Digital Image Processing Image Restoration Digital Image Processing

Content Introduction Image degradation/restoration model Noise models Restoration by spatial filtering Estimation of degradation functions Inverse filtering Wiener filtering Geometric transformation

Introduction Objective of image restoration The distortion is due to to recover a distorted image to the original form based on idealized models. The distortion is due to Image degradation in sensing environment e.g. random atmospheric turbulence Noisy degradation from sensor noise. Blurring degradation due to sensors e.g. camera motion or out-of-focus Geometric distortion e.g. earth photos taken by a camera in a satellite

Introduction Enhancement Restoration Concerning the extraction of image features Difficult to quantify performance Subjective; making an image “look better” Restoration Concerning the restoration of degradation Performance can be quantified Objective; recovering the original image

Image degradation / restoration model

Noise models Assuming that noise is independent of spatial coordinates, and uncorrelated with respect to the image content

Noise models

Adding Gaussian Noise at Matlab t_gaus = imnoise (t, ‘gaussian’); imshow(t_gaus);

Noise models

Adding Salt and Pepper Noise at Matlab t_sp = imnoise (t, ‘salt & pepper’); imshow(t_sp);

Noise models Other common noise models Rayleigh noise Gamma noise Exponential noise Uniform noise

Noise Models Rayleigh Noise Gamma(Erlang) Noise Exponential Noise

Noise models paper salt -3-levels -simple constant areas (spans from black to white) paper salt

Additive Noise Histograms

Additive Noise Histograms

Periodic Noise Noise components Are generated due to electrical or electromechanical interference during image acquisition Periodic noise can be reduced in via frequency domain

Adding Periodic Noise at Matlab tw = imread(filename); t = rgb2gray(tw); s = size(t); [x,y] = meshgrid(1:s(1), 1:s(2)); p = sin(x/3+y/5)+1; t_pn = (im2double(t)+p’/2)/2; imshow(t_pn);

Restoration by spatial filtering Noise is unknown Spatial filtering is appropriate when only additive noise is present

Restoration of Salt and Pepper Noise by LPF a3 = fspecial(‘average’); t_sp_a3 = filter2 (a3, t_sp); a7 = fspecial(‘average’, [7,7]); t_sp_a7 = filter2 (a7, t_sp);

Restoration of Salt and Pepper Noise by Median Filter T_sp_m3 = medfilt2(t_sp);

Restoration by spatial filtering

Restoration by spatial filtering

Restoration by spatial filtering

Restoration by spatial filtering Q is the order of filter

Restoration by spatial filtering Noise level is Mean =0 Variance = 400

Restoration by spatial filtering Mean filters (noise reduced by blurring) Arithmetic mean filter and geometric mean filter are well suited for random noise such as Gaussian noise Contraharmonic mean filter is well suited for impulse noise Disadvantage: must know pepper noise or salt noise in advance

Restoration by spatial filtering

Restoration by spatial filtering wrong

Restoration by spatial filtering -- Repeated passes of median filter tend to blur the image. -- Keep the number of passes as low as possible.

Restoration by spatial filtering Fig. 8 next page

Restoration by spatial filtering Pepper noise Salt noise

Median and alpha-trimmed filter performed better High level of noise  large filter Median and alpha-trimmed filter performed better Alpha-trimmed did better than median filter

Restoration by spatial filtering Filters discussed so far Do not consider image characteristics Adaptive filters to be discussed Behaviors based on statistical characteristics of the subimage under a filter window Better performance More complicated Adaptive, local noise reduction filter Adaptive median filter

Restoration by spatial filtering

Restoration by spatial filtering

Restoration by spatial filtering

Restoration by spatial filtering Adaptive filtering

Restoration by spatial filtering

Restoration by spatial filtering Is Z_med impulse? Is Z_xy impulse?

Restoration by spatial filtering

Periodic Noise Reduction (Frequency Domain Filtering) Band-Reject Filters Ideal Band-reject Filter -D(u,v) =distance from the origin of the centered freq. rectangle -W =width of the band -D0=Radial center of the band.

Periodic Noise Reduction (Frequency Domain Filtering) Butterworth Band-Reject Filter of order n Gaussian Band-Reject Filter

Periodic Noise Reduction (Frequency Domain Filtering)

Periodic Noise Reduction (Frequency Domain Filtering) Band-Pass Filters Opposite operation of a band-reject fiter

Periodic Noise Reduction (Frequency Domain Filtering) Notch Filters Rejects (or passes) frequencies in predefined neighborhoods about a center frequency Ideal Must appear in symmetric pairs about the origin. Butterworth Gaussian

Periodic Noise Reduction (Frequency Domain Filtering) Notch Filters Ideal Center frequency components Shift with respect to the center

Notch pass filter Horizontal lines of the noise pattern I can be seen

Optimum Notch Filtering Several pairs of components are present  more than just one sinusoidal component

Optimum Notch Filtering

Estimation of degradation functions

Estimation of degradation functions

Estimation of degradation functions

Estimation of degradation functions

Estimation of degradation functions

Estimation of degradation functions (model bases) Mathematical model for uniform linear motion between the image and the sensor during image acquisition Let x0(t) and y0(t) denote time varying components of motion in the x- and y-directions Degradation model

Estimation of degradation functions (model bases) Uniform Linear motion in the x and y direction x0(t)=at/T and y0(t)=bt/T where the image has been displaced by a total distance a in the x-direction and b in the y-direction

Estimation of degradation functions (model bases)

Inverse filtering

Inverse filtering

Inverse filtering Degradation function Curtain of noise Cutting off values of the ratio outside a radius of 40, 70,85.

Wiener filtering

Wiener filtering

Wiener filtering White noinse

Wiener filtering

Geometric transformations Objective: to eliminate geometric distortion that occurs when an image is captured Examples of geometric distortion Pincushion distortion (associated with zoom lenses)

Geometric transformations

Geometric transformations Two steps in geometric transformation Spatial transformation: rearrangement of pixels on the image plane Gray-level interpolation: assignment of gray levels to pixels in the spatially transformed image

Geometric transformations

Geometric transformations • Solution To formulate the spatial relocation of pixels by the use of the corresponding tiepoints Tiepoints: a subset of pixels whose locations in the input (distorted) and output (restored) images are known.

Geometric transformations

Geometric transformations

Geometric transformations

Geometric transformations

Geometric transformations