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Transcript of Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients...
![Page 1: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/1.jpg)
Image Transforms
主講人:虞台文
![Page 2: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/2.jpg)
Content Overview Convolution Edge Detection
– Gradients– Sobel operator– Canny edge detector– Laplacian
Hough Transforms Geometric Transforms
– Affine Transform– Perspective Transform
Histogram Equalization
![Page 3: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/3.jpg)
Image Transforms
Overview
![Page 4: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/4.jpg)
Image Transform Concept
T[]T[]
![Page 5: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/5.jpg)
Image Transform Concept
T[]T[]
( , )I x y( , )I x y ( , )O x y( , )O x y
( , ) ( , )TO x y I x y
![Page 6: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/6.jpg)
Image Transforms
Convolution
![Page 7: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/7.jpg)
Image Convolution
( , )g x y
( , )f x y ( * )( , )f g x y*
g(x,y) is known as convolution kernel.
( * )( , ) ( , ) ( , )v u
f g x y f u v g x u y v
![Page 8: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/8.jpg)
Image Convolution
( , )g x y
( , )f x y ( * )( , )f g x y*
g(x,y) is known as convolution kernel.
( * )( , ) ( , ) ( , )v u
f g x y f u v g x u y v
( * )( , ) ( , ) ( , )y h x w
v y h u x w
f g x y f u v g x u y v
height 2h + 1width 2w + 1
![Page 9: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/9.jpg)
Image Convolution
g(x,y) is known as convolution kernel.
( * )( , ) ( , ) ( , )y h x w
v y h u x w
f g x y f u v g x u y v
height 2h + 1width 2w + 1
![Page 10: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/10.jpg)
Some Convolution Kernels
![Page 11: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/11.jpg)
OpenCV Implementation Image Filter
void cvFilter2D( const CvArr* src, CvArr* dst, const CvMat* kernel, CvPoint anchor=cvPoint(-1, -1));
void cvFilter2D( const CvArr* src, CvArr* dst, const CvMat* kernel, CvPoint anchor=cvPoint(-1, -1));
![Page 12: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/12.jpg)
Deal with Convolution Boundaries
void cvCopyMakeBorder( const CvArr* src, CvArr* dst, CvPoint offset, int bordertype, CvScalar value=cvScalarAll(0));
void cvCopyMakeBorder( const CvArr* src, CvArr* dst, CvPoint offset, int bordertype, CvScalar value=cvScalarAll(0));
![Page 13: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/13.jpg)
Image Transforms
Edge Detection
![Page 14: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/14.jpg)
Edge Detection
Convert a 2D image into a set of curves– Extracts salient features of the scene– More compact than pixels
![Page 15: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/15.jpg)
Origin of Edges
depth discontinuity
surface color discontinuity
illumination discontinuity
surface normal discontinuity
Edges are caused by a variety of factors
![Page 16: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/16.jpg)
Edge Detection
How can you tell that a pixel is on an edge?
![Page 17: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/17.jpg)
Edge Types
Step Edges
Roof Edge Line Edges
![Page 18: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/18.jpg)
Real Edges
Noisy and Discrete!
x
I
We want an Edge Operator that produces:– Edge Magnitude– Edge Orientation– High Detection Rate and Good Localization
![Page 19: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/19.jpg)
Derivatives of Image in 1D
( )I x
2 ( ) ( )I x I x
( )I I x
Edges can be characterized as either:– local extrema of I(x)– zero-crossings of 2I(x)
1D image
gradient
Laplacian
![Page 20: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/20.jpg)
2D-Image Gradient
( , ) ,T
I II x y
x y
,T
x yI I
![Page 21: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/21.jpg)
2D-Image Gradient ( , ) , ,
TT
x y
I II x y I I
x y
Gives the direction of most rapid change in intensity
Gradient direction:
Edge strength:
,0T
xI I
0,T
yI I
,T
x yI I I
1tan /y xI I
2 2x yI I I
![Page 22: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/22.jpg)
Classification of Points
To precisely locate the
edge, we need to thin.
Ideally, edges should be
only one point thick.
