Total variation denoising

Image:TVD 1D Example.png|thumb|right|300px|Application of 1D total-variation denoising to a signal obtained from a single-molecule experiment. == Regularization properties ==

The regularization parameter \lambda plays a critical role in the denoising process. When \lambda=0, there is no smoothing and the result is the same as minimizing the sum of squared errors. As \lambda \to \infty, however, the total variation term plays an increasingly strong role, which forces the result to have smaller total variation, at the expense of being less like the input (noisy) signal. Thus, the choice of regularization parameter is critical to achieving just the right amount of noise removal. == 2D signal images ==

We now consider 2D signals y, such as images. The total-variation norm proposed by the 1992 article is : V(y) = \sum_{i,j} \sqrt{|y_{i+1,j} - y_{i,j}|^2 + |y_{i,j+1} - y_{i,j}|^2} and is isotropic and not differentiable. A variation that is sometimes used, since it may sometimes be easier to minimize, is an anisotropic version : V_\operatorname{aniso}(y) = \sum_{i,j} \sqrt{|y_{i+1,j} - y_{i,j}|^2} + \sqrt{|y_{i,j+1} - y_{i,j}|^2} = \sum_{i,j} |y_{i+1,j} - y_{i,j}| + |y_{i,j+1} - y_{i,j}|. The standard total-variation denoising problem is still of the form : \min_y [ \operatorname E(x, y) + \lambda V(y)], where E is the 2D L2 norm. In contrast to the 1D case, solving this denoising is non-trivial. A recent algorithm that solves this is known as the primal dual method. Due in part to much research in compressed sensing in the mid-2000s, there are many algorithms, such as the split-Bregman method, that solve variants of this problem. == Rudin–Osher–Fatemi PDE ==

Suppose that we are given a noisy image f and wish to compute a denoised image u over a 2D space. ROF showed that the minimization problem we are looking to solve is: : \min_{u\in\operatorname{BV}(\Omega)} \; \|u\|_{\operatorname{TV}(\Omega)} + {\lambda \over 2} \int_\Omega(f-u)^2 \, dx where \operatorname{BV}(\Omega) is the set of functions with bounded variation over the domain \Omega, \operatorname{TV}(\Omega) is the total variation over the domain, and \lambda is a penalty term. When u is smooth, the total variation is equivalent to the integral of the gradient magnitude: : \|u\|_{\operatorname{TV}(\Omega)} = \int_\Omega\|\nabla u\| \, dx where \|\cdot\| is the Euclidean norm. Then the objective function of the minimization problem becomes:\min_{u\in\operatorname{BV}(\Omega)} \; \int_\Omega\left[\|\nabla u\| + {\lambda \over 2}(f-u)^2 \right] \, dxFrom this functional, the Euler-Lagrange equation for minimization – assuming no time-dependence – gives us the nonlinear elliptic partial differential equation: {{Equation box 1|cellpadding|border|indent=:|equation= \begin{cases} \nabla\cdot\left({\nabla u\over{\|\nabla u\|}} \right ) + \lambda(f-u) = 0, \quad &u\in\Omega \\ {\partial u\over{\partial n}} = 0, \quad &u\in\partial\Omega \end{cases} |border colour=#0073CF|background colour=#F5FFFA}}For some numerical algorithms, it is preferable to instead solve the time-dependent version of the ROF equation:{\partial u\over{\partial t}} = \nabla\cdot\left({\nabla u\over{\|\nabla u\|}} \right ) + \lambda(f-u) == Applications ==

The Rudin–Osher–Fatemi model was a pivotal component in producing the first image of a black hole. == See also ==