A DIRECTIONAL TOTAL VARIATION ˙ Ilker Bayram Electronics and Telecommunication Eng. Dept. Istanbul Technical University ibayram@itu.edu.tr Mustafa E. Kamasak Computer Eng. Dept. Istanbul Technical University kamasak@itu.edu.tr ABSTRACT Total variation (TV) is an isotropic image prior that penal- izes the abrupt changes in the images in all directions. In this paper, we modify TV so as to make it more suitable for im- ages with a dominant direction. Specifically, we describe the implementation of a directional TV, and we demonstrate its utility for image denoising. We show that image denoising with the directional TV prior can be more effective compared to the regular (isotropic) TV for images with a dominant di- rection. Index Terms— total variation, directional total variation, image denoising 1. INTRODUCTION Total variation (TV) penalizes abrupt changes in images. It is a very effective signal prior for piecewise smooth images as in Fig. 1a. However, TV is an isotropic functional and is not very suitable for images with a dominant direction, like the one in Fig. 1b. For such images, one could, in principle, scale the image in order to reduce the dominance of the direction. However, for discrete-space images, scaling requires interpo- lation and therefore it is not very feasible. In this paper, we describe a different approach to define a directional TV. We also study a related denoising problem for discrete-space im- ages and provide an algorithm for its solution. The total variation (TV) of a discrete-space image f is defined as, TV(f )=  i,j  ( Δ 1 f (i, j ) ) 2 + ( Δ 2 f (i, j ) ) 2 (1) where Δ 1 and Δ 2 denote horizontal and vertical difference operators, (possibly) defined as, Δ 1 f (i, j )= f (i, j ) − f (i − 1,j ), (2) Δ 2 f (i, j )= f (i, j ) − f (i, j − 1). (3) We can rewrite this as, TV(f )=  i,j ‖Δf (i, j )‖ 2 =  i,j sup t∈B2 〈Δf (i, j ),t〉 (4) (a) (b) Fig. 1. Total variation is a simple and effective prior for piece- wise smooth images as in (a). We describe a directional total variation for images with a dominant direction as in (b). where Δ is the linear operator defined as, Δf (i, j )=  Δ 1 f (i, j ) Δ 2 f (i, j )  (5) and B 2 is the unit ball of the ℓ 2 norm. Henceforth, we use Δ to denote the matrix that represents the linear mapping de- fined in (5). Total variation is isotropic because it is invariant under a rotation of the image (or, equivalently, the components of Δ f ). This is a consequence of the ℓ 2 norm (or B 2 ) appearing in (4). We can obtain a directional total variation by replacing B 2 with some other set. In particular, if we use an ellipse, E α,θ oriented along the angle θ, with a unit length minor axis and a major axis of length α> 1, (see Fig.2), the resulting norm TV α,θ (f )=  i,j sup t∈E α,θ 〈Δf (i, j ),t〉 (6) is more sensitive to variations along θ. Given this new total variation, we would like to have al- gorithms that use this pseudo-norm as a regularizer. In this paper, we study the denoising problem, f ∗ = argmin f 1 2 ‖y − f ‖ 2 2 + λ TV α,θ (f ), (7) 20th European Signal Processing Conference (EUSIPCO 2012) Bucharest, Romania, August 27 - 31, 2012 © EURASIP, 2012 - ISSN 2076-1465 265