TI
Non-zeroedge width
I T
( , ) , ,T
T
x y
I II x y I I
x y
![Page 23: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/23.jpg)
The Sobel Operators
( , ) , ,T
T
x y
I II x y I I
x y
-1 0 1
-2 0 2
-1 0 1
1 2 1
0 0 0
-1 -2 -1
x xI S y yI S
Sobel (3 x 3):
Sobel (5 x 5):
-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
Good LocalizationNoise SensitivePoor Detection
Poor LocalizationLess Noise SensitiveGood Detection
![Page 24: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/24.jpg)
OpenCV Implementation The Sobel Operators
( , ) , ,T
T
x y
I II x y I I
x y
void cvSobel( const CvArr* src, CvArr* dst, int xorder, int yorder, int aperture_size = 3);
void cvSobel( const CvArr* src, CvArr* dst, int xorder, int yorder, int aperture_size = 3);
![Page 25: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/25.jpg)
OpenCV Implementation The Scnarr Operator
( , ) , ,T
T
x y
I II x y I I
x y
void cvSobel( const CvArr* src, CvArr* dst, int xorder, int yorder, int aperture_size = 3);
void cvSobel( const CvArr* src, CvArr* dst, int xorder, int yorder, int aperture_size = 3);
aperture_size
CV_SCHARR
![Page 26: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/26.jpg)
Demonstration
( , )I x y
( , ) , ,T
T
x y
I II x y I I
x y
xI
yI2 2x yI I
![Page 27: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/27.jpg)
Exercise
DownloadTest Program
DownloadTest Program
![Page 28: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/28.jpg)
Effects of Noise
Where is the edge?Where is the edge?
Consider a single row or column of the image
![Page 29: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/29.jpg)
Solution: Smooth First
f
g
*g f
*x g f
![Page 30: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/30.jpg)
Solution: Smooth First
f
g
*g f
*x g f
Where is the edge?Where is the edge?
![Page 31: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/31.jpg)
Derivative Theorem of Convolution
( , )g x y I *I g
2 2
2 2
1( , ) exp
2 2
x yg x y
Gaussian:
( , ) ( , )h x y g x y
2 2
4 2
2 2
4 2
( , )exp
2 2
( , )exp
2 2
g x y x x y
x
g x y y x y
y
![Page 32: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/32.jpg)
Derivative Theorem of Convolution
saves us one operation. * *x xg f g f
x g
f
*x g f
![Page 33: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/33.jpg)
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.
![Page 34: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/34.jpg)
Optimal Edge Detection: Canny
Based on the first derivative of a Gaussian
Detection/Localization trade-off– More smoothing improves detection– And hurts localization.
![Page 35: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/35.jpg)
Stages of the Canny algorithm
Noise reductionSize of Gaussian filter
Finding the intensity gradient of the image Non-maximum suppression Tracing edges through the image and hyste
resis thresholdingHigh thresholdLow threshold
![Page 36: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/36.jpg)
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 hyste
resis thresholding– High threshold– Low threshold
![Page 37: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/37.jpg)
OpenCV Implementation The Canny Operator
void cvCanny( const CvArr* img, CvArr* edges, double lowThresh, double highThresh, int apertureSize = 3);
void cvCanny( const CvArr* img, CvArr* edges, double lowThresh, double highThresh, int apertureSize = 3);
![Page 38: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/38.jpg)
Example: Canny Edge Detector
DownloadTest Program
DownloadTest Program
![Page 39: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/39.jpg)
Review:Derivatives of Image in 1D
( )I x
2 ( ) ( )I x I x
( )I I x
Edges can be characterized as either:– local extrema of I(x)– zero-crossings of 2I(x)
1D image
gradient
Laplacian
![Page 40: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/40.jpg)
Laplacian
2 22
2 2
( , ) ( , )( , ) ( , )
I x y I x yI x y f x y
x y
A scalar isotropic.
Edge detection: Find all points for which
2I(x, y) = 0
No thinning is necessary.
Tends to produce closed edge contours.
