| In this thesis we study two different models for PDE-based image processing. Both model the removal of noise, also referred to as smoothing, from digital images while retaining essential features, such as edges, and both take the restored image, represented as a function defined on a rectangle Ω ⊂ Rn, to be the solution to a minimization problem over BV space.; The first model uses an adaptive total variation (ATV) functional defined on BV space. We first define the ATV functional for functions that are not necessarily in any Sobolev space. This space is the α- BV space, where a is a chosen function to locally control the amount of smoothing. Then we derive important approximation and compactness theorems concerning functions in α-BV. Having defined our functional and proven existence and uniqueness of a solution, we then study the associated time evolution problem. Here we define a weak solution u( x, t) to this problem and prove its existence, uniqueness, stability, and asymptotic behavior as t → ∞. We prove that u(x, t) weakly converges in L 2(Ω) to the solution u∞ of the original stationary problem. In addition, we demonstrate some numerical results of the time evolution ATV model as well as prove the existence of a solution for an updated ATV functional. Also discussed is an updated version, where the parameter function α depends on the solution u and not on initial noisy image.; The second model uses a functional which smoothes the image where its gradient norm is below a certain threshold ε, that is where |∇ u| < ε, using either the Laplacian or a regularized p-Laplacian for 1 < p < 2, and retains edges where its gradient norm is above the threshold (|∇u| ≥ ε). We in fact prove that the solution u is smooth where |∇ u| < ε. |
