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Image Transforms

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Image Transforms Content Overview Convolution Edge Detection Gradients Sobel operator Canny edge detector Laplacian Hough Transforms Geometric ... – PowerPoint PPT presentation

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Title: Image Transforms


1
Image Transforms
  • ??????

2
Content
  • Overview
  • Convolution
  • Edge Detection
  • Gradients
  • Sobel operator
  • Canny edge detector
  • Laplacian
  • Hough Transforms
  • Geometric Transforms
  • Affine Transform
  • Perspective Transform
  • Histogram Equalization

3
Image Transforms
  • Overview

4
Image Transform Concept
T
5
Image Transform Concept
T
6
Image Transforms
  • Convolution

7
Image Convolution
g(x,y) is known as convolution kernel.
8
Image Convolution
g(x,y) is known as convolution kernel.
height 2h 1 width 2w 1
9
Image Convolution
g(x,y) is known as convolution kernel.
height 2h 1 width 2w 1
10
Some Convolution Kernels
11
OpenCV Implementation ? Image Filter
void cvFilter2D( const CvArr src, CvArr
dst, const CvMat kernel, CvPoint
anchorcvPoint(-1, -1) )
12
Deal with Convolution Boundaries
void cvCopyMakeBorder( const CvArr src,
CvArr dst, CvPoint offset, int bordertype,
CvScalar valuecvScalarAll(0) )
13
Image Transforms
  • Edge Detection

14
Edge Detection
  • Convert a 2D image into a set of curves
  • Extracts salient features of the scene
  • More compact than pixels

15
Origin of Edges
surface normal discontinuity
depth discontinuity
surface color discontinuity
illumination discontinuity
Edges are caused by a variety of factors
16
Edge Detection
How can you tell that a pixel is on an edge?
17
Edge Types
18
Real Edges
Noisy and Discrete!
  • We want an Edge Operator that produces
  • Edge Magnitude
  • Edge Orientation
  • High Detection Rate and Good Localization

19
Derivatives of Image in 1D
? 1D image
? gradient
? Laplacian
  • Edges can be characterized as either
  • local extrema of ?I(x)
  • zero-crossings of ?2I(x)

20
2D-Image Gradient
21
2D-Image Gradient
  • Gives the direction of most rapid change in
    intensity
  • Gradient direction
  • Edge strength

22
Classification of Points
  • To precisely locate the edge, we need to thin.
  • Ideally, edges should be only one point thick.

23
The Sobel Operators
-1 0 1
-2 0 2
-1 0 1
1 2 1
0 0 0
-1 -2 -1
Good Localization Noise Sensitive Poor Detection
Sobel (3 x 3)
-1 -2 0 2 1
-2 -3 0 3 2
-3 -5 0 5 3
-2 -3 0 3 2
-1 -2 0 2 1
1 2 3 2 1
2 3 5 3 2
0 0 0 0 0
-2 -3 -5 -3 -2
-1 -2 -3 -2 -1
Sobel (5 x 5)
Poor Localization Less Noise Sensitive Good
Detection
24
OpenCV Implementation ? The Sobel Operators
void cvSobel( const CvArr src, CvArr dst,
int xorder, int yorder, int aperture_size
3 )
25
OpenCV Implementation ? The Scnarr Operator
void cvSobel( const CvArr src, CvArr dst,
int xorder, int yorder, int aperture_size
3 )
aperture_size
CV_SCHARR
26
Demonstration
27
Exercise
Download Test Program
28
Effects of Noise
Consider a single row or column of the image
Where is the edge?
29
Solution Smooth First
30
Solution Smooth First
Where is the edge?
31
Derivative Theorem of Convolution
Gaussian
32
Derivative Theorem of Convolution
  • saves us one operation.

33
Optimal Edge Detection Canny
  • Assume
  • Linear filtering
  • Additive iid Gaussian noise
  • An "optimal" edge detector should have
  • Good Detection
  • Filter responds to edge, not noise.
  • Good Localization
  • detected edge near true edge.
  • Single Response
  • one per edge.

34
Optimal Edge Detection Canny
  • Based on the first derivative of a Gaussian
  • Detection/Localization trade-off
  • More smoothing improves detection
  • And hurts localization.

35
Stages of the Canny algorithm
  • Noise reduction
  • Size of Gaussian filter
  • Finding the intensity gradient of the image
  • Non-maximum suppression
  • Tracing edges through the image and hysteresis
    thresholding
  • High threshold
  • Low threshold

36
Parameters of Canny algorithm
  • Noise reduction
  • Size of Gaussian filter
  • Finding the intensity gradient of the image
  • Non-maximum suppression
  • Tracing edges through the image and hysteresis
    thresholding
  • High threshold
  • Low threshold

37
OpenCV Implementation ? The Canny Operator
void cvCanny( const CvArr img, CvArr
edges, double lowThresh, double highThresh,
int apertureSize 3 )
38
Example Canny Edge Detector
Download Test Program
39
ReviewDerivatives of Image in 1D
? 1D image
? gradient
? Laplacian
  • Edges can be characterized as either
  • local extrema of ?I(x)
  • zero-crossings of ?2I(x)

40
Laplacian
  • A scalar ? isotropic.
  • Edge detection Find all points for which
  • ?2I(x, y) 0
  • No thinning is necessary.
  • Tends to produce closed edge contours.

