This document summarizes techniques for least mean square filtering and geometric transformations. It discusses minimum mean square error (Wiener) filtering, constrained least squares filtering, and geometric mean filtering for noise removal. It also covers spatial transformations, nearest neighbor gray level interpolation, and bilinear interpolation for geometric correction of distorted images. Examples are provided to demonstrate geometric distortion, nearest neighbor interpolation, and bilinear transformation.

1. LEAST MEAN SQUARE
AND
GEOMETRIC TRANFORMATION
PRESENTED BY,
K.LALITHAMBIGA,
II- M.Sc (CS & IT),
Nadar Saraswathi College of
Arts and Science, Theni.

2. SYNOPSIS
 Minimum Mean Square Error Filtering
or
 Least Square Error Filtering
 Constrained Least Square Filtering
 Geometric Transformation
 Spatial Transformation
 Geometric Distortion
 Gray Level Interpolation
 Nearest Neighbor Gray Level Interpolation
 Bilinear Interpolation

3. MINIMUM MEAN SQUARE ERROR (WIENER)
FILTERING
 In most images, adjacent pixels are highly correlated, while the
gray level of widely separated pixels are only loosely correlated.
 Therefore, the autocorrelation function of typical images
generally decreases away from the origin.
 Power spectrum of an image is the Fourier transform of its
autocorrelation function, therefore we can argue that the power
spectrum of an image generally decreases with frequency.
 Typical noise sources have either a flat power spectrum or one
that decreases with frequency more slowly than typical image
power spectrum.
 Therefore, the expected situation is for the signal to dominate the
spectrum at low frequencies, while the noise dominates the high
frequencies.

4. MINIMUM MEAN SQUARE ERROR (WIENER)
FILTERING
 The estimate ƒ of the uncorrupted image ƒ such that the
mean square error between them is minimized .
 The minimum of the error function is given in the
frequency domain by the expression
e²=E{(ƒ-ƒ )²}ˆ
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5. MINIMUM MEAN SQUARE ERROR
(WIENER) FILTERING
 Degradation model
 Wiener filter
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6. MINIMUM MEAN SQUARE ERROR
(WIENER) FILTERING

7. MINIMUM MEAN SQUARE ERROR (WIENER)
FILTERING

8. WIENER FILTERING - PROBLEMS
 The power spectra of the under graded image and noise
must be known.
 Weights all errors equally regardless of their location in the
image, while the eye is considerably more tolerant of errors
in the dark areas and high-gradient areas in the image.
 In minimizing the mean square error, Wiener filter also
smooth the image more than the eye would prefer

9. CONSTRAINED LEAST SQUARES
FILTERING
Only the mean and variance of the noise is required
g-vector by using the image elements in first row of g(x,y)
-dimensions
H –The matrix H then has dimensions MNX MN
The degradation model in vector-matrix form
The objective function
Subject to the constraint
111   MNMNMNMNMN ηfHg
21
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1
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x
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yxfC
ηHfg 
ηf,
22
ηHfg 

10. CONSTRAINED LEAST SQUARES
FILTERING
The frequency domain solution to this optimization
problem
The Fourier transform of the function
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11. CONSTRAINED LEAST SQUARES
FILTERING - EXAMPLE

12. Low noise: Wiener and CLS generate
equal results.
High noise: CLS outperforms Wiener if λ
is properly selected.
It is easier to select the scalar value for λ
than to approximate the SNR which is
seldom constant

13. GEOMETRIC MEAN FILTER
 The geometric mean filter is a member of a set of
nonlinear filters that are used to remove Gaussian noise.
 It operates by replacing each pixel by the geometric mean of
the values in its neighborhood.
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14. GEOMETRIC MEAN FILTER
 and being positive, real constants.
The geometric mean filter consists of the 2 expressions in
brackets raised to the powers  and 1- 
=1-this filter reduces to the inverse filter.
=0-this filter raised to the same power.
Its is also called parametric wiener filter.
 =½ and  =1 this filter also called as spectrum
equalization filter.

15. GEOMETRIC TRANSFORMATIONS
 The geometric transformations modify the spatial
relationships between pixels in an image.
 The geometric transformations often are called rubber-
sheet transformations
 Geometric transformation consists of 2 basic operations:
1. A Spatial Transformations –rearrangement of pixels on
the image plane.
2. Gray-level interpolation –assignment of gray levels to
pixels in the spatially transformed image.

16. SPATIAL TRANSFORMATION
 Assume the original image f(x,y) is subject to geometric
distortion yielding g(x’,y’)
 Spatial transformation &
 Coordination transformation
 The most frequently to overcome this difficulty is to formulate
the spatial relocation of pixels by the use of tiepoints,
 Subset of pixels whose location
 Input – Distorted
 Output – Corrected
• Function need 8 or more points
to find {ci; 1  i  8}
x´=r(x,y) y´=s(x,y)
8765
4321
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cxycycxcyxsy
cxycycxcyxrx
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17. GRAY LEVEL INTERPOLATION
 Spatial transformations establish a correspondence between
a point (x’, y’) in the distorted image g(x’,y’) and original
image f(x,y).
 To correct the geometric transformation, one needs to
estimate gray values of f(x,y),
 If x’ and y’ are integers, then
 If x’ and y’ are fraction numbers, but fall within the b order
of the original image, then interpolation will be needed to
find
 The gray-level interpolation is based on a nearest neighbor
approach. The method is called zero order interpolation.
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)','(),(ˆ yxgyxf 
),(ˆ yxf

18. NEAREST NEIGHBOR GRAY LEVEL
INTERPOLATION
   )','(),(ˆ yxgyxf 

19. BILINEAR INTERPOLATION
 Estimate the value of =g(x’,y’)) using
four nearest neighbors when x’ and y’ are
fractional numbers.
 Substitute g(x1,y1), g(x1,y2), g(x2,y1), g(x2,y2)
into above equation and solve for a, b, c, d.
It’s 4 equations and 4 unknowns.
(x1,y2)
(x1,y1)
(x2,y2)
(x2,y1)
   
   
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yyyyy
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'''
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21
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(x’,y’)
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20. a. An image with 25 regularly spaced
tiepoints.
b. Geometric distortion by
rearranging the tiepoints
c. Distorted image, nearest neighbor
interpolation
d. Restored image, NN
e. Distorted image, bilinear
transformation
f. Restored image, BT
EXAMPLES

21. EXAMPLES
a. Original image
b. Distorted image using bilinear
transform
c. Difference between a and b
d. Geometrically restored image
using bilinear transform for
gray level interpolation