MAP Estimation with Gaussian Mixture Markov Random Field Model for Inverse Problems

Transcription

1 MAP Estimation with Gaussian Miture Markov Random Field Model for Inverse Problems Ruoqiao Zhang 1, Charles A. Bouman 1, Jean-Baptiste Thibault 2, and Ken D. Sauer 3 1. Department of Electrical and Computer Engineering,Purdue University 2. GE Healthcare Technologies 3. Department of Electrical Engineering, University of Notre Dame

2 Better Prior Models Why do we need better prior models? Better prior models will be needed as data becomes sparser Models must be adaptive to different classes of images Low, mid, and high level representations are needed What is needed? More epressive models of images Trained on real data (scientific/medical data) Computationally efficient to implement Promising recent approaches: Dictionary learning; ksvd; Non-local means; BM3D; Bilateral filters Many of these are not really consistent prior models Do not quantify multivariate distribution of image

3 Mission statement: Formulate a single, consistent, robust, and epressive prior model for any image,, that can be used in computationally efficient Bayesian estimation algorithms. p θ = 1 z ep{ u } θ - parameterizes model So we need to construct u()

4 Modeling Patches with Gaussian Miture P s MM patch at location s Gaussian miture model (GMM) for image patches K 1 1 = π k ( 2π ) p/2 k=0 g P s B k ep 2 P µ 2 s k B k Advantage: We can approimate any distribution with GMM

5 Advantage of GMM Patch Model 1 prior manifold Single GMM component Advantage of multivariate Gaussian miture Can model any distribution with enough GM components Capture multivariate distribution of a patch Model interaction between density and teture 2

6 GMM with 22 Image Patch Dual energy CT eample 12 clusters. Display 2 dimensions out of 8 Water/iodine decomposition color-coded scatter plot m2,i m1,i

7 Question? How to build a consistent image model out of GMM patches?

8 Model 1: Non-Overlapping Tiling with GMM Patches Tile image with non-overlapping patches Image distribution p 0 () = s S 0 g( P s ) Energy function u() = s S 0 V ( P s ) V ( P s ) = logg( P s )

9 Model 1: Non-Overlapping Tiling with GMM Patches Tile image with non-overlapping patches Energy function 1 = log π k V P s u() = K 1 k=0 s S 0 ( 2π ) p/2 V ( P s ) B k sums over non-overlapping patches ep 2 P µ 2 s k B k

10 Model 2: Non-Overlapping Tiling with GM Patches M 2 different tilings with non-overlapping patches p 0 () = g P s p 0 () = g P s p M 2 () = g P s s S 0 s S 1 s S M 2 Form a single distribution using the Product of Eperts p() = 1 z M 2 p i i=0 1/M 2 = 1 z s S g P s 1/M 2

11 Model 2: Non-Overlapping Tiling with GM Patches M 2 different tilings with non-overlapping patches p 0 () = g P s p 0 () = g P s p M 2 () = g P s s S 0 s S 1 s S M 2 Product of Eperts energy function u() = 1 M 2 V ( P s ) s S

12 Final GM-MRF Model Prior model Energy function Log GMM p = 1 z u() = 1 M 2 { } ep u() V ( P s ) s S corrects for patch overlap sums over all patches K 1 1 = log π k ( 2π ) p/2 k=0 V P s B k ep 2 P µ 2 s k B k

13 Final GM-MRF Model GM-MRF prior model V P s = 1 z ep 1 V ( P M 2 s ) p K 1 1 = log π k ( 2π ) p/2 k=0 s S B k ep 2 P µ 2 s k B k OK, but Is this really an MRF? Yes, with an (2M-1)(2M-1) neighborhood. How do I train the model? Just use your favorite GMM app to fit to patch data. How do I use this? Hmm, good point. We ll give you a surrogate function.

14 MAP Estimation with GM-MRF Model MAP estimate MAP estimate with surrogate prior where ˆ = argmin ˆ = argmin u + u { log p y } + u( ; ) { log p y } = u ( ; ) u( ; ) u is the current state of Perform surrogate optimization iteratively, updating with each iteration How do we find u ;?

15 Lemma: Surrogate Functions for Logs of Eponential Mitures Each v k is quadratic, so the resulting surrogate function, q ;!, is also quadratic

16 Surrogate Prior for GM-MRF Model Original energy function u = 1 η Surrogate energy function where the weights are given by The weights, V P s u( ;! ) = 1 w k = s S K 1 1 = log π k ( 2π ) p/2 k=0!w k K 1 s S k=0 V P s w k P s µ k 2η Bk + c! { } { } µ j 1/2 π k B k ep 1 2 P s " µ k K 1 j=0 B k 1/2 1 π j B j ep 2 P s " ep 2 P 2 s µ k B k, are soft classifications into the GMM classes 2 Bk 2 2 B j : current state of

17 Eperiments Denoising eperiments with the GM-MRF model 1. high dosage CT images with artificially added white noise; 2. low dosage CT images, containing real reconstruction noise. Compared with the following methods q-ggmrf model (8-point neighborhood, p=2, q=1.2, c=10) K-SVD method (77 patch, 512 dictionary entries) BM3D method (88 patch) The GM-MRF model was trained from clean high dosage CT images, with 30 subclasses and patch size 55. Parameters adjusted for lowest RMSE values (Eperiment 1) and comparable noise level in homogeneous region (Eperiment 2) for all methods.

18 MAP classification with Learned GM-MRF Color-codes the most probable subclass for each patch with the learned GMM parameters Shows that the GMM parameters capture different materials along with different edges materials edges

19 Eperiment 1: High Dosage CT Images GM-MRF model achieves - lowest RMSE - less salt/pepper noise and sharper edges than q-ggmrf model - less aggressive and preserves more details in soft tissues than K-SVD and BM3D original GM-MRF K-SVD Methods RMSE (HU) noisy GM-MRF Q- GGMRF noisy q-ggmrf BM3D K-SVD BM3D 17.25

20 Eperiment 2: Low Dosage CT Images GM-MRF achieves - sharper edges than q- GGMRF model - less artifacts and better teture in soft tissues than K-SVD and BM3D GM-MRF q-ggmrf original K-SVD BM3D

21 Eperiment 2: Low Dosage Difference Images GM-MRF model shows the ability to regularize different materials/structures differently: - more regularization in soft tissue - less regularization in bone/lung tissue GM-MRF original K-SVD original q-ggmrf original BM3D original

22 Conclusions GM-MRF (Gaussian Miture MRF) Is and MRF Can be trained for any image Captures full multivariate distribution of image How is the GM-MRF used? Is constructed with POE trick (geometric mean of distributions) Surrogate function for an miture distribution Medical applications It can capture both mean and teture characteristics for medical applications MAP optimization looks like it uses an adaptive quadratic prior