.993216 5.125181 20 2.737294 - 4.140568 4.347224 5.327073 2.958785 6.020157 5.048000 21 2.718129 - 4.056518 4.096061 5.352228 2.930524 5.841179 4.944443 22 2.729428 - 4.151032 4.189607 5.234871 2.912600 5.911826 4.785452 23 2.746764 - 4.033381 4.220265 5.233908 2.910188 6.027016 4.796527 24 2.745650 - 4.002668 4.192032 5.183343 2.885713 6.070908 4.767665 25 2.734945 - 3.953484 4.184487 5.233879 2.864371 6.094308 4.862391 26 2.736192 - 3.999540 4.171921 5.144291 2.862345 5.849265 4.756981 27 2.748772 - 3.934195 4.145893 5.071000 2.850037 5.879921 4.802422 28 2.737918 - 3.911794 4.065598 5.074897 2.831346 6.081284 4.695708 29 2.732123 - 4.004441 3.906136 5.068943 2.827667 5.851877 4.453587 30 2.747672 - 3.884325 4.033484 5.090059 2.823598 5.811532 4.399318 31 2.744320 - 3.912122 4.012542 5.091320 2.805714 5.905230 4.520255 32 2.735354 - 3.895737 3.866368 5.111089 2.788945 5.701329 4.315735 33 2.741739 - 3.903576 3.832810 5.108865 2.734001 6.061327 4.523477 34 2.744239 - 3.775784 3.862657 5.117916 2.747728 6.029439 4.484844 35 2.753015 - 3.849150 3.825256 5.129606 2.763275 6.076796 4.361021 36 2.746948 - 3.770820 3.779227 5.152292 2.765591 6.126031 4.459760 Table 1: Table of the log losses of all the models depending on the number of patterns in the dataset *Note that DAN4x4 can only performs until 16 patterns, because it only has 16 transducers. For more results, an example of the amplitude acoustic patterns generated by the OneHot4x4 with a dataset of 36 images can be seen in the supplemental information at Fig. 28. A graph to understand better how the loss evolve as the number of patterns increase for all the different models can be seen a Fig. 18. 22 Figure 18: All losses for dataset with different number of patterns Here are some conclusions of this comparison: As we could have expected for the array amp slice the performance maintains stable along the number of patterns: Figure 19: ArrayAmp Log Loss for different number of patterns dataset That is because there is no training and each pattern is optimized individually which means that if we want to tune the phased array for 36 patterns we have to do 36 optimizations, thus means that if we want to do more and more patterns the computational time skyrockets. Also we can see that using a phased array of 16x16 transducers has better results than a phased array of 4x4 which make a lot sense. Another interesting aspect which is logical is that the results for the models that uses an holographic plate get worse as we increase the number of patterns. That is because we are forcing the holographic plate to try to satisfy a greater number of patterns each time and this 23 results in an overall general decrease in the quality of the patterns. There will be a time when we could say that the plate has reached his limit, that is, there are too many patterns for that plate. Also we can see that with 16x16 emitters the workload of the holographic plate is reduce (thanks to having more emitters) and can take more patterns. Figure 20: EDDN Log Loss for different number of patterns dataset Despite of that, it seems that for the One-Hot Model with 16x16 emitters, the plate has not reached his ’limit’ and still adding more patterns does not lead to a reduction in the overall general log loss, as it can be seen in Fig. 21: Figure 21: One-Hot Model Log Loss for different number of patterns dataset For more results, see the evolution of figure with number 1 shape, when training One Hot 4x4 with different amount of patterns in the dataset in the supplementary information at Fig. 29. Surprisingly for DAN is backwards, the model with 4x4 performs betters (at least up to the number of pattern it can take) than the model with 16x16 emitters: 24