Guy Gilboa, Michael Moeller, Martin Burger, “Nonlinear Spectral Analysis via One-homogeneous Functionals – Overview and Future Prospects”, accepted to Journal of Mathematical Imaging and Vision (JMIV), 2016.

Abstract:

We present in this paper the motivation and theory of nonlinear spectral representations, based on convex regularizing functionals. Some comparisons and analogies are drawn to the fields of signal processing, harmonic analysis and sparse representations. The basic approach, main results and initial applications are shown. A discussion of open problems and future directions concludes this work.

Michael Moeller, Julia Diebold, Guy Gilboa, Daniel Cremers, “Learning Nonlinear Spectral Filters for Color Image Reconstruction”, ICCV 2015.

Abstract:

This paper presents the idea of learning optimal filters for color image reconstruction based on a novel concept of nonlinear spectral image decompositions recently proposed by Gilboa. The general idea is to use total variation regularization along with Bregman iterations to represent the input data as the sum over image layers containing features at different scales. Filtered images can be obtained by weighted linear combinations of the different frequency layers. We show that learning the optimal weights can significantly improve the results in comparison to the standard variational approach, and can achieve state-of-the-art results. While we focus on the problem of image denoising, our general framework extends to a number of image reconstruction tasks.

O. Katzir and G. Gilboa, “A Maximal Interest-Point Strategy Applied to Image Enhancement with External Priors”, IEEE International Conference on Image Processing (ICIP-2015) , pp. 1324-1328, 2015.

Abstract:

Curvature is a fundamental component in differential geometry. It is used extensively in signal, image and shape processing, as a feature and in segmentation flows and regularization processes. In this paper we extend the notion of curvature in two ways. First, we present the Menger curvature which goes beyond classical curves and Riemannian manifolds to general metric spaces and is rigorously defined on a variety of discrete settings. We further extend the curvature to become a non-local entity using an adaptive, non-local integration measure, allowing curvature to be computed in a robust manner. Examples on natural and textural images highlight potential applications of these new concepts.

Tags: external priors , image enhancement , Interest points , triangle patches

G. Gilboa, E. Appleboim, E. Saucan and Y.Y. Zeevi, “On the Role of Non-local Menger Curvature in Image Processing”, IEEE International Conference on Image Processing (ICIP-2015) , pp. 4337-4341, 2015.

Abstract:

Curvature is a fundamental component in differential geometry. It is used extensively in signal, image and shape processing, as a feature and in segmentation flows and regularization processes. In this paper we extend the notion of curvature in two ways. First, we present the Menger curvature which goes beyond classical curves and Riemannian manifolds to general metric spaces and is rigorously defined on a variety of discrete settings. We further extend the curvature to become a non-local entity using an adaptive, non-local integration measure, allowing curvature to be computed in a robust manner. Examples on natural and textural images highlight potential applications of these new concepts

Tags: Menger curvature , non-local curvature

M. Burger, L. Eckardt, G. Gilboa, M. Moeller, “A spectral framework for one-homogeneous functionals”, Proc. Scale Space and Variational Methods in Computer Vision (SSVM), 2015.

Abstract:

This paper discusses a generalization of spectral representations related to convex one-homogeneous regularization functionals, e.g. total variation or $\ell^1$-norms. Those functionals serve as a substitute for a Hilbert space structure (and the related norm) in classical linear spectral transforms, e.g. Fourier and wavelet analysis. We discuss three meaningful definitions of spectral representations by scale space and variational methods and prove that (nonlinear) eigenfunctions of the regularization functionals are indeed atoms in the spectral representation. Moreover, we verify further useful properties related to orthogonality of the decomposition and the Parseval identity.

The spectral transform is motivated by total variation and further developed to higher order variants. Moreover, we show that the approach can recover Fourier analysis as a special case using an appropriate $\ell^1$-type functional and discuss a coupled sparsity example.

