1. Geometric and Point Transforms

2. Copyright GA Tagliarini, PhD
Standard Angles
Rotation or reflection through standard angles such as 90°, 180°, or 270° can be done most conveniently via direct index.
Intermediate angle rotations employ matrix manipulations of the indices
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3. Rotation By Matrix Multiplication
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4. Rotating The Standard Basis
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5. Graphically Portraying The Standard Basis Rotation
<0, 1>
<-sin(θ), cos(θ)>
-sin(θ)
<cos(θ), sin(θ)>
cos(θ) θ
sin(θ) θ
<1, 0>
cos(θ)
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6. Rotating An Arbitrary Point <x, y>
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7. Geometrically Interpreting Rotation
<xcos(θ)-ysin(θ), xsin(θ)+ycos(θ)> = <rcos(α+θ), rsin(α+θ)> Why??
<0, 1> r
y or y<0, 1>
<x, y> = <r cos(α), r sin(α)> r θ α
x or x<1, 0>
<1, 0>
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8. Observations About Rotation
Depending upon the rotation angle and image dimension, special processing may be required for some pixels
Matrix multiplication applied to pixel indices provides access to pixel values for the destination image
Working from the source to the destination may omit pixels
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9. Copyright GA Tagliarini, PhD
Spatial Filtering
Common types of filtering
Smoothing: averaging may be done
Over multiple images to de-noise the source
To reduce noise or soften the appearance in a single image
Speckle reducing
Edge enhancing
Often implemented by point operations using neighborhood data and a filter
Filters are commonly represented with rectangular arrays of numbers
The numeric values are the coefficients of the filter
The arrays may be referred to as masks, kernels, templates, windows
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10. Smoothing Filter: Averaging In An Image
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11. Geometrical Interpretation
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Geometrical Interpretation s a b c d e f g h i
1/9 r c
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Copyright GA Tagliarini, PhD

12. Averaging Observations
Average filtering will reduce the clarity of sharp edges. (Why?)
The form of the filter is typically adapted at the boundaries of the image to accommodate computation. This 3x3 filter becomes:
2x2 at the image corners (with coefficients = ¼)
2x3 along the top and bottom, and 3x2 along the left and right sides (with coefficients = 1/6)
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13. Averaging Observations (continued)
Averaging (and other) filters are commonly square with odd side length (so that they possess a center weight).
The center weight may be zero.
The weights are not required all to have the same value.
The sum of the weights will be 1.
Not all filters will be symmetric.
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14. Example Averaging Filters
Some other averaging filters:
1/8 1 = 1 2 3 7 11 17
1/16
2/16
4/16 1 2 4 =
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15. Speckle Reduction By Median Filtering
Median filters are another type of spatial filter.
Suppose a neighborhood of odd side length, sl = 2m+1 for pixel (r, c) in an image s, denoted N(s, r, c, sl).
To apply a median filter:
Find the median intensity, mi, of the pixels in N(s, r, c, m); i.e., mi = median( N(s, r, c, sl))
Set d(r, c) = mi
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16. Copyright GA Tagliarini, PhD
Median Filtering
The median can be found by sorting the pixel intensities in N(s, r, c, sl)
It is NOT necessary to sort the list of pixel intensities completely. (Why?)
Exercises
Implement monochrome and multi-spectral median filters.
What happens when a median filter is applied recursively?
How does this differ from the result of recursively applying a smoothing filter?
