Haar Wavelets, Gradients and Approximate TV Regularization

TL;DR

This work shows how this TV regularization can be approximately performed even in arbitrary dimensions by applying appropriate shrinkage to selected and properly weighted Haar wavelet coefficients, all of which depends even on the dimensionality of the data.

Abstract

We show how total variation regularization of images in arbitrary dimensions can be approximately performed by applying appropriate shrinkage to some Haar wavelets coefficients. The approach works directly on the wavelet coefficients and is therefore suited for the application on large volumes from computed tomography.

Figures (11)

• Figure 1: Overview rendering of the complete mummy dataset at a coarse resolution. Image: Fraunhofer IIS/Christoph Heinzl
• Figure 2: Rendering of full-resolution details of the mummy skull and teeth. All levels of detail are obtained from the locally decompressed wavelet dataset only. Image: Fraunhofer IIS/Christoph Heinzl
• Figure 3: The level image (top) and their vector fields of gradients for multiple levels $n$(bottom) for ${f(x,y) = |x|+|y|}$ where $x,y>0$
• Figure 4: The level image (top) and their vector fields of gradients for multiple levels $n$(bottom) for ${f(x,y) = x^2 + y^2}$ where $x,y>0$
• Figure 5: Simple synthetic image (top) and the resulting gradient field (based on a low-resolution version for increased visual clarity), with the normalization factor ${2^{n(1+s/2) + 2} = 2^{2n+2}}$(middle) and with a normalization $2^{n+2}$ that compensates the resolution effects (bottom). In the images, the three colors stand for the three highest resolutions

Theorems & Definitions (18)

• Lemma 1
• proof
• Theorem 2
• proof
• Corollary 3
• Remark 1: Wavelet coefficient decay
• Theorem 4
• Remark 2: Normalization of coefficients
• Lemma 5
• proof