![Page 41: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/41.jpg)
Laplacian
2 22
2 2
( , ) ( , )( , )
I x y I x yI x y
x y
![Page 42: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/42.jpg)
Discrete Laplacian Operators
2 22
2 2
( , ) ( , )( , )
I x y I x yI x y
x y
010
141
010
111
181
111
121
242
121
![Page 43: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/43.jpg)
OpenCV Implementation The Discrete Laplacian Operators
2 22
2 2
( , ) ( , )( , )
I x y I x yI x y
x y
void cvLaplace( const CvArr* src, CvArr* dst, int apertureSize = 3);
void cvLaplace( const CvArr* src, CvArr* dst, int apertureSize = 3);
![Page 44: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/44.jpg)
Example
2 22
2 2
( , ) ( , )( , )
I x y I x yI x y
x y
2 ( , )I x y( , )I x y
![Page 45: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/45.jpg)
Laplician for Edge Detection
2 22
2 2
( , ) ( , )( , )
I x y I x yI x y
x y
2 ( , )I x y
Find zero-crossing on the Laplacian image.
![Page 46: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/46.jpg)
Zero Crossing Detection
2 22
2 2
( , ) ( , )( , )
I x y I x yI x y
x y
There is a little bug in the above algorithm.
Try to design your own zero-crossing detection algorithm.
![Page 47: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/47.jpg)
Example:Laplician for Edge Detection
DownloadTest Program
DownloadTest Program
2 22
2 2
( , ) ( , )( , )
I x y I x yI x y
x y
![Page 48: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/48.jpg)
Laplacian for Image Sharpening
2 22
2 2
( , ) ( , )( , )
I x y I x yI x y
x y
2 ( , )I x y( , )I x y
*w
![Page 49: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/49.jpg)
Example:Laplacian for Image Sharpening
2 22
2 2
( , ) ( , )( , )
I x y I x yI x y
x y
( , )I x y Sharpened Image
![Page 50: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/50.jpg)
Laplacian of Gaussian (LoG)
( , )g x y 2I 2*I g
2 2
2 2
1( , ) exp
2 2
x yg x y
Gaussian:
2( , ) ( , )h x y g x y
2 2 2 22
4 2 2
1( , ) ( , ) 1 exp
2 2
x y x yh x y g x y
![Page 51: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/51.jpg)
Some LoG Convolution Kernels
2 2 2 22
4 2 2
1( , ) ( , ) 1 exp
2 2
x y x yh x y g x y
0 0 0 0 1 1 1 0 0 0 0
0 0 1 2 3 3 3 2 1 0 0
0 1 2 4 5 5 5 4 2 1 0
0 2 4 5 1 2 1 5 4 2 0
1 3 5 1 14 24 14 1 5 3 1
1 3 5 2 24 40 24 2 5 3 1
1 3 5 1 14 24 14 1 5 3 1
0 2 4 5 1 2 1 5 4 2 0
0 1 2 4 5 5 5 4 2 1 0
0 0 1 2 3 3 3 2 1 0 0
0 0 0 0 1 1 1 0 0 0 0
0 1 1 2 2 2 1 2 0
1 2 4 5 5 5 4 2 1
1 4 5 3 0 3 5 4 1
2 5 3 12 24 12 3 5 2
2 5 0 24 40 24 0 5 2
2 5 3 21 24 12 3 5 2
1 4 5 3 0 3 5 4 1
1 2 4 5 5 5 4 2 1
0 1 1 2 2 2 1 2 0
0 0 1 0 0
0 1 2 1 0
1 2 16 2 1
0 1 2 1 0
0 0 1 0 0
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Example:LoG for Edge Detection
by LoG
by Laplacian
![Page 53: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/53.jpg)
Image Transforms
Hough Transforms
![Page 54: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/54.jpg)
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
![Page 55: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/55.jpg)
HT for Line Detection
x
y
y m bx
m
b
(m, b)
A line in xy-plane is a point in mb-plane.
![Page 56: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/56.jpg)
HT for Line Detection
x
y 1 1y m bx
m
b
( , )x y
(m1, b1)2 2y m bx
(m2, b2)
3 3y m bx
(m3, b3)
All lines passing through a point in xy-plane is a line in mb-plane.
