CAP 5415 Computer Vision Fall 2012

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1 CAP 5415 Computer Vision Fall 2012 Hough Transform Lecture-18 Sections 4.2, 4.3 Fundamentals of Computer Vision

2 Image Feature Extraction Edges (edge pixels) Sobel, Roberts, Prewit Laplacian of Gaussian Cann Interest Points Harris SIFT Descriptors SIFT HOG Alper Yilmaz, Fall 2004

3 Shape Features Straight Lines Circles and Ellipses Arbitrar Shapes Alper Yilmaz, Fall 2004

4 How to Fit A Line? mx c

5 How to Fit A Line? Least squares Fit (over constraint) RANSAC (constraint) Hough Transform (under constraint)

6 Least Squares Fit Standard linear solution to estimating unknowns. If we know which points belong to which line Or if there is onl one line mx c f x, m, Minimize E c i i f x, m, c i Take derivatives wrt m and c set them to 0 2

7 Line Fitting c mx c mx c mx c mx n n D A n B n c m x x x AD B B A A A D A D A A A B A A A AD A B A T T T T T T T T 1 1 1

8 RANSAC: Random Sampling and Consensus 1. Randoml select two points to fit a line 2. Find the error between the estimated solution and all other points. If the error is less than tolerance, then quit, else go to step (1).

9 Line Fitting: Segmentation Several Lines How do we Know which points belong to which lines?

10 Hough Transform METHOD AND MEANS FOR RECOGNIZING COMPLEX PATTERNS, Paul V. C. Hough et al Inventors: Paul V. C. Hough, Paul V. C. Hough Current U.S. Classification: 382/281; 342/176; 342/190; 382/202 Alper Yilmaz, Fall 2004

11 Line Fitting: Hough Transform Line equation mx c m is slope, c is - intercept Rewrite this equation c ( x) m For particular edge point i this becomes c ( x ) m i i This is an equation of a line in (c,m) space. Alper Yilmaz, Mubarak Shah, Fall 2011

12 Line Fitting: Hough Transform c ( x ) m i i

13 Hough Transform Algorithm for Fitting Straight Lines

14 Polar Form of Equation of Line c i ( x) m j Problematic for vertical lines m and c grow to infinit p x cos sin Use from gradient

15 Image Gradient Alper Yilmaz, Fall 2004 x x x S S S S S S tan ) ( ), ( direction magnitude Vector Gradient x, x, x,

16 Hough Transform for Polar Form of Equation of Line

17 Line Fitting p Alper Yilmaz, Mubarak Shah, Fall 2011

18 Line Fitting

19 Line Fitting Examples ideal nois ver nois

20 Noise Factor This is the number of votes that the real line of 20 points gets with increasing noise

21 Noise Factor as the noise increases in a picture without a line, the number of points in the max cell goes up, too

22 Difficulties What is the increments for and p. too large? We cannot distinguish between different lines too small? noise causes lines to be missed

23 Circle Fitting Similar to line fitting Three unknowns ( x x o ) 2 ( o ) 2 r 2 0 Construct a 3D accumulator arra A Dimensions: x 0, 0, r Fix one of the parameters and loop for the others Increment corresponding entr in A. Find the local maxima in A

24 More Practical Circle Fitting Use the tangent direction at the edge point Compute x 0, 0 given x,, r x 0 0 x r cos r sin

26 Examples Alper Yilmaz, Fall 2004

27 Generalized Hough Transform Used for shapes with no analtical expression Requires training Object of known shape Generate model R-table Similar approach to line and circle fitting during detection

28 Generating R-table Compute centroid For each edge compute its distance to centroid r x x x Find edge orientation (gradient angle) Construct a table of angles and r values 0 0 (x o, o ) (x,)

29 Generating R-table Ф1 Ф2 Ф3 Ф4 r1, r2, r3 r14, r21, r23 r41, r42, r33 r10, r12, r13 (x,) Ф2 (x o, o ) r1 Ф1 (x,)

30 Detecting shape known Edge points (x,) Gradient angle at ever edge point R-table of the shape needs to be determined For each edge point find store it in corresponding row of R-table Create an accumulator arra of 2D (x,)

32 Rotation and Scale Invariance Rotation around Z-axis x x cos xsin Scaling x sx s sin cos Rotation+scaling x s s x cos xsin sin cos

33 Rotation and Scale Invariance Replace equations 4.13 and 4.14 in Algorithm 4.8 b following and loop for scale and rotation angles. Alpr Yilmaz, Mubarak Shah, Fall 2011

Types of Edges. Why Edge Detection? Types of Edges. Edge Detection. Gradient. Edge Detection

Types of Edges. Why Edge Detection? Types of Edges. Edge Detection. Gradient. Edge Detection Why Edge Detection? How can an algorithm extract relevant information from an image that is enables the algorithm to recognize objects? The most important information for the interpretation of an image