DIP9-Segmentation2

a Hough Transform pair. A point in image space and a line in parameter space are a Hough Transform pair.
Original space Parameter Space
yi axi b
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x,y p

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Hough Transform
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Hough Transform
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Hough Transform
Hough Transform: 20 colinear points
•p, θ representation of line •Maximum accumulator value is 20
Edge Image (Canny)
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Hough Transform
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Hough transform detecting circles
Although the focus so far has been on straight lines, the hough transform is applicable to any function of the form g(v,c)=0, where v is a vector of coordinates and c is a vector of coefficients. For example, the points lying on the circle
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Hough Transform
1. They belonged to one of the three accumulator cells with the highest count 2. No gaps were longer than five pixels
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( x c1 ) ( y c2 ) c
2 2
2 3
can be detected by using the approach just discussed. The only difference is that there are three parameter.
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Hough Transform
a can be huge for near vertical lines, and there is no representation for a vertical line. Solution – using polar representation for lines.
Hough transform detecting circles
The procedure is to increment c1 and c2, solve for the c3 that satisfy:
( x c1 ) ( y c2 ) c
2 2
2 3
Then updated the accumulator corresponding to the cell associated with the triplet (c1,c2,c3) Clearly, the complexity of the hough transform is proportional to the number of coordinates and coefficients in a given functional representation
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Hough Transform
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Hough Transform
Quantize the parameter space (an array) and let the generation of a line be the increment of a counter at all cells in the parameter space that lie along the line. Histogram the resulting parameter space (accumulators). Peaks in this accumulator array occur at parameter locations corresponding to lines with significant support.
*A solution: Transform
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Hough Transform
yi axi b y j ax j b
Original space
Hough 1962
Hough Transform
b xi a yi b x ja y j
Parameter Space
b xi a yi
' ' '
yi axi b
b x ja y j
'
yi xi a b
' '
' '
y j x ja b
Points-Lines Duality
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Hough Transform
Consider a straight line. The HT maps all the points on the line onto a single point in the parameter space. A line in image space and a point in parameter space are
Eliminates isolated short segments, link little breaks.
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Global Processing
Global Processing Find lines, curves, or parts of lines or curves in an input image.
– Edge linking and boundary detection – Thresholding – Region based segmentation
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Edge Link and Boundary Detection
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Local Processing
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Hough transform detecting circles
Hough Transform
Note: the parametric solution represents the whole line and not the segment
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Hough Transform
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Hough Transform
Gray Scale Image
Local Processing
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Local Processing
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Local Processing
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Local Processing
Use vertical and horizontal Sobel operators Link conditions: gradient value > 25; gradient direction difference < 15
Hough Transform
Calculating steps of hough transform: 1. In parameter space, quantize a and b, and give out (amin,amax) and (bmin,bmax). 2. Set an accumulator A: A is A(amin:amax, bmin:bmax),set A to zero at the beginning. 3. For a given point (xi,yi) in the original space, let a equal to each allowed values on the aaxis, and solve for the corresponding b using the equation: b xi a yi 4. The resulting b are then rounded off to the nearest allowed value in the b-axis. If a choice of ap results in solution bq, we let A(p,q)=A(p,q)+1 5. At the end of this procedure, a value Q in A(i,j) corresponds to Q points in the xy space lying on the line y=aix+bj.
Digital Image Processing
Image Segmentation
Luguangm@
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Contents
Today we will begin looking at automated image analysis by examining the thorny issue of image segmentation:
x cos y sin
In polar representation, a point in the image correspond to sinusoidal curve in the parameter space x cos <=90 y sin -90<=