Image Restoration

1. Image Restoration
Image Restoration refers to a class of methods that aim to remove
or reduce the degradations that have occurred while the digital
image was being obtained. All natural images when displayed
have gone through some sort of degradation:
during display mode
during acquisition mode, or
during processing mode
The degradations may be due to
sensor noise
blur due to camera misfocus
relative object-camera motion
random atmospheric turbulence
others

2. In most of the existing image restoration methods we assume that
the degradation process can be described using a mathematical
model.
Objective of image restoration 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

3. Comparison of enhancement and restoration
Image restoration differs from image enhancement in that the latter is concerned
more with accentuation or extraction of image features rather than restoration
of degradations.
Image restoration problems can be quantified precisely, whereas enhancement
criteria are difficult to represent mathematically.

9. Image degradation / restoration model

10. A general model of a simplified digital image
degradation process
A simplified version for the image restoration process model is
y(i, j) = H[ f (i, j)]+ n(i, j)
where
y(i, j) the degraded image
f (i, j) the original image
H an operator that represents the degradation process
n(i, j) the external noise which is assumed to be image-
independent

11. Restoration methods could be classified as follows:
 deterministic: we work with sample by sample processing of
the observed (degraded) image
 stochastic: we work with the statistics of the images involved
in the process
 non-blind: the degradation process is known
 blind: the degradation process is unknown
 semi-blind: the degradation process could be considered
partly known

12. Image Restoration
Image Restoration Methods (Deterministic-Linear)
 Deterministic methods use a degradation model having degradation function H
 Within that the linear methods uses a linear restoration model
 Deterministic-Linear image restoration is classified into:
1. Un-constrained restoration
a) Inverse filter
b) Pseudo-Inverse filter
2. Constrained restoration using Lagrange multiplier -

14. Image Restoration
Image Restoration Methods (Deterministic-Linear)
1. Un-constrained restoration
a) Inverse filter
 Here we know the exact Point Spread Function h(m,n)
 We neglect the random noise
 By degradation model:
 For simplicity
 The noise is given by
 For neglecting the noise we need to find an
(1)
(2)
minimize the noise function:
(3)
 The operation is such that we are not giving any constraints for during
minimization, hence the name Un constrained restoration (4)

15.  Take the partial derivative of and equate to zero and solve for
(5)
(6)
 Taking Fourier transform on eqn: 6
(7)
 Taking Inverse Fourier transform on eqn: 7
Inverse filter-Disadvantages
 Not possible to obtain the inverse
 To exist the inverse the H matrix should be non-singular
 Noise will degrade the performance of inverse filter
 It has tendency to amplify the noise

17. Image Restoration Methods (Deterministic-Linear)
1. Un-constrained restoration
b) Pseudo Inverse filter
 When the H matrix is non invertible we go for the Pseudo inverse filter
 We know that the inverse filter is given as:
 Where
 i.e.
(1)
represents the point spread function
Tends to zero at higher frequencies so that the inverse filtering
becomes a multiplication with infinity (Denominator of eqn (1) becomes zero)
 Then the noises get amplified vigorously
 To avoid this we use the Pseudo-inverse filtering given as:
 Proper selection of will give good restoration

19. Image Restoration Methods (Deterministic-Linear)
2. Constrained restoration with Lagrange multiplier
(Constrained Least Square Filter)
The lost information in the degraded image can be mitigated by constraining
the restoration
Constraints add information to the process
Constraints can have additional information about the original scene
So that the restoration become more faithful
“Constrained restoration is a process of obtaining a meaningful restoration by
biasing the solution towards the minimiser of some specified constraint function”
Constrained Least Square Filter is a regularization technique which adds the
Lagrange multiplier to control the balance between noise artifacts and consistency
with the observed data

20. Image Restoration Methods (Deterministic-Linear)
2. Constrained restoration with
Lagrange multiplier (Constrained Least Square
Filter)
Constrained Least Square Filter is given as:
 Where is the Laplacian filter and has larger amplitudes at high
frequencies where the noise will dominates
 Which reduces noise at high frequencies
 Proper choice of and will reduce noise by minimizing higher
order derivatives

21. Minimum Mean Square Error (MMSE) Filtering (Wiener Filter)
Inverse filter has two main drawbacks:
1.At higher frequencies the noises start dominating
2.It assumes the noise to be minimum
To counteract these problems we use Wiener filter. Which includes both
degradation function H and statistical characteristics of noise
Introduced by N Wiener in 1942
The objective is to minimize the MSE :
Where E{.} is the expectation or mean
Assumptions:
1.Image and noise are uncorrelated (one or other has zero mean)
2.Intensity levels in the estimated image are a linear function of intensities in
degraded image g

22. Minimum Mean Square Error (MMSE) Filtering (Wiener Filter)
In frequency domain Wiener filter/MMSE filter/LSE (Least Square Error)
filter can be given as:
 Where

23. Minimum Mean Square Error (MMSE) Filtering (Wiener Filter)
Features:
1. Excellent noise removal at all frequencies
2. With lower orders of noise variances the estimate will be very close to original
3. Since it is an unconstraint one it simple
Original Wiener Filtered
Degraded

24. Geometric Operations
• Scale - change image content size
• Rotate - change image content orientation
• Reflect - flip over image contents
• Translate - change image content position
• Affine Transformation
– general image content linear geometric transformation

25. Geometric Transformations-Spatial Transformations
 Used when the degradation affects the geometry of an image. E.g. Angle, width
, length or volume etc.

26. Geometric transformations
• Geometric transformations are common in
computer graphics, and are often used in
image analysis.
• Geometric transforms permit the elimination
of geometric distortion that occurs when an
image is captured.
• If one attempts to match two different images
of the same object, a geometric
transformation may be needed.
• Examples?

27. Geometric Transformations
• A geometric transform consists of two basic
steps ...
– Step1: determining the pixel co-ordinate
transformation
• mapping of the co-ordinates of the moving image
pixel to the point in the fixed image.
Fixed Image Moving Image
(x,y)
T(x,y)

28. Geometric transformations
– Step2: determining the brightness of the points in the
digital grid of the transformed image.
• brightness is usually computed as an interpolation of the
brightnesses of several points in the neighborhood.
Fixed Image Moving Image xformed Moving Image
(x,y)
T(x,y)

29. Rotation Operation: Example
I = imread('ic.tif');
J = imrotate(I,35,'bilinear');
imshow(I)
figure, imshow(J)

31. Affine Transformation
• An affine transformation maps variables (e.g. pixel
intensity values located at position in an input image)
into new variables (e.g. in an output image) by applying a
linear combination of translation, rotation, scaling
operations.
• Significance: In some imaging systems, images are
subject to geometric distortions. Applying an affine
transformation to a uniformly distorted image can
correct for a range of perspective distortions.