Image Degradation Model (Linear/Additive),,,,,,,, g x y h x y f x y x y G uv H uv F uv N uv 1
Source of noise Image acquisition (digitization) Image transmission Spatial properties of noise Statistical behavior of the gray level values of pixels Noise parameters, correlation with the image Frequency properties of noise Fourier spectrum Ex. white noise (a constant Fourier spectrum) 2
Noise Model p z 1 2 2 exp z 2 2 2 2 2 z a p z zaexp uza b b b b1 az az 1! p z az ae u z p z e u z b 1 p z u z a u z b b a p z P za P zb a b 3
Test Pattern Histogram has three Spikes! 4
Noisy Images Gaussian Rayleigh Gamma 5
Noisy Images Exponential Uniform Salt & Pepper 6
Periodic Noise: Electronic Devices 7
Periodic noise Observe the frequency spectrum Random noise with PDFs Case 1: imaging system is available Capture images of flat environment Case 2: noisy images available Take a strip from constant area Draw the histogram and observe it Measure the mean and variance 8
Medical Example: MRI Artifact: Phantom: Phantom Gibbs Noise 9
Medical Example: CT Metal Artifact: 10
Noise Estimation: Shape: Histogram of a subimage (Background) 11
Noise only spatial filter: g(x,y)=f(x,y)+η(x,y) Adaptive, local noise reduction: If is small, return g(x,y) If L>>, return value close to g(x,y) If L, return the arithmetic mean m L 2 2 L f ˆ x, y g x, y g x, y m L 12
Example: Original Noisy A- Mean G- Mean Local 13
Linear Degradation: gx, y H f x, y x, y L-System: H f x, y f, H x, y dd hx, y,, H x, y H f x, y f, hx, ydd LSI-Syste m: gx, y f x, yhx, y x, y Guv, Fuv, Huv, Nuv, 14
Degradation Estimation: Image Observation: Look at the image and Experiments: Acquire image using well defined object (Flat, pinhole, and etc.) Modeling: Introduce certain model for certain degradation using physical knowledge. 15
Degradation (Using Observation/PSF) Original Object Degraded Object H s uv, G uv, Hs u, v Hs u, v Fˆ u, v A s PSF 16
Medical Image Analysis and Processing Atmospheric Turbulence: 17
Modeling of turbulence in atmospheric images: 2 2, exp H u v k u v 56 18
Motion Blurring Modeling:, 0, 0 T g x y f x x t y y t dt 0 T j 2 ux vy G u, v f x x 0 t, y y 0 t dt e dxdy 0 T j 2 ux t vy t 0 0 G uv, F uv, e dt F uv, H uv, 0 19
Linear one/two dimensional motion blurring: T x 0 tat T, tmax T H u, v sinuae ua x t at T, y t bt T 0 0 T H u, v sin ua vb e ua vb jua j uavb 20
Motion Blurring Example: 21
MR Motion Artifact: 22
Motion Blurring Discrete Modeling: 23
Inverse Filtering: Without Noise: Hˆ u, v ˆ G u, v F u, v H u, v F u, v F u, v Hˆ u, v Problem of division by zero! 24
Inverse Filtering: With Noise: ˆ ˆ G u, v F u, v H u, v N u, v N u, v F u, v F u, v Hˆ uv, H uv, Hˆ uv, Problem of division by zero! Impossible to recover even if H(.,.) is known!! 25
Pseudo Inverse (Constrained) Filtering: Set infinite (large) value to zero; Multiply H(u,v) by a I/G/B LPF G u, v ˆ H u, v H ˆ, ˆ H u v F u, v G u, v ˆ H u, v H HTHR G u, v Hˆ u, v H ˆ ˆ F u, v H u, v T Hˆ u, v H THR THR THR THR HTHR 0 26
Full Band 60 Band 70 Band 85 Band 27
Phase Problem: Look at this formulation: 1 ˆ Hu, v ˆ, ˆ H u v Fu, v Gu, v 1 Hˆ u, v HTHR H H THR THR We preserve the Correct Phase! 28
Phase Problem: 29
Wiener Filtering : f(x,y) Degradation g(x,y) g(x,y) De Degradation f(x,y),,,,,, Wu v Gu v,, ˆ,,,., 2, g xy sxy xy Guv Suv Nuv Fˆ u, v,., E uv F uv F uv F uv W uv G uv E E u v E F WG F WG 30
Wiener Filtering in 2D case: W 2 E E u, v P WP W W P WP 2,, 2 FF GG FG GF E E u, v PFG u, v 0 Wu, v P u, v P u, v E X : Spectral Estimation XX P u, v E XY : Cross Spectral Estimation XY P u v P u v XY YX GG 31
Wiener Filtering in 2D case: Special Cases: Noise Only:,,,,,, g x y f x y x y G u v F u v N u v Uncorrelated Noise and Image: W u, v FF PFF u, v,, P u v P u v NN 32
Degradation plus Noise:,,,,,,,, g x y f x y h x y x y G uv F uv H uv N uv Uncorrelated Noise and Image: W P H FF u, v 2 PFF H PNN H 1 H 1 P H P 2 2 H 2 NN H 2 PFF FF 2 H H P P NN FF H P NN SNR 1 33
Degradation plus Noise: White Noise 1 H H H 2 2 K Select Interavtively 34
Wiener Filter is known as: Wiener-Hopf Minimum Mean Square Error Least Square Error Problems with Wiener: P FF P NN 35
Phase in Wiener Filter: W H W 2 2 PNN 1 H H PFF P 1 H H 2 H NN PFF W H No Phase compensation! 36
Wiener Filter vs. Inverse Filter: 1 H H H 0 W lim W 0 2 H 2 P P NN NN H H 0 H 0 P FF 37
Full Inverse Pseudo Inverse Wiener 38
Inverse Motion Blurring +Noise Wiener Noise Decrease 39
Iterative Wiener Filter: We formulate for noise only case: 0. i = 0 1. P 2. W i FF i1 i i i1 i1 i1 3. F W G 4. P GG FF FF NN i1 i1 = E F FF = P P P P 2 5. Repeat 2,3,4 until convergence. 40
Adaptive Wiener Filter: Image are Non Stationary! Need Adaptive WF which is locally optimal. Assume small region which image are stationary Image Model in each region: Noise Image:,,, f x y x y x y f : zero-mean white noise with unit variance!, : Constant over each region. f f f g x, y f x, y v x, y, v : Constant over each region 41
Local Wiener Filter in each region: W a u, v P f,, f xy f xyg xy x y 2 ff f 2 2 ff Pvv f v 2 2 f v fˆ x, y g x, y w x, y f a f ˆ f x, ygx, yf f 2 f 2 2 f v, : Low-pass filtered on noisy image.,,, Zero mean assumption g x, y, : Hi-pass filtered on noisy image. f P 2 wa x y x y ˆ xy, f x, y HPx, y LPx, y 2 f 2 2 f xy, v 42
Parameter Estimation: 2 g 2 v Local Noisy Image Variance Variance in a smooth image region or background xy, xy, 2 2 2 f g v 43
Results: 44
Results: 45
Results: 46
Results: 47
Matlab Command: deconvblind: Restore image using blind deconvolution deconvlucy: Restore image using accelerated Richardson Lucy algorithm deconvreg: Restore image using Regularized filter deconvwnr: Restore image using Wiener filter wiener2: Perform 2 D adaptive noise removal filtering edgetaper: Taper the discontinuities along the image edges 48