Isotropic and Anisotropic Total Variation From a Matrix Point of View

Total variation (TV) proves to be an effective measure in the areas of image processing. We analyze the TV in our technical report [1] from a matrix point of view. It is shown that the anisotropic TV can be equivalently computed by evaluating the l1 norm of a matrix-vector product,

$\text{TV}_{\ell_1}(\mathbf{X}) =\|\mathbf{W}\mathbf{x}\|_{\ell_1}$

where $\mathbf{x}$ is generated by putting all the elements of $\mathbf{X}$ column-wisely as a vector (implemented in MATLAB as x=X(:) ). The equation formulated above resembles the computation of the $\ell_1$ norm of wavelet coefficients of $\mathbf{x}$ , under a wavelet transformation matrix $\mathbf{W}$ . For an image $\textbf{X} \in R^{M \times N}$ , however, we should note that matrix $\textbf{W}$ is of size $2MN \times MN$ , which is not a square matrix. This is the fundamental difference between the total variation of an image $\mathbf{X}$ and the $\ell_1$ norm of the wavelet coefficients of $\mathbf{x}$ where the wavelet transformation matrix involved is square and most likely orthogonal if it is an orthogonal wavelet transformation.

Two MATLAB files in computing the isotropic TV and the anisotropic TV have also been prepared.

[1] J. Yan, Isotropic and Anisotropic Total Variation From a Matrix Point of View, Dept. of Electrical and Computer Engineering, University of Victoria, Victoria, BC, Canada, Apr. 2011. pdf code1 code2

Explore posts in the same categories: Uncategorized

This entry was posted on April 12, 2011 at 4:16 pm and is filed under Uncategorized. You can subscribe via RSS 2.0 feed to this post's comments. You can comment below, or link to this permanent URL from your own site.

Lego's Blog