We generalize the total variation restoration model, introduced by Rudin, Osher, and Fatemi in 1992, to matrix-valued data, in particular, to diffusion tensor images (DTIs). Our model is a natural extension of the color total variation model proposed by Blomgren and Chan in 1998. We treat the diffusion matrix D implicitly as the product D = LL(T), and work with the elements of L as variables, i...

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 ...

A constrained optimization type of numerical algorithm for removing noise from images is presented. The total variation of the image is minimized subject to constraints involving the statistics of the noise. The constraints are imposed using Lagrange multipliers. The solution is obtained using the gradient-projection method. This amounts to solving a time dependent partial differential equation...

Fig. 3: FA maps from the original (left), and the denoised (right) DTI data set. Magnified views of a ROI (bottom) demonstrate feature preservation in fine structures. Fig. 1: A numerical example of spatially variant regularization. (a) A numerical test image. (b) Noisy test image. (c) TV denoising with λ=20. (d) TV denoising with λ=10. (e) λ map: λ=10 (dark region) and λ=20 (bright region). (f...

Motivated by its practical success, we show that the twodimensional total variation denoiser satisfies a sharp oracle inequality that leads to near optimal rates of estimation for a large class of image models such as bi-isotonic, Hölder smooth and cartoons. Our analysis hinges on properties of the unnormalized Laplacian of the twodimensional grid such as eigenvector delocalization and spectral...

The objectives of this chapter are: (i) to introduce a concise overview of regularization; (ii) to define and to explain the role of a particular type of regularization called total variation norm (TV-norm) in computer vision tasks; (iii) to set up a brief discussion on the mathematical background of TV methods; and (iv) to establish a relationship between models and a few existing methods to s...

Let n be an integer and A0, . . . , Ak random subsets of {1, . . . , n} of fixed sizes a0, . . . , ak, respectively chosen independently and uniformly. We provide an explicit and easily computable total variation bound between the distance from the random variable W = | ∩j=0 A j|, the size of the intersection of the random sets, to a Poisson random variable Z with intensity λ = EW . In particul...

Quasi–stationary distributions, as discussed by Darroch & Seneta (1965), have been used in biology to describe the steady state behaviour of population models which, while eventually certain to become extinct, nevertheless maintain an apparent stochastic equilibrium for long periods. These distributions have some drawbacks: they need not exist, nor be unique, and their calculation can present p...

The recently introduced second order total generalised variation functional TGVβ,α has been a successful regulariser for image processing purposes. Its definition involves two positive parameters α and β whose values determine the amount and the quality of the regularisation. In this paper we report on the behaviour of TGVβ,α in the cases where the parameters α, β as well as their ratio β/α bec...