A line in xy-plane is a point in mb-plane.
b my x
![Page 57: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/57.jpg)
HT for Line Detection
x
y 1 1y m bx
m
b
( , )x y
(m1, b1)2 2y m bx
(m2, b2)
3 3y m bx
(m3, b3)
All lines passing through a point in xy-plane is a line in mb-plane.A line in xy-plane is a point in mb-plane.
b my x ( , )x y
Given a point in xy-plane, we draw a line in mb-plane.
b my x
![Page 58: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/58.jpg)
HT for Line Detection
x
y
m
b
A line in xy-plane is a point 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.
Given a point in xy-plane, we draw a line in mb-plane.
y m bx (m, b)
![Page 59: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/59.jpg)
HT for Line Detection
x
y
m
b
A line in xy-plane is a point 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.
Given a point in xy-plane, we draw a line in mb-plane.
y m bx (m, b)
How to implement?How to implement?
Is mb representation suitable?Is mb representation suitable?
![Page 60: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/60.jpg)
HT Line Detection by -representation
x
y
( cos , sin )
( , )x y
cos sinx y (, )
A line in xy-plane is a point in -plane.
![Page 61: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/61.jpg)
HT Line Detection by -representation
x
y
A line in xy-plane is a point in -plane.
( , )x y1
2
3 4
43
2
1
cos sinx y
All lines passing through a point in xy-plane is a curve in -plane.
![Page 62: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/62.jpg)
HT Line Detection by -representation
x
y
A line in xy-plane is a point in -plane.
( , )x y1
2
3 4
43
2
1
cos sinx y
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.
( , )x y
cos sinx y
![Page 63: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/63.jpg)
HT Line Detection by -representation
x
y
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.
(, )
![Page 64: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/64.jpg)
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.
![Page 65: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/65.jpg)
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);
CvSeq* cvHoughLines2( CvArr* image, void* line_storage, int method, double rho, double theta, int threshold, double param1 = 0, double param2 = 0);
![Page 66: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/66.jpg)
Example:Hough Line Transform
DownloadTest Program
DownloadTest Program
![Page 67: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/67.jpg)
Hough Circle Transform
2 2 2( ) ( )x ya b r Circle equation:
x
y
r
a
b
0 0( , )x y
image space
0 0( , )x y
parameter space
![Page 68: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/68.jpg)
Hough Circle Transform
2 2 2( ) ( )x ya b r Circle equation:
x
y
r
a
b
0 0( , )x y
image space
0 0( , )x y
parameter space
Cost ineffective & time consuming
![Page 69: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/69.jpg)
Hough Gradient Method
2 2 2( ) ( )x ya b r Circle equation:
x
y
image space
0 0( , )x y
( , )a b
Parametric form: 0
0
cos
sin
x
y
a r
b r
0
0
cos
sin
a r
b r
x
y
0 0tan tanxb ya
![Page 70: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/70.jpg)
Hough Gradient Method
2 2 2( ) ( )x ya b r Circle equation:
x
y
image space
0 0( , )x y
( , )a b
Parametric form: 0
0
cos
sin
x
y
a r
b r
0
0
cos
sin
a r
b r
x
y
0 0tan tanxb ya
The value of can be obtained from the edge detection
process.
The value of can be obtained from the edge detection
process.
![Page 71: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/71.jpg)
Hough Gradient Method
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), incre
ment 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.
Circle equation:
x
y
image space
0 0( , )x y
( , )a b
0 0tan tanxb ya
0 0tan tanxb ya
2 2 2( ) ( )x ya b r
![Page 72: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/72.jpg)
OpenCV Implementation Hough Circle Transform
CvSeq* cvHoughCircles( CvArr* image, void* circle_storage, int method, double dp, double min_dist, double param1=100, double param2=100 int min_radius=0, int max_radius=0);
CvSeq* cvHoughCircles( CvArr* image, void* circle_storage, int method, double dp, double min_dist, double param1=100, double param2=100 int min_radius=0, int max_radius=0);
![Page 73: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/73.jpg)
Example:Hough Circle Transform
DownloadTest Program
DownloadTest Program
![Page 74: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/74.jpg)
Image Transforms
Geometric Transforms
![Page 75: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/75.jpg)
Geometric Transforms Stretch, Shrink, Warp, and Rotate
![Page 76: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/76.jpg)
Scaling , Rotation, Translation
0
0x
y
s
s
x x
y y
cos sin
sin cos
x x
y y
x x x
y y y
Scaling
Rotation
Translation
![Page 77: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/77.jpg)
Scaling , Rotation + Translation
0
0x
y
s
s
x x
y y
cos sin
sin cos
x x
y y
x x x
y y y
Scaling
Rotation
Translation
+Translation
+Translation
x
y
x
y
![Page 78: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/78.jpg)
Homogeneous Coordinate
xx
y
w
w
wy
1
xx
yy
![Page 79: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/79.jpg)
Scaling , Rotation + Translation
0
0x
y
x x x
y y y
s
s
cos sin
sin cos
x x x
y y y
Scaling
Rotation
+Translation
+Translation
0
01
x
y
s
s
xx
yy
cos sin
sin cos1
xx
yy
2 3 matrix
2 3 matrix
![Page 80: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/80.jpg)
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 tran
sformation is as a 2-by-3 matrix.
