Image Transforms

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Image Transforms

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Content

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

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Image Transforms

• Overview

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Image Transform Concept
T
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Image Transform Concept
T
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Image Transforms

• Convolution

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

• Edge Detection

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Edge Detection

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

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Origin of Edges
surface normal discontinuity
depth discontinuity
surface color discontinuity
illumination discontinuity
Edges are caused by a variety of factors
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Edge Detection
How can you tell that a pixel is on an edge?
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Edge Types
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Real Edges
Noisy and Discrete!

• We want an Edge Operator that produces
• Edge Magnitude
• Edge Orientation
• High Detection Rate and Good Localization

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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)

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2D-Image Gradient
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2D-Image Gradient

• Gives the direction of most rapid change in
  intensity
• Gradient direction
• Edge strength

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Classification of Points

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

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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
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OpenCV Implementation ? The Sobel Operators
void cvSobel( const CvArr src, CvArr dst,
int xorder, int yorder, int aperture_size
3 )
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OpenCV Implementation ? The Scnarr Operator
void cvSobel( const CvArr src, CvArr dst,
int xorder, int yorder, int aperture_size
3 )
aperture_size
CV_SCHARR
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Demonstration
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Exercise
Download Test Program
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Effects of Noise
Consider a single row or column of the image
Where is the edge?
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Solution Smooth First
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Solution Smooth First
Where is the edge?
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Derivative Theorem of Convolution
Gaussian
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Derivative Theorem of Convolution

• saves us one operation.

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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.

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Optimal Edge Detection Canny

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

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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

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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

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OpenCV Implementation ? The Canny Operator
void cvCanny( const CvArr img, CvArr
edges, double lowThresh, double highThresh,
int apertureSize 3 )
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Example Canny Edge Detector
Download Test Program
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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)

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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.

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

• Hough Transforms

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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

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HT for Line Detection
A line in xy-plane is a point in mb-plane.
(m, b)
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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)
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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)
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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)
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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)
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HT Line Detection by ??-representation
A line in xy-plane is a point in ??-plane.
(?, ?)
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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.
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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.
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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.
(?, ?)
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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.
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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 )
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ExampleHough Line Transform
Download Test Program
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Hough Circle Transform
Circle equation
image space
parameter space
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Hough Circle Transform
Cost ineffective time consuming
Circle equation
image space
parameter space
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Hough Gradient Method
Circle equation
Parametric form
image space
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Hough Gradient Method
The value of can be obtained from the edge
detection process.
Circle equation
Parametric form
image space
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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
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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 )
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ExampleHough Circle Transform
Download Test Program
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Image Transforms

• Geometric Transforms

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Geometric Transforms ?Stretch, Shrink, Warp, and
Rotate
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Scaling , Rotation, Translation
Scaling
Rotation
Translation
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Scaling , Rotation Translation
Scaling
Translation
Rotation
Translation
Translation
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Homogeneous Coordinate
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Scaling , Rotation Translation
Scaling
2?3 matrix
Translation
Rotation
Translation
2?3 matrix
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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
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Affine Transformation
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GetAffineTransform
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Get Affine Transform
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Get 2D Rotation Matrix
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WarpAffine
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GetQuadrangleSubPix
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Example Affine Transform
Download Test Program
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GetQuadrangleSubPix
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Sparse Affine Transformation
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Perspective Transform
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Perspective Transform
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Perspective Transform
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Affine Transform vs. Perspective Transform
Affine Transform
Perspective Transform
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Get Perspective Transform
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WarpPerspective
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Sparse Perspective Transformation
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Image Transforms

• Histogram Equalization

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Graylevel Histogram of Image
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Goal of Histogram Equalization
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Goal of Histogram Equalization
Image Enhancement
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Method ? Graylevel Remapping
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Probability Theory
pdf
cdf
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Example Gaussian
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Example Gaussian
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Demonstration
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OpenCV Implementation
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Example
Download Test Program