f(x,y) is the original image H is the degradation process (or function) n(x,y) represents noise g(x,y) is the obtained degraded image p q

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1 Image Restoration Image Restoration G&W Chapter The Degradation Model browse through the contents 5.11 Geometric Transformations Goal: Reconstruct an image that has been degraded in some way Main idea: Model the degradation using (a priori) information about the degradation process and apply inverse filtering Image restoration is an objective process Image enhancement is a subjective process Ingela Nyström ingela@cb.uu.se Degradation Model Theory n(x,y) f(x,y) H + g(x,y) p q f(x,y) is the original image H is the degradation process (or function) n(x,y) represents noise g(x,y) is the obtained degraded image g(x,y)=h(f(x,y))+n(x,y) 1

2 Degradations Degradation can be caused by defects of the optical lens (compare the Hubble telescope!) non linearity of the sensor graininess of the film material relative motion between object and camera wrong lens focus atmospheric turbulence especially in remote sensing and astronomy et cetera Restoration Model Two groups of restoration techniques g(x,y) Restoration process ^ f(x,y) The restoration processshouldminimizeshould the difference between the original image f(x,y) and ^ the restored image f(x,y) 2

3 Restoration of noisy images Reduction of Type 1 noise We start with only the noise n(x,y) and therefore assume that H is the identity g(x,y)=f(x,y)+ n(x,y) f(x,y) n(x,y) We will look at two main types of noise: 1) Spatially independent and uncorrelated to pixel values 2) Spatially periodic + g(x,y) Spatial mean filtering Median filtering Filtering adapted to the local variance Reduction of Type 2 noise Frequency filtering 3

4 Frequency Filters Notch filter example Periodic noise reduction by frequency domain filtering Lowpass smoothing filter High frequencies are attenuated; noise and edges are blurred Highpass sharpening filter Low frequencies are attenuated; edges are enhanced Bandreject filter used in image restoration Specific frequencies are rejected Notch filter used in image restoration Rejects specific frequencies in specific directions Two notch filters Notch filter result Butterworth notch reject filters of order 5 Quite amazing restoration 4

5 Degradation Function We now move on to the degradation function H and therefore assume that the noise n(x,y) = 0 g(x,y) = H(f(x,y)) or g(x,y) = h(x,y)*f(x,y) or G(u,v) = H(u,v)F(u,v) Since the degradation is modeled as a convolution, the restoration process is generally called deconvolution The Point Spread Function (PSF) All optical imaging systems blur a point of light. This is called the point spread function (PSF). Single impulse signal Two impulse signals Output blurred by the PSF A single output signal is seen due to blurring by the PSF We assume that H is linear and position invariant H[af 1 (x,y) + bf 2 (x,y)] = ah[f 1 (x,y)] + bh[f 2 (x,y)] If the impulse response (how a point is imaged) is known, the response to any input signal can be calculated How to estimate H? Relative motion of camera and object When the true H is not known, then the restoration ti is performed as a blind deconvolution by image observation experimentation mathematical modeling 5

6 Inverse Filtering Inverse Filtering II Wiener Filtering Wiener Filtering II 6

7 Geometric Restoration Matching between two images typically consists of two steps: 1. Pixel coordinate transformation 2. Grey level interpolation Geometric distortions can be due to, e.g.: Tilted surface rectification in satellite and aerial limages Inhomogeneous magnetic field in MR images Motion of object between successive images Pixel Coordinate Transformation original image transformation new image f(x,y) T g(x,y) = T[f(x,y)] The transformation T is usually bi linear: describing rubber sheet transformations affine: including geometrical transformations, e.g.: rotation translation scaling skewing Bi linear Transformation Rotation Mark the corresponding points in the different images, set up an equation system and solve it analytically x3, y3 x 2, y 2 x 3, y 3 x1, y1 x 4, y 4 x4, y4 x 1 x1, y 1 ' x = c * x + c * y + c * x y + c ' y = c * x + c * y + c * x y + c unknown, 8 equations => the coefficients can be found 7

8 Translation Scaling Skewing Grey level Interpolation 8

9 Inverse Geometric Transformation Grey level Interpolation x = T y = T 1 x 1 y ( x', y') ( x', y') For each pixel in the output image (x,y ), we compute the pixel coordinate (x,y) in the input image from which the pixel originates (x,y)is in general a non integer position A change of scale example: x = (1/a) x and y = (1/b) y The non integer coordinate position (x,y) in the original image is surrounded by 4 neighbours Common interpolation approaches are nearest neighbour neighbour bi linear bi cubic (x,y) Nearest neighbour Interpolation Nearest neighbour Interpolation The output pixel (x,y ) is assigned dthe grey level lof the input pixel closest to (x,y) (x,y) The coordinate values are rounded to nearest integers, so the position error is at most half a pixel. Straight line boundaries may appear step like after transformation 9

10 Bi linear Interpolation Bi linear Interpolation The grey level assigned to (x,y ) is a combination of the 4 neighbouring values of (x,y) Δy Δx V1 V3 Bi linear interpolation can cause blurring due to its averaging nature. The problem of step like straight boundaries with the nearest neighbour interpolation is reduced. First in x direction: Top=(1 Δx)V1+ ΔxV3 Bottom= (1 Δx)V2+ ΔxV4 Then in y direction: G=(1 Δy)Top+ ΔyBottom V2 (x,y) V4 Bi cubic Interpolation The 16 closest neighbours are used. Very nice results, but rather high computational complexity. (x,y) 10