41
Laplacian
42
Discrete Laplacian Operators
43
OpenCV Implementation ?The Discrete Laplacian
Operators
void cvLaplace( const CvArr src, CvArr
dst, int apertureSize 3 )
44
Example
45
Laplician for Edge Detection
Find zero-crossing on the Laplacian image.
46
Zero Crossing Detection
There is a little bug in the above algorithm.
Try to design your own zero-crossing detection
algorithm.
47
ExampleLaplician for Edge Detection
Download Test Program
48
Laplacian for Image Sharpening
49
ExampleLaplacian for Image Sharpening
Sharpened Image
50
Laplacian of Gaussian (LoG)
Gaussian
51
Some LoG Convolution Kernels
52
ExampleLoG for Edge Detection
by Laplacian
by LoG
53
Image Transforms
  • Hough Transforms

54
Goal of Hough Transforms
  • A technique to isolate the curves of a given
    shape / shapes in a given image
  • Classical Hough Transform
  • can locate regular curves like straight lines,
    circles, parabolas, ellipses, etc.
  • Generalized Hough Transform
  • can be used where a simple analytic description
    of feature is not possible

55
HT for Line Detection
A line in xy-plane is a point in mb-plane.
(m, b)
56
HT for Line Detection
A line in xy-plane is a point in mb-plane.
All lines passing through a point in xy-plane is
a line in mb-plane.
(m2, b2)
(m1, b1)
(m3, b3)
57
HT for Line Detection
A line in xy-plane is a point in mb-plane.
All lines passing through a point in xy-plane is
a line in mb-plane.
Given a point in xy-plane, we draw a line in
mb-plane.
(m2, b2)
(m1, b1)
(m3, b3)
58
HT for Line Detection
A line in xy-plane is a point in mb-plane.
Given a point in xy-plane, we draw a line in
mb-plane.
A line in xy-plane is then transformed in to a
set of lines in mb-plane, which intersect at a
common point.
(m, b)
59
HT for Line Detection
A line in xy-plane is a point in mb-plane.
Given a point in xy-plane, we draw a line in
mb-plane.
How to implement?
Is mb representation suitable?
A line in xy-plane is then transformed in to a
set of lines in mb-plane, which intersect at a
common point.
(m, b)
60
HT Line Detection by ??-representation
A line in xy-plane is a point in ??-plane.
(?, ?)
61
HT Line Detection by ??-representation
A line in xy-plane is a point in ??-plane.
All lines passing through a point in xy-plane is
a curve in ?? -plane.
62
HT Line Detection by ??-representation
A line in xy-plane is a point in ??-plane.
All lines passing through a point in xy-plane is
a curve in ?? -plane.
Given a point in xy-plane, we draw a curve in ??
-plane.
63
HT Line Detection by ??-representation
A line in xy-plane is a point in ??-plane.
Given a point in xy-plane, we draw a curve in ??
-plane.
A line in xy-plane is then transformed in to a
set of curves in ?? -plane, which intersect at a
common point.
(?, ?)
64
HT Line Detection by ??-representation
A line in xy-plane is a point in ??-plane.
Given a point in xy-plane, we draw a curve in ??
-plane.
A line in xy-plane is then transformed in to a
set of curves in ?? -plane, which intersect at a
common point.
65
OpenCV Implementation ? Hough Line Transform
CvSeq cvHoughLines2( CvArr image, void
line_storage, int method, double rho,
double theta, int threshold, double param1
0, double param2 0 )
66
ExampleHough Line Transform
Download Test Program
67
Hough Circle Transform
Circle equation
image space
parameter space
68
Hough Circle Transform
Cost ineffective time consuming
Circle equation
image space
parameter space
69
Hough Gradient Method
Circle equation
Parametric form
image space
70
Hough Gradient Method
The value of can be obtained from the edge
detection process.
Circle equation
Parametric form
image space
71
Hough Gradient Method
Circle equation
  • Quantize the parameter space for the parameters a
    and b.
  • Zero the accumulator array M(a, b).
  • Compute the gradient magnitude G(x, y) and angle
    ?(x, y).
  • For each edge (x0, y0) point in G(x, y),
    increment all points in the accumulator array
    M(a, b) along the line
  • Local maxima in the accumulator array correspond
    to centers of circles in the image.

image space
72
OpenCV Implementation ? Hough Circle Transform
CvSeq cvHoughCircles( CvArr image, void
circle_storage, int method, double dp,
double min_dist, double param1100, double
param2100 int min_radius0, int
max_radius0 )
73
ExampleHough Circle Transform
Download Test Program
74
Image Transforms
  • Geometric Transforms

75
Geometric Transforms ?Stretch, Shrink, Warp, and
Rotate
76
Scaling , Rotation, Translation
Scaling
Rotation
Translation
77
Scaling , Rotation Translation
Scaling
Translation
Rotation
Translation
Translation
78
Homogeneous Coordinate
79
Scaling , Rotation Translation
Scaling
2?3 matrix
Translation
Rotation
Translation
2?3 matrix
80
Affine Transformation
  • An affine transformation is any transformation
    that can be expressed in the form of a matrix
    multiplication followed by a vector addition.
  • In OpenCV the standard style of representing such
    a transformation is as a 2-by-3 matrix.

2?3 matrix
81
Affine Transformation
82
GetAffineTransform
83
Get Affine Transform
84
Get 2D Rotation Matrix
85
WarpAffine
86
GetQuadrangleSubPix
87
Example Affine Transform
Download Test Program
88
GetQuadrangleSubPix
89
Sparse Affine Transformation
90
Perspective Transform
91
Perspective Transform
92
Perspective Transform
93
Affine Transform vs. Perspective Transform
Affine Transform
Perspective Transform
94
Get Perspective Transform
95
WarpPerspective
96
Sparse Perspective Transformation
97
Image Transforms
  • Histogram Equalization

98
Graylevel Histogram of Image
99
Goal of Histogram Equalization
100
Goal of Histogram Equalization
Image Enhancement
101
Method ? Graylevel Remapping
102
Probability Theory
pdf
cdf
103
Example Gaussian
104
Example Gaussian
105
Demonstration
106
OpenCV Implementation
107
Example
Download Test Program
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