Image Transforms

1
Image Transforms

• Transforming images to images

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Classification of Image Transforms

• Point transforms
• modify individual pixels
• modify pixels locations
• Local transforms
• output derived from neighbourhood
• Global transforms
• whole image contributes to each output value

3
Point Transforms

• Manipulating individual pixel values
• Brightness adjustment
• Contrast adjustment
• Histogram manipulation
• equalisation
• Image magnification

4
Grey Scale Manipulation

• Brightness modifications
• Contrast modifications
• Histogram manipulation

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Brightness Adjustment

• Add a constant to all values
• g g k
• (k 50)

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Contrast Adjustment

• Scale all values by a constant
• g gk
• (k 1.5)

7
Image Histogram

• Measure frequency of occurrence of each
  grey/colour value

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Histogram Manipulation

• Modify distribution of grey values to achieve
  some effect

9
Equalisation/Adaptive Equalisation

• Specifically to make histogram uniform

10
Equalisation Transform

• Equalised image has n x m/l pixels per grey level
• Cumulative to level j
• jnm/l pixels
• Equate to a value in input cumulative histogram
  Ci
• Ci jnm/l
• j Cil/nm
• Modifications to prevent mapping to 1.

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Thresholding

• Transform grey/colour image to binary
• if f(x, y) gt T output 1
• else 0
• How to find T?

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Threshold Value

• Manual
• User defines a threshold
• P-Tile
• Mode
• Other automatic methods

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P-Tile

• If we know the proportion of the image that is
  object
• Threshold the image to select this proportion of
  pixels

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Mode

• Threshold at the minimum between the histograms
  peaks.

15
Automated Methods

• Find a threshold ? such that
• (Start at ? 0 and work upwards.)

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Image Magnification

• Reducing
• new value is weighted sum of nearest neighbours
• new value equals nearest neighbour
• Enlarging
• new value is weighted sum of nearest neighbours
• add noise to obscure pixelation

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Local Transforms

• Convolution
• Applications
• smoothing
• sharpening
• matching

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Convolution Definition

• Place template on image
• Multiply overlapping values in image and template
• Sum products and normalise
• (Templates usually small)

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Example
Image
Template
Result
. . . . . ... 3 5 7 4 4 4
5 8 5 4 4 6 9 6 4 4 6 9 5 3
4 5 8 5 4 . . . . . ...
. . . . . ... . . . . .
... . 6 6 6 . . 6 7 6 . .
6 7 6 . . . . . . ... . .
. . . ...
1 1 1 1 2 1 1 1 1
Divide by template sum
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Separable Templates

• Convolve with n x n template
• n2 multiplications and additions
• Convolve with two n x 1 templates
• 2n multiplications and additions

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Example

• Laplacian template
• Separated kernels

0 1 0 -1 4 1 0 1 0
-1 2 -1
-1 2 -1
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Composite Filters

• Convolution is distributive
• Can create a composite filter and do a single
  convolution
• Not convolve image with one filter and convolve
  result with second.
• Efficiency gain

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Applications

• Usefulness of convolution is the effects
  generated by changing templates
• Smoothing
• Noise reduction
• Sharpening
• Edge enhancement
• Template matching
• A later lecture

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Smoothing

• Aim is to reduce noise
• What is noise?
• How is it reduced
• Addition
• Adaptively
• Weighted

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Noise Definition

• Noise is deviation of a value from its expected
  value
• Random changes
• x ? x n
• Salt and pepper
• x ? max, min

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Noise Reduction

• By smoothing
• S(x n) S(x) S(n) S(x)
• Since noise is random and zero mean
• Smooth locally or temporally
• Local smoothing
• Removes detail
• Introduces ringing

27
Adaptive Smoothing

• Compute smoothed value, s
• Output s if s x gt T
• x otherwise

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Median Smoothing

• Median is one value in an ordered set
• 1 2 3 4 5 6 7 ? median 4
• 2 3 4 5 6 7 ? median 4.5

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Original
Smoothed
Median Smoothing
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Gaussian Smoothing

• To reduce ringing
• Weighted smoothing
• Numbers from Gaussian (normal) distribution are
  weights.

