2. September 28, 19981 SEGMENTATION ITERATIVE ALGORITHMS HOUGH TRANSFORM

3. September 28, 19982 ITERATIVE ALGORITHMS ON IMAGES Start with an initial image. Apply a rule which transforms individual pixel grey values according to neighboring grey values. At each step reapply this same rule. After ‘alot’ of iterations the effect of this rule on the resulting image should be negligible and the iterations are said to converge. FOR k=0 to 999 IMAGE = F( IMAGE ) k+1 k

4. September 28, 19983 ITERATIVE ALGORITHMS ON IMAGES A general example of the form of an iterative rule at the (k+1)st iteration: I1 I2 I3 I4 I0 I5 I6 I7 I8 Local Image Neighborhood I0 = F (I1,I2,I3,I4,I5,I6,I7,I8,I0 ) k+1 k k k k k k k k k As a purely illustrative example, the convolution operation is an example of an iterative rule

5. September 28, 19984 DISCRETE CONVOLUTION Template ‘Kernel’ T1 T2 T3 T4 T5 T6 T7 T8 T9 I1 I2 I3 I4 I5 I6 I7 I8 I9 Image I = T1 x I1 + T2 x I2 + T3 x I3 + T4 x I4 + T5 x I5 + T6 x I6 + T7 x I7 + T8 x I8 + T9 x I9 3x3 Template Local Image Neighborhood

6. September 28, 19985 ITERATIVE ALGORITHMS ON IMAGES Specific simple example of an iterative procedure on an image SUCCESSIVE CONVOLUTION AVERAGING (Caveat: This is an illustrative example and not necessarily useful) What Happens ?? ANSWER: Successive blurring at each iteration until eventually after a certain number of iterations the image becomes a constant grey value equal to the average of all pixels.

7. September 28, 19986 ITERATIVE ALGORITHMS ON IMAGES SOMETHING MORE USEFUL 64 92 87 34 72 75 98 56 68 74 28 92 83 65 65 73 74 72 120 141 93 127 78 82 81 88 137 143 187 156 58 77 77 89 130 81 102 117 87 32 74 54 94 104 114 121 48 72 56 67 68 48 73 74 99 90 70 81 91 66 73 61 87 91 0 0 0 0 0 0 255 255 255 255 0 0 0 0 0 0 SIMULATED ANNEALING RELAXATION LABELING

8. September 28, 19987

9. 8

10. 9 GRAYLEVEL HISTOGRAM PIXEL GRAYVALUE # OF PIXELS

11. September 28, 199810

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14. September 28, 199813 HOUGH ALGORITHM Choose an analytic form f(x,y,a1,a2,…,an) and choose a range of values for parameters a1, a2, a3,….,an. Create accumulator array A(a1,a2,…,an) which represents direct match of f(x,y,a1,a2,…,an) with binary image. Local for local maximum which exceeds certain threshold.

15. September 28, 199814 GENERALIZED HOUGH ALGORITHM (Xc,Yc)  R-TABLE    1 2 n r1,r2,…,,rn