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.

G. Gilboa, “A spectral approach to total variation”, Scale-Space and Variational Methods, SSVM 2013, LNCS 7893, p. 36-47, 2013.

Abstract:

The total variation (TV) functional is explored from a spectral perspective. We formulate a TV transform based on the second time derivative of the total variation flow, scaled by time. In the transformation domain disks yield impulse responses. This transformation can be viewed as a spectral domain, with somewhat similar intuition of classical Fourier analysis. A simple reconstruction formula from the TV spectral domain to the spatial domain is given. We can then design low-pass, high-pass and band-pass TV filters and obtain a TV spectrum of signals and images.

  Matlab Code

G. Gilboa, “Expert regularizers for task specific processing”, Scale-Space and Variational Methods, SSVM 2013, LNCS 7893, p. 24-35, 2013

Abstract:

This study is concerned with constructing expert regularizers for specific tasks. We discuss the general problem of what is desired from a regularizer, when one knows the type of images to be processed. The aim is to improve the processing quality and to reduce artifacts created by standard, general-purpose, regularizers, such as total-variation or nonlocal functionals. Fundamental requirements for the theoretic expert regularizer are formulated. A simplistic regularizer is then presented, which approximates in some sense the ideal requirements.

G. Gilboa, S. Osher, “Nonlocal linear image regularization and supervised segmentation”, SIAM Multiscale Mod. Simul. (MMS), Vol. 6, No. 2, pp. 595-630, 2007.

Abstract

A nonlocal quadratic functional of weighted differences is examined. The weights are based on image features and represent the affinity between different pixels in the image. By prescribing different formulas for the weights, one can generalize many local and nonlocal linear denoising algorithms, including the nonlocal means filter and the bilateral filter. In this framework one can easily show that continuous iterations of the generalized filter obey certain global characteristics and converge to a constant solution. The linear
operator associated with the Euler-Lagrange equation of the functional is closely related to the graph Laplacian. We can thus interpret the steepest descent for minimizing the functional as a nonlocal diffusion process. This formulation allows a convenient framework for nonlocal variational minimizations, including variational denoising, Bregman iterations and the recently proposed inverse-scale-space.

It is also demonstrated how the steepest descent flow can be used for segmentation. Following kernel based methods in machine learning, the generalized diffusion process is used to propagate sporadic initial user’s information to the entire image. Unlike classical variational segmentation methods the process is not explicitly based on a curve length energy and thus can cope well with highly non-convex shapes and corners. Reasonable robustness to noise is still achieved.

 

  Matlab Code for non-local diffusion

Tags: NLH1 , Nonlocal diffusion , nonlocal regularizers

G. Gilboa, “Nonlinear Scale Space with Spatially Varying Stopping Time”, PAMI, Vol. 30, No. 12, pp. 2175-2187, 2008.

Abstract:

A general scale space algorithm is presented for denoising signals and images with spatially varying dominant scales. The process is formulated as a partial differential equation with spatially varying time. The proposed adaptivity is semi-local and is in conjunction with the classical gradient-based diffusion coefficient, designed to preserve edges. The new algorithm aims at maximizing a local SNR measure of the denoised image. It is based on a generalization of a global stopping time criterion presented recently by the author and colleagues. Most notably, the method works well also for partially textured images and outperforms any selection of a global stopping time. Given an estimate of the noise variance, the procedure is automatic and can be applied well to most natural images.

Tags: nonlinear diffusion , nonlinear scale-space , stopping time