Tags: inverse scale space , one homogeneous , spectral TV , tgv

D. Horesh, G. Gilboa, “Multiscale orientation detection using the total variation transform”, Proc. Scale Space and Variational Methods in Computer Vision (SSVM), 2015.

Abstract:

Multi-level texture separation can considerably improve texture analysis, a significant component in many computer vision tasks.
This paper aims at obtaining precise local texture orientations of images in a multiscale manner, characterizing the main obvious ones as well as the very subtle ones.
We use the total variation spectral framework to decompose the image into its different textural scales. Gabor filter banks are then employed to detect prominent orientations within the multis-cale representation. A necessary condition for perfect texture separation is given, based on the spectral total-variation theory.
We show that using this method we can detect and differentiate a mixture of overlapping textures and obtain with high fidelity a multi-valued orientation representation of the image.

J.-F. Aujol, G. Gilboa, N. Papadakis, “Non local total variation spectral framework”, Proc. Scale Space and Variational Methods in Computer Vision (SSVM), 2015.

Abstract:

Eigenvalue analysis based on linear operators has been extensively used in signal and image processing to solve a variety of problems such as segmentation, dimensionality reduction and more.
Recently, nonlinear spectral approaches, based on the total variation functional have been proposed. In this context, functions for which the nonlinear eigenvalue problem $ \lambda u \in \partial J(u)$ admits solutions,

are studied.
When $u$ is the characteristic function of a set $A$, then it is called a calibrable set. If $\lambda >0$ is a solution of the above problem, then $1/\lambda$ can be interpreted as the scale of $A$.

However, this notion of scale remains local, and it may not be adapted for non-local features. For this we introduce in this paper the definition of non-local scale related to the non-local
total variation functional. In particular, we investigate sets that evolve with constant speed under the non-local total variation flow.
We prove that non-local calibrable sets have this property. We propose an onion peel construction to build such sets.
We eventually confirm our mathematical analysis with some simple numerical experiments.

Tags: calibrable sets , Non-local , nonlinear eigenvalue problem , scale , total variation

G. Gilboa, “A Total Variation Spectral Framework for Scale and Texture Analysis”, SIAM Journal on Imaging Sciences, Vol. 7, No. 4, pp. 1937–1961, 2014.

Abstract:

A new total variation (TV) spectral framework is presented. A TV transform is proposed which can be interpreted as a spectral domain, where elementary TV features, like disks, approach impulses. A reconstruction formula from the pectral to the spatial domain is given, allowing the design of new filters. The framework formulates a new representation of images which can enhance the understanding of scales in the $L^1$ sense and improve the analysis and processing of textures. An example of a texture processing application illustrates possible benefits of this new framework.

  Matlab Code

G. Gilboa, “Nonlinear Band-Pass Filtering Using the TV Transform”, Proc. European Signal Processing Conference (EUSIPCO-2014), Lisbon, pp. 1696 – 1700, 2014.

Abstract:

A distinct family of nonlinear filters is presented. It is based on a new formalism, defining a nonlinear transform based on the TV-functional. Scales in this sense are related to the size of the object and its contrast. Edges are very well preserved and selected scales of the object can be either selected, removed or enhanced. We compare the behavior of the filter to other filters based on Fourier and wavelets transforms and present its unique qualities.

  Matlab Code

Tags: nonlinear BPF , spectral TV , TV transform

G. Gilboa, “A Total Variation Spectral Framework for Scale and Texture Analysis”, CCIT Report 833, 2013.

Abstract:

A new total variation (TV) spectral framework is presented. A TV transform is proposed which can be interpreted as a spectral domain, where elementary TV features, like disks, approach impulses. A reconstruction formula from the pectral to the spatial domain is given, allowing the design of new filters.The framework allows deeper understanding of scales in an $L^1$ sense and the ability to better analyze and process textures. An example of a texture processing application illustrates possible benefits of this new framework.