Feature and texture distillation via neural network training

. Encoded-diffraction hybrid systems — optical encoding and simple electronic decoding — offers feature distillation via model training. Additionally, the most faithfully reconstructed images are not the ones that are best classified. We parametrize our results with singular value decomposition (SVD) entropy, a proxy for image complexity.


Introduction
There is recently significant interest in hybrid opticalelectronic systems that leverage optical encoding and simple back-end electronic decoding of data.Optics provides high-speed, massively parallel, critical lowpower data processing while electronics provides the variable sensor sampling, nonlinear logistic and thresholding needed for decision-making and data classification.While hybrid image processing with simple neural networks have achieved significant milestones [1], there is currently data that suggests it is necessary to have complex computing architectures in order to analyse more complex images [2].Here, we explore the issues of image complexity, fractal texture analysis, and the potential for dimension reduction via neural network model training.
Image complexity often refers to the degree by which an image can be compressed.The upper bound for compression is the quantity famously dubbed Shannon entropy and is rooted in the statistics of information, defined as HShan = -i pi log pi where pi is the probability of a positive bit.This measure characterizes time-domain signal data and is also used to characterize image or multidimensional data.In fact, Shannon entropy reveals little about an image since a shuffle of the image pixels yields the same classical value for Shannon entropy.While other entropy measures have been proposed as a proxy for image complexity [3], creating a generalizable image complexity measure remains a difficult problem [4].Recently, with optical neural networks and Fourierdomain inputs, we have shown that the choice of generalized training images leads to learned preferences that distil different salient features of the reconstructed images based on spectral correlations [5].In this work, we have also shown that certain images are more easily learned than others, and faster learning times correspond to lower image complexity.However, we have also shown that complicated neural networks may be avoided, even when images are complex if we are only concerned with image classification.That is, the most faithfully reconstructed images are not always best classified.Our results indicate that simpler optical neural network classifiers are available to analyse complex datasets via dimensionality reduction.Our analysis focuses on the application of SVD-entropy as a proxy for image complexity HSVD = i i log i where i are the singular values.When normalized by the size of the matrix or range of matrix data, HSVD characterizes quantity of eigenimages needed to reconstruct an image in a manner that is consistent with definitions of image complexity and also invariant with fractal dimension [6].
We train image datasets with different fractal and random training datasets including speckle patterns.We show that the resulting model distils and achieves image segmentation not only for objects (which we have achieved previously [7]) but also for textures.Finally, when we reclassify these images, we achieve order-ofmagnitude improvements in the accuracies, many of which rival decade-old computer vision state-of-the-art.

Results
1 shows results training models for image reconstruction with synthetic fractal datasets.The resulting CIFAR-10 image data background textures are highlighted.Figure 2 shows training and subsequent classification with image classification using HSVD and a spatial frequency density, k.While the classification of raw images achieves only a few percent accuracy, the reconstructed, filtered images achieve 30-58% accuracy.The parameterization was introduced in [5].

Conclusion
We achieve feature distillation and segmentation with texture using learned models that are trained with synthetic fractal and random data.We show that simple hybrid optical neural networks may analyse complex images via reduced dimensions from training.

Figure 2 :
Figure 2: (A) A learned model is used to reconstruct test images (MNIST Digits, MNIST Fashion, CIFAR-10, and CIFAR-100).These reconstructed images are then used to train 2-layer classification NNs.Note that the (B) Average Structural Similarity Index (SSIM) of reconstructed test images.Each square corresponds to the SSIM of the particular dataset reconstructed using a model trained with a different dataset.Representative images of the datasets used are shown in a grid.(C) Classification accuracies of a 2-layer NN trained on images reconstructed and represented in (B).Models that were trained on the ground truth images themselves rather than the reconstructed images achieved 94\% -MNIST Digits, 86\% -MNIST -Fashion, 26\% -CIFAR-10, 8.2\% -CIFAR-100.