信号与数据稀疏性建模重点.pptVIP

* * The evolution of the data Model Signal Models Smooth Piecewise smooth Smooth withpoint singularities ? Signal model: a mathematical description of the behavior we expect from a “good” (uncontaminated ) signal in our system Machine Learning * Mathematics Signal Processing New Emerging Models Sparseland and Example-Based Models Wavelet Theory Signal Transforms Multi-Scale Analysis Approximation Theory Linear Algebra Optimization Theory Denoising Compression Inpainting Blind Source Separation Demosaicing Super-Resolution * The Sparseland Model Task: model image patches of size 10×10 pixels. We assume that a dictionary of such image patches is given, containing 256 atom images. The Sparseland model assumption: every image patch can be described as a linear combination of few atoms. α1 α2 α3 Σ * The Sparseland Model/Transform We start with a 10-by-10 pixels patch and represent it using 256 numbers – This is a redundant representation. However, out of those 256 elements in the representation, only 3 are non-zeros – This is a sparse representation. Bottom line in this case: 100 numbers representing the patch are replaced by 6 (3 for the indices of the non-zeros, and 3 for their entries). Properties of this model: Sparsity and Redundancy. α1 α2 α3 Σ * Problems With Sparseland Sparse and Redundant Representations Theory Numerical Problems Applications (image processing) * Difficulties With Sparseland Problem 1: Given an image patch, how can we find its atom decomposition ? A simple example: There are 2000 atoms in the dictionary The signal is known to be built of 15 atoms possibilities If each of these takes 1nano-sec