EE565 Advanced Image Processing Copyright Xin Li Why do we Need Image Model in the first place? Any image processing algorithm has to work on a collection (class) of images instead of a single one Mathematical model gives us the abstraction of common properties of the images within the same class Model is our hypothesis and images are our observation data In physics, can F=ma explain the relationship between force and acceleration?  In image processing, can this model fit this class of images?

Content EE565 Advanced Image Processing Copyright Xin Li regression models transform models patch models PDE models synthesisdenoisingcompression… theory LS method PCA/KLT NN/kNN Diff. Geo. 2D Kalman filtering, Gaussian MRF, Edge-directed prediction Pyramid-based synthesis, wavelet thresholding, EZW/SPIHT Image quilting, Nonlocal-mean/BM3D, ??? Reaction-diffusion, MCD/PMD, ???

EE565 Advanced Image Processing Copyright Xin Li Theory Thread: Statistical vs. Deterministic They are different languages invented by mathematicians to facilitate the communication of scientific results (just like English vs. Chinese spoken by people in different countries) None is better than other – pick up the one you feel most comfortable with We adopt a statistical language most of the time in this class

EE565 Advanced Image Processing Copyright Xin Li The Curse of Dimensionality Even for a small-size image such as 64- by-64, we need to model it by a random process in 4096-dimensional space (R 4096 ) whose covariance matrix is sized by 4096-by-4096 More importantly, we ask ourselves: do we need to consider all pixels simultaneously?

EE565 Advanced Image Processing Copyright Xin Li A Simple Idea: Locality The conditional pdf is determined by a local neighborhood N past samples In PDE-based framework, locality refers to the modeling based on local gradients (approximation of low-order derivatives)

EE565 Advanced Image Processing Copyright Xin Li Parametric vs. Nonparametric non-parametric sampling Input image XkXk Parametric model

EE565 Advanced Image Processing Copyright Xin Li Spatial vs. Wavelet

EE565 Advanced Image Processing Copyright Xin Li Complete vs. Overcomplete

EE565 Advanced Image Processing Copyright Xin Li Marginal PDF of wavelet coefficients where Laplacian Gaussian P: shape parameter : variance parameter

EE565 Advanced Image Processing Copyright Xin Li Joint PDF of Wavelet Coefficients Neighborhood I(Q): {Left,Up,cousin and aunt} X= Y= Joint pdf of two correlated random variables X and Y Can you use this model to interpret why EZW works?

EE565 Advanced Image Processing Copyright Xin Li Good Bad Spatially Fixed vs. Adaptive Models

EE565 Advanced Image Processing Copyright Xin Li Locality Revisited: “relativity theory” for image processing Input image N past samples The definition of local neighborhood has to be relative

EE565 Advanced Image Processing Copyright Xin Li Application I: Image Denoising Spatial domain denoising techniques Conventional Wiener filtering Spatially adaptive Wiener filtering Wavelet domain denoising Wavelet thresholding: hard vs. soft Wavelet-domain adaptive Wiener filtering From local to nonlocal denoising

EE565 Advanced Image Processing Copyright Xin Li Linear Frequency Weighting FT Power spectrum |X| 2

EE565 Advanced Image Processing Copyright Xin Li Spatially Adaptive Wiener Filtering of Wavelet Coefficients Basic assumption: image source is modeled by a nonstationary Gaussian process Signal variance is locally estimated from the windowed noisy observation data T T N=T 2 Recall

EE565 Advanced Image Processing Copyright Xin Li Wavelet Thresholding DWT IWTThresholding YX ~ Hard thresholding Soft thresholding Noisy signal denoised signal

EE565 Advanced Image Processing Copyright Xin Li Spatially Adaptive Wiener Filtering in Wavelet Domain Wavelet high-band coefficients are modeled by a Gaussian random variable with zero mean and spatially varying variance Apply Wiener filtering to wavelet coefficients, i.e., estimated in the same way as spatial-domain (Slide 15)

EE565 Advanced Image Processing Copyright Xin Li Translation Invariant Denoising Noisy image T ce T ce -1 ThresholdingWD = shift(m K,n K ) WD shift(-m K,-n K ) shift(m 1,n 1 ) WD shift(-m 1,-n 1 ) Avg denoised image  (m k,n k ): a pair of integers, k=1-K (K: redundancy ratio)

EE565 Advanced Image Processing Copyright Xin Li Further Improvements Gaussian scalar mixture (GSM) based denoising (Portilla et al.’ 2003) Instead of estimating the variance, it explicitly addresses the issue of uncertainty with variance estimation Hidden Markov Model (HMM) based denoising (Romberg et al.’ 2001) Build a HMM for wavelet high-band coefficients (refer to the posted paper)

EE565 Advanced Image Processing Copyright Xin Li Nonlocal Patch-based Denoising WD T T -1 ThresholdingWD = Noisy patches Denoised patches

EE565 Advanced Image Processing Copyright Xin Li Application II: Texture Syntehsis Spatial-domain models Parametric autoregressive model Nonparametric resampling based Wavelet-domain models Histogram matching based Parametric models based joint-statistics

EE565 Advanced Image Processing Copyright Xin Li Spatial-Domain Parametric Texture Synthesis

EE565 Advanced Image Processing Copyright Xin Li Nonparametric Texture Synthesis

EE565 Advanced Image Processing Copyright Xin Li Wavelet-domain Histogram Matching

EE565 Advanced Image Processing Copyright Xin Li Wavelet-Domain Parametric Texture Models original synthesized

EE565 Advanced Image Processing Copyright Xin Li Other Applications Interpolation Spatial-domain covariance-based models PDE-based (nonlinear diffusion) models Compression Statistical modeling of wavelet coefficients Dual to wavelet-based image denoising Object recognition From NN/kNN to sparsity-based method

EE565 Advanced Image Processing Copyright Xin Li Summary on Theory Image models are at the foundation of any image processing algorithm Statistical models help us deal with the uncertainty in observation data Appropriate image representation (e.g., prediction/transform) facilitates the modeling task Spatial adaptation is important – to have a good model for a wide class of images Localized models are popular and powerful but nonlocal models might prevail later

EE565 Advanced Image Processing Copyright Xin Li Summary on Practice MATLAB provides a user friendly platform for testing your ideas You can see what you have done Experimental efficiency is important Avoid loops and test small-size images C/C++ programming skills are a plus Efficient implementation could make a difference