# Talks

## Graph-Based Signal Processing

## Graph-Based Signal Processing

Compression of frontal facial images is an appealing and important application. Recent work has shown that specially tailored algorithms for this task can lead to performance far exceeding JPEG2000. This paper proposes a novel such compression algorithm, exploiting our recently developed redundant tree-based wavelet transform. Originally meant for functions defined on graphs and cloud of points, this new transform has been shown to be highly effective as an image adaptive redundant and multi-scale decomposition. The key concept behind this method is reordering of the image pixels so as to form a highly smooth 1D signal that can be sparsified by a regular wavelet. In this work we bring this image adaptive transform to the realm of compression of aligned frontal facial images. Given a training set of such images, the transform is designed to best sparsify the whole set using a common feature-ordering. Our compression scheme consists of sparse coding using the transform, followed by entropy coding of the obtained coefficients. The inverse transform and a post-processing stage are used to decode the compressed image. We demonstrate the performance of the proposed scheme and compare it to other competing algorithms.

What if we take all the overlapping patches from a given image and organize them to create the shortest path by using their mutual Euclidean distances? This suggests a reordering of the image pixels in a way that creates a maximal 1D regularity. What could we do with such a construction? In this talk we consider a wider perspective of the above, and introduce a wavelet transform for graph-structured data. The proposed transform is based on a 1D wavelet decomposition coupled with a pre-reordering of the input so as to best sparsify the given data. We adopt this transform to image processing tasks by considering the image as a graph, where every patch is a node, and edges are obtained by Euclidean distances between corresponding patches. We show several ways to use the above ideas in practice, leading to state-of-the-art image denoising, deblurring, inpainting, and face-image compression results.

Images are 2D signals, and should be processed as such — this is the common belief in the image processing community. Is it truly the case? Around thirty years ago, some researchers suggested to convert images into 1D signals, so as to harness well-developed 1D tools such as adaptive-filtering and Kalman- estimation techniques. These attempts resulted with poorly performing algorithms, strengthening the above belief. Why should we force unnatural causality between spatially ordered pixels? Indeed, why? In this talk I will present a conversion of images into 1D signals that leads to state-of-the-art results in series of applications – denoising, inpainting, compression, and more. The core idea in our work is that there exists a permutation of the image pixels that carries in it most of the “spatial content”, and this ordering is within reach, even if the image is corrupted. We expose this permutation and use it in order to process the image as if it is a one-dimensional signal, treating successfully a series of image processing problems.

Images, video, audio, text documents, financial data, medical information, traffic info – all these and many others are data sources that can be effectively processed. Why? Is it obvious? In this talk we will start by discussing “modeling” of data as a way to enable their actual processing, putting emphasis on sparsity-based models. We will turn our attention to graph-structured data and propose a tailored sparsifying transform for its dimensionality reduction and subsequent processing. We shall conclude by showing how this new transform becomes relevant and powerful in revisiting … classical image processing tasks..

What if we take all the overlapping patches from a given image and organize them to create the shortest path by using their mutual distances? This suggests a reordering of the image pixels in a way that creates a maximal 1D regularity. What could we do with such a construction? In this talk we consider a wider perspective of the above, and introduce a wavelet transform for graph-structured data. The proposed transform is based on a 1D wavelet decomposition coupled with a pre-reordering of the input so as to best sparsify the given data. We adopt this transform to image processing tasks by considering the image as a graph, where every patch is a node, and edges are obtained by Euclidean distances between corresponding patches. We show several ways to use the above ideas in practice, leading to state-of-the-art image denoising, deblurring, and inpainting results.

Images, video, audio, text documents, financial data, medical information, traffic info — all these and many others are data sources that can be effectively processed. Why? Is it obvious? In this talk we will start by discussing “modeling” of data as a way to enable their actual processing, putting emphasis on sparsity-based models. We will turn our attention to graph-structured data and propose a tailored sparsifying transform for its dimensionality reduction and subsequent processing. We shall conclude by showing how this new transform becomes relevant and powerful in revisiting … classical image processing tasks.

What if we take all the overlapping patches from a given image and organize them to create the shortest path by using their mutual distances? This suggests a reordering of the image pixels in a way that creates a maximal 1D regularity. Could we repeat this process in several scales? What could we do with such a construction? In this talk we consider a wider perspective of the above line of questions: We introduce a wavelet transform that is meant for data organized as a connected-graph or as a cloud of high-dimensional points. The proposed transform constructs a tree that applies a 1D wavelet decomposition filters, coupled with a pre-reordering of the input, so as to best sparsify the given data. We adopt this transform to image processing tasks by considering the image as a graph, where every patch is a node, and vertices are obtained by Euclidean distances between corresponding patches. We show three ways to use the above ideas in practice – adopt only the patch-reordering, use the obtained wavelet transform as a sparsifying process, and a third approach were this transform is used as a regularizer. State-of-the-art image denoising, deblurring, and inpainting results are obtained with the proposed schemes.

What if we take all the overlapping patches from a given image and organize them to create the shortest path by using their mutual distances? This suggests a reordering of the image pixels in a way that creates a maximal 1D regularity Could we repeat this process in several “scales” ? What could we do with such a construction? In this talk we consider a wider perspective of the above line of questions: We introduce a wavelet transform that is meant for data organized as a connected-graph or as a cloud of high dimensional points. The proposed transform constructs a tree that applies a 1D wavelet decomposition filters, coupled with a pre-reordering of the input, so as to best sparsify the given data. We adopt this transform to image processing tasks by considering the image as a graph, where every patch is a node, and vertices are obtained by Euclidean distances between corresponding patches. State of- the-art image denoising results are obtained with the proposed scheme.