00 01 0
10 11 1
a a bx
a bya
x
y
00 01 0
10 11 1 1
a a b
a a b
x
y
2 3 matrix
![Page 81: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/81.jpg)
Affine Transformation
00 01 0
10 11 1 1
xa a bx
ay
y a b
![Page 82: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/82.jpg)
GetAffineTransform
00 01 0
10 11 1 1
xa a bx
ay
y a b
![Page 83: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/83.jpg)
Get Affine Transform
00 01 0
10 11 1 1
xa a bx
ay
y a b
![Page 84: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/84.jpg)
Get 2D Rotation Matrix
![Page 85: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/85.jpg)
WarpAffine00 01 0
10 11 1 1
xa a bx
ay
y a b
![Page 86: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/86.jpg)
GetQuadrangleSubPix
![Page 87: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/87.jpg)
Example: Affine Transform
DownloadTest Program
DownloadTest Program
![Page 88: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/88.jpg)
GetQuadrangleSubPix
![Page 89: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/89.jpg)
Sparse Affine Transformation
![Page 90: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/90.jpg)
Perspective Transform
![Page 91: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/91.jpg)
Perspective Transform
![Page 92: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/92.jpg)
Perspective Transform
![Page 93: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/93.jpg)
Affine Transform vs. Perspective Transform
00 01 0
10 11 1 1
xa a bx
ay
y a b
00 01 0
10 11 1
0 0 11 1
a a bx x
by a ya
Affine Transform:
x w
y w
w
Perspective Transform:
00 01 0
10 11 1
20 21 1 1
a a b
a a b
a
y
w a
x x
y
/
/
x x w
y y w
![Page 94: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/94.jpg)
Get Perspective Transform
00 01 0
10 11 1
20 21 1 1
x w x x
y
a a b
a a yb
a a
w y
w w
![Page 95: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/95.jpg)
WarpPerspective
00 01 0
10 11 1
20 21 1 1
x w x x
y
a a b
a a yb
a a
w y
w w
![Page 96: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/96.jpg)
Sparse Perspective Transformation
00 01 0
10 11 1
20 21 1 1
x w x x
y
a a b
a a yb
a a
w y
w w
![Page 97: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/97.jpg)
Image Transforms
Histogram Equalization
![Page 98: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/98.jpg)
Graylevel Histogram of Image
![Page 99: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/99.jpg)
Goal of Histogram Equalization
![Page 100: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/100.jpg)
Goal of Histogram Equalization
Image Enhancement
![Page 101: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/101.jpg)
Method Graylevel Remapping
0 1
fX(x)
x 0 1
fY(y)
y
y
xX Y
![Page 102: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/102.jpg)
Probability Theory
y
xX Y
( )Xf xpdf
( )XF xcdf( )XY F X ~ (0,1)Y U
( )Xy F x
![Page 103: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/103.jpg)
Example: Gaussian
![Page 104: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/104.jpg)
Example: Gaussian
![Page 105: Image Transforms 主講人:虞台文. Content Overview Convolution Edge Detection – Gradients – Sobel operator – Canny edge detector – Laplacian Hough Transforms.](https://reader035.fdocument.pub/reader035/viewer/2022081418/56649c925503460f9494dcd8/html5/thumbnails/105.jpg)
Demonstration
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OpenCV Implementation
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Example
DownloadTest Program
DownloadTest Program