EE565 Advanced Image Processing Copyright Xin Li 20081 Different Frameworks for Image Processing Statistical/Stochastic Models: Wiener’s MMSE estimation.

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Presentation transcript:

EE565 Advanced Image Processing Copyright Xin Li Different Frameworks for Image Processing Statistical/Stochastic Models: Wiener’s MMSE estimation (e.g., denoising  filtering) Transform-Based Models: Fourier/Wavelets transform ( e.g., denoising  thresholding ) Variational PDE Models: Evolve image according to local derivative/geometric info, (e.g. denoising  diffusion) Concepts are related mathematically: Brownian motion – Fourier Analysis --- Diffusion Equation

EE565 Advanced Image Processing Copyright Xin Li PDE-based Image Processing  Image as a surface  Image Interpolation Implication of artifacts on surface area Minimal surface solution via mean curvature diffusion  Image inpainting Variational formulation Energy minimization solution  Image denoising From linear to nonlinear diffusion Perona-Malik diffusion

EE565 Advanced Image Processing Copyright Xin Li PDE-based Image Interpolation* Bilinear Interpolation PDE-based post-processing Low-resolution image Intermediate result High-resolution image

EE565 Advanced Image Processing Copyright Xin Li Image as a Surface 3D visualizationsingle-edge image If image can be viewed as a surface, it is then natural to ask: can we apply geometric tools to process this surface (or its equivalent image signals)?

EE565 Advanced Image Processing Copyright Xin Li Geometric Formulation  Image I: R 2 →R may be viewed as a two-dimensional surface in three-dimensional space, i.e., G: symmetric and positive definite matrix

EE565 Advanced Image Processing Copyright Xin Li Key Motivation  Why these concepts are useful for image processing? Image surface containing artifacts do not have minimal surface minimize S(M) leads to Euler-Lagrange Equation: (A)

EE565 Advanced Image Processing Copyright Xin Li Minimal Surface Unit normal of this surface is Mean curvature is Theorem Surfaces of zero mean curvature have minimal areas (B) Exercise: Derive (B) from (A) by direct calculation

EE565 Advanced Image Processing Copyright Xin Li Mean Curvature Diffusion Diffusion equation Discrete Implementation NOT straightforward! Reference: MegaWave 2.0 software We will discuss more numerical implementation next

EE565 Advanced Image Processing Copyright Xin Li Experiment Result Before post-processing After post-processing

EE565 Advanced Image Processing Copyright Xin Li Further Diffusion After 3 iterationsAfter 10 iterations

EE565 Advanced Image Processing Copyright Xin Li PDE-based Image Processing  Image as a surface  Image Interpolation Implication of artifacts on surface area Minimal surface solution via mean curvature diffusion  Image inpainting Variational formulation Energy minimization solution  Image denoising From linear to nonlinear diffusion Perona-Malik diffusion

EE565 Advanced Image Processing Copyright Xin Li Image Inpainting Extended inpainting domain Assumption: inpainting domain is local and does not contain texture (complimentary to texture-synthesis based inpainting techniques) Image example

EE565 Advanced Image Processing Copyright Xin Li Total Variation Key idea: it is L 1 instead of L 2 norm (minimizing L 2 will not preserve edges) Clean (TV small) noisy (TV large)

EE565 Advanced Image Processing Copyright Xin Li Variational Problem Formulation Restored image degraded image Rational: The first term describes the smoothness constraint within the extend inpainting domain The second term describes the observation constraint Total variation (TV)

EE565 Advanced Image Processing Copyright Xin Li How to obtain the corresponding PDE? Euler-Lagrangian Equation Where TV-inpainting:

EE565 Advanced Image Processing Copyright Xin Li Numerical Implementation of PDEs m n n+1 n-1 m+1 m-1 or

EE565 Advanced Image Processing Copyright Xin Li Inpainting Example (Courtesy: Jackie Shen, UMN MATH)

EE565 Advanced Image Processing Copyright Xin Li PDE-based Image Processing  Image as a surface  Image Interpolation Implication of artifacts on surface area Minimal surface solution via mean curvature diffusion  Image inpainting Variational formulation Energy minimization solution  Image denoising From linear to nonlinear diffusion Perona-Malik diffusion

EE565 Advanced Image Processing Copyright Xin Li Geometry-driven PDEs x y I(x,y) image I image I viewed as a 3D surface (x,y,I(x,y))

EE565 Advanced Image Processing Copyright Xin Li Simplest Case: Laplace Equation Linear Heat Flow Equation: scale A Gaussian filter with zero mean and variance of t Isotropic diffusion:

EE565 Advanced Image Processing Copyright Xin Li Example t=0 t=1 t=2

EE565 Advanced Image Processing Copyright Xin Li Example (Cont.) t=4t=8t=16

EE565 Advanced Image Processing Copyright Xin Li From Isotropic to Anisotropic  Gaussian filtering (isotropic diffusion) could remove noise but it would blur images as well  Ideally, we want Filtering (diffusion) within the object boundary No filtering across the edge orientation  How to achieve such “ anisotropic diffusion ” ?

EE565 Advanced Image Processing Copyright Xin Li Perona-Malik ’ s Idea Isotropic diffusion: edge stopping function

EE565 Advanced Image Processing Copyright Xin Li Pursuit of Appropriate g Define 1D case: Encourage diffusion: Discourage diffusion: Edge slope decreases Edge slope increases

EE565 Advanced Image Processing Copyright Xin Li Examples K Choice-I Choice-II

EE565 Advanced Image Processing Copyright Xin Li Discrete Implementation

EE565 Advanced Image Processing Copyright Xin Li Numerical Examples

EE565 Advanced Image Processing Copyright Xin Li Scale-space with Anisotropic Diffusion original P-M filter (K=16,100 iterations)

EE565 Advanced Image Processing Copyright Xin Li P-M Filter for Image Denoising Noisy image (PSNR=28.13) P-M filtered image (PSNR=29.83)

EE565 Advanced Image Processing Copyright Xin Li Variational Interpretation

EE565 Advanced Image Processing Copyright Xin Li Comparison between Wavelet - based and PDE-based denoising  Wavelet theory Strength: offers a basis to distinguish signals from noise (signal behaves as significant coefficients while noise will not) Weakness: ignore geometry  Diffusion theory Strength: geometry-driven Weakness: localized model (poor to realize global trend in the signal)