Straiht Line Detection Any straiht line in 2D space can be represented by this parametric euation: x; y; ; ) =x cos + y sin, =0 To nd the transorm o a

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1 Houh Transorm E186 Handout Denition The idea o Houh transorm is to describe a certain line shape straiht lines, circles, ellipses, etc.) lobally in a parameter space { the Houh transorm domain. We assume the shape can be described by an euation x; y; 1 ; ; n )=0 where 1 ; ; n are a set o n parameters needed to describe the shape. The orward transorm Every point x; y) in the spatial domain is transormed to a hypersurace in the n-dimensional parameter space specied by the above euation. The inverse transorm Every point 1 ; ; n ) in the parameter space describes by the same euation a specic curve, the shape o interest, in the 2D imae plane. 1

2 Straiht Line Detection Any straiht line in 2D space can be represented by this parametric euation: x; y; ; ) =x cos + y sin, =0 To nd the transorm o a certain point x 0 ;y 0 ) in the parameter space, we solve the euation above or and et = x 0 cos + y 0 sin, x 0 = x y0 2 cos + sin ) x y0 2 x y0 2 = r 0 cos 0 cos + sin 0 sin ) = r 0 cos 0, ) where 8 < : r 0 4 = x y = tan,1 y 0 =x 0 ) Now we see that the transorm o the point x 0 ;y 0 ) is a sinusoidal curve in the parameter space. And a iven point 0 ; 0 ) in the parameter space can be inverse transormed back to the spatial domain to represent a straiht line specied by the euation x cos 0 + y sin 0, 0 =0 To detect all straiht lines in the imae, the ollowin alorithm can be used. Make available an n = 2 dimensional array y 0 H k ; l ); 0 k; l m) or the parameter space. Here m represents the resolution o the parameter space. Find the radient imae o the iven imae: x i ;y j )=jx i ;y j )j6 x i ;y j ); 0 i; j n) where n is the resolution o the imae. 2

3 For any pixel satisyin jx i ;y j )j >T s, increment all elements on the curve = x i cos + y i sin in the parameter space: or all = x i cos + y j sin ; H; ) =H; )+1; In the parameter space, any element H; ) >T h represents a straiht line detected in the imae. This alorithm can be improved by makin use o the radient direction 6, as in this case, the radient direction is the same as the anle. Thereore or any point jx; y)j > T s, we only need to increment the elements on a small sement o the sinusoidal curve. The third step in the alorithm can be modied as For any pixel satisyin jx i ;y j )j >T s, or all satisyin 6 x i ;y j ), 6 x i ;y j )+ = x i cos + y j sin ; H; ) =H; )+1; where denes a small rane in to allow some room or error in 6. 3

4 Make Use o 6 In eneral, the radient direction 6 can be used to reduce the computation in the parameter space. We rst note that the tanent direction o any unction x; y) is related to the derivative dy=dx tan = dy 0 dx =, xx; y) yx; 0 y) 4 = x; y) where 0 xx; y) and 0 yx; y) are the partial derivatives o the unction x; y) with respect to x and y, respectively. We also note that at an arbitrary point on the curve specied by the euation x; y) = 0, the radient direction is always perpendicular to the tanent direction: = 6 2 i.e. tan = tan 6 2 )=,cot 6 Now the radient direction 6 can be used to establish a second euation, in addition to the oriinal euation describin the shape, and thereby reducin the dimensions o the hyper-surace in the parameter space that needs to be incremented by 1. x; y; 1 ; ; n )=0 x; y; 1 ; ; n )=,cot 6 4

5 Detection o Circles A circle in the imae can be described by x; y; x 0 ;y 0 ;r)=x, x 0 ) 2 +y, y 0 ) 2, r 2 =0 where x 0, y 0, and r are three parameters which span a 3D parameter Houh space. Any point x; y) in the imae corresponds to a cone shaped surace in the 3D parameter space. To use 6, consider the derivative 0 dy dx =, xx; y) yx; 0 y) =,x, x 0 y, y 0 Now we only need to increment those elements in the parameter space that satisy both o the ollowin euations x, x0 ) 2 +y, y 0 ) 2, r 2 =0 x, x 0 )=y, y 0 )=cot 6 eometrically these two simultaneous euations represent the intersection o a cone specied by the rst euation and a plane specied by the second euation which passes throuh the axis o the cone. Solvin these euations or x 0 and y 0, we et x 0 = x r cos 6 y 0 = y r sin 6 and the alorithm or detectin circles: For any pixel satisyin jx i ;y j )j > T s, increment all elements satisyin the two simultaneous euations above; or all r x0 = x i r cos 6 y 0 = y j r sin 6 Hx 0 ;y 0 ;r)=hx 0 ;y 0 ;r)+1; In the parameter space, any element Hx 0 ;y 0 ;r) > T h circle with radius r located at x 0 ;y 0 )intheimae. represents a 5

6 Detection o Ellipses Here we assume the axes o the ellipses are in parallel with the coordinates o the imae space, i.e., the euations speciyin the ellipses are in the ollowin standard orm x; y; x 0 ;y 0 ;a;b)= x, x 0) 2 + y, y 0) 2, 1=0 a 2 b 2 where x 0, y 0, a and b are our parameters which span a 4D parameter Houh space. To use 6, consider 0 dy dx =, xx; y) yx; 0 y) =,x, x 0)=a y, y 0 )=b Now we only need to increment those elements in the parameter space that satisy both o the ollowin euations 8 < : x,x0) 2 a 2 + y,y 0) 2 x,x0)=a = cot 6 y,y0)=b b 2, 1=0 Solvin these euations or x 0 and y 0, we et x 0 = x i a cos 6 y 0 = y j b sin 6 and the alorithm or detectin circles: For any pixel satisyin jx i ;y j )j > T s, increment all elements satisyin the two simultaneous euations above; or all a and all b 6 6 x 0 = x a cos y 0 = y b sin Hx 0 ;y 0 ;a;b)=hx 0 ;y 0 ;a;b)+1; In the parameter space, any element Hx 0 ;y 0 ;a;b) >T h ellipse detected in the imae. represents an 6