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17. Copyright GA Tagliarini, PhD
Contrast Enhancement
Polynomial or root transformations
Log transformations
Histogram equalization
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18. Polynomial And Root Transforms
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19. Copyright GA Tagliarini, PhD
Log Transforms
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20. Key Insight: Changing Contrast
f(x)
When f’>1, a unit change in x results more than a unit change in y
When f’<1, a unit change in x results less than a unit change in y
When f’≈1, a unit change in x results approximately a unit change in y y
f’<1
f’≈1 x
f’>1
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21. An Typical Image Histogram
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22. Overlaying The Histograms
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23. Copyright GA Tagliarini, PhD
Cumulative Histogram
Similar to a probability density function
Increases contrast where the most frequently occurring pixel intensities lie
Used for monochrome images but separate application to RGB color planes leads to surreal images
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24. Copyright GA Tagliarini, PhD
Sharpening
Unlike smoothing filters, the goal is to enhance detail that has been blurred
Averaging filters typically use a scaled weighted sum (an integration)
Sharpening filters commonly employ differences (derivatives)
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25. Copyright GA Tagliarini, PhD
Derivatives
Recall that the derivative of a function f at a point a may be defined as when the limit exists.
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26. Copyright GA Tagliarini, PhD
Graphically a x
f(x)-f(a)
x-a
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27. Copyright GA Tagliarini, PhD
With Digital Samples
The closest approximation is when x=a+1
So x-a=1
Hence f(a+1)-f(a) is an estimate of f’(a) a x
f(x)-f(a)
x-a
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28. One Dimensional Derivative…
f(x)
f(x+1)
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29. One Dimensional Second Derivative…
f(x-1)
f(x)
f(x+1)
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30. Laplacian 1 -4 Which gives rise to the mask 11/30/20181 -4
Which gives rise to the mask
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31. Enhancement Using The Laplacian
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32. A One-Dimensional Example
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33. Copyright GA Tagliarini, PhD
Example Detail f 50 45 40 35 30 25 20 60
100
140
180
220
120
200 f' -5
-100
-180
f'' 5
-140
-360 f# 15
260
-80
380
-160
scaled f# [20, 220] 98 94 92 90 89 87 85
116
131
146
161
176
clipped f#
clipped f#
255
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34. Alternative Laplacian Masks
Implementing the computational simplification use:
To employ diagonal derivatives in addition to x-y use -1 5 -1 9
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35. Example With f, f’, f’’, and Sharpened f
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36. Example With Sharpening
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37. Gradient Operators For Enhancement
Robert’s cross-gradient (2x2)
Prewitt’s gradient (3x3)
Sobel operator (3x3)
These use the magnitude of a gradient estimate to create image enhancements
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38. Copyright GA Tagliarini, PhD
Gradient
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39. Creating And Using Gradient Estimates
Often, the sum of the absolute values of Gx and Gy are used
Computationally, |Gx| and |Gy| take much less time to compute than squaring and finding a square root of a sum.
Hence, the gradient magnitude may be estimated by
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40. Using Gradient Operators
Estimate the gradient magnitudes and create a gradient magnitude map.
Often the gradient map is half-toned using an anecdotally determined threshold. The gradient map may be used directly to find edges.
The half-toned map (or its negative) can be “OR-ed” with the original for interpretation.
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41. Robert’s Cross-Gradient Operators
Robert’s cross-gradient is estimated using |grad(f)| = |z9-z5|+|z8-z6|
Using the masks below to estimate derivatives: z1 z2 z3 z4 z5 z6 z7 z8 z9 -1 1 -1 1
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42. Copyright GA Tagliarini, PhD
Robert’s Gradient Map
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43. Copyright GA Tagliarini, PhD
Prewitt’s Gradient
Estimate Prewitt’s gradient using: |grad(f)| = |(z3+z6+z9)-(z1+z4+z7)|+ |(z7+z8+z9)-(z1+z2+z3)|
Use the masks below: z1 z2 z3 z4 z5 z6 z7 z8 z9 -1 1 -1 1
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44. Prewitt’s Gradient Map
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45. Copyright GA Tagliarini, PhD
Sobel Masks
Sobel’s gradient is estimated using |grad(f)| = |z3-z1|+|z6-z4|+|z9-z7|
Using the masks below to estimate derivatives: z1 z2 z3 z4 z5 z6 z7 z8 z9 -1 1 -2 2 -1 -2 1 2
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46. Using Sobel’s Gradient
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47. Using Sobel’s Gradient With Wavelet
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