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Sharpening

• What is it?
• Enhancing discontinuities
• Edge detection
• Why do it?
• Perceptually important
• Computationally important

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Edge Definition
33
Edge Types

• Step edge
• Line edge
• Roof edge
• Real edges

34
First Derivative, Gradient Edge Detection

• If an edge is a discontinuity
• Can detect it by differencing

35
Roberts Cross Edge Detector

• Simplest edge detector
• Inaccurate localisation

36
Prewitt/Sobel Edge Detector
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Edge Detection

• Combine horizontal and vertical edge estimates

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Problems

• Enhanced edges are noise sensitive
• Scale
• What is local?

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Canny/Deriche Edge Detector

• Require
• edges to be detected
• accurate localisation
• single response to an edge
• Solution
• Convolve image with Difference of Gaussian (DoG)

40
Example Results
41
Second Derivative Operators Zero Crossing

• Model HVS
• Locate edge to subpixel accuracy
• Convolve image with Laplacian of Gaussian (LoG)
• Edge location at crossing of zero axis

42
Example Results
43
Global Transforms

• Computing a new value for a pixel using the whole
  image as input
• Cosine and Sine transforms
• Fourier transform
• Frequency domain processing
• Hough transform
• Karhunen-Loeve transform
• Wavelet transform

44
Cosine/Sine

• A halfway solution to the Fourier Transform
• Used in image coding

45
Fourier

• All periodic signals can be represented by a sum
  of appropriately weighted sine/cosine waves

46
Transformed section of BT Building image.
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Frequency Domain Filtering

• Convolution Theorem
• Convolution in spatial domain
• is equivalent to
• Multiplication in frequency domain

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Smoothing

• Suppress high frequency components

49
Sharpening

• Suppress low frequency components

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Hough Transform

• To detect curves analytically
• Example
• straight lines

51
Straight Line

• y mx c
• gradient m, intercept c
• c -mx y
• gradient -x, intercept y
• ALL points in (x, y) transform to a straight line
  in (c, m)
• Can therefore detect collinear points

52
Analytic Curve Finding

• Alternative representation
• to avoid infinities
• Other curves
• higher dimensional accumulators

53
Performance Improvement Techniques

• Look at pairs of points
• Use edge orientation

54
Karhunen-Loeve (Principal Component)

• A compact method of representing variation in a
  set of images
• PCs define a co-ordinate system
• 1st PC records most of variation
• 2nd PC records most of remainder
• etc

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Method

• Take a set of typical images
• Compute mean image and subtract from each sample
• Transform images into columns
• Group images into a matrix
• Compute covariance matrix
• Compute eigenvectors
• these are the PCs
• eigenvalues show their importance

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Uses

• Compact representation of variable data
• Object recognition

57
Wavelet

• A hierarchical representation

58
Example
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Uses

• Hierarchical representation
• Multiresolution processing
• Coding

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Geometric Transformations

• Definitions
• Affine and non-affine transforms
• Applications
• Manipulating image shapes

61
Affine TransformsScale, Shear, Rotate, Translate
Length and areas preserved.

Change values of transform matrix elements
according to desired effect. a, e ? scaling b, d
? shearing a, b, d, e ? rotation c, f ?
translation
62
Affine Transform Examples
63
Warping Example
Ansell Adams Aspens
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Image Resampling

• Moving source to destination pixels
• x and y could be non-integer
• Round result
• can create holes in image
• Manipulate in reverse
• where did warped pixel come from
• source is non-integer
• interpolate nearest neighbours

65
You Should Know

• Point transforms
• scaling, histogram manipulation,thresholding
• Local transforms
• edge detection, smoothing
• Global transforms
• Fourier, Hough, Principal Component, Wavelet
• Geometrical transforms