Almost Vanishing Polynomials and an Application to Hough Transforms Maria-Laura Torrente joint work with M.C. Beltrametti Dipartimento di Matematica Universit` a di Genova RICAM, Linz November 2013 M. Torrente AVP and Applications to HT RICAM, Linz 1 / 29
Introduction Let’s start from the application. M. Torrente AVP and Applications to HT RICAM, Linz 2 / 29
Introduction Let’s start from the application. In the analysis of digital images, e.g. medical and astronomical images, the problem of automated recognition of special curves is very important. M. Torrente AVP and Applications to HT RICAM, Linz 2 / 29
The Hough Transform The main tool is based on the Hough Transform technique. HT is a technique mainly used in image analysis, computer vision, and digital image processing. The purpose of HT is to identify, in a given image, (approximate) instances of a certain class of shapes. M. Torrente AVP and Applications to HT RICAM, Linz 3 / 29
The Hough Transform The main tool is based on the Hough Transform technique. HT is a technique mainly used in image analysis, computer vision, and digital image processing. The purpose of HT is to identify, in a given image, (approximate) instances of a certain class of shapes. Originally (1962, Hough) HT was concerned with identification of lines in images; later on (1972 Duda & Hart, 1981 Ballard) HT was extended to identify circles and ellipses; many refinements have been investigated since then. HT exploits the duality between image space and parameter space; result is achieved through a voting procedure in the parameter space. M. Torrente AVP and Applications to HT RICAM, Linz 3 / 29
Detecting Aligned Points Suppose we want to detect aligned points in a given picture. M. Torrente AVP and Applications to HT RICAM, Linz 4 / 29
Detecting Aligned Points Suppose we want to detect aligned points in a given picture. Represent a straight line as y = ax + b ( not the best representation!). M. Torrente AVP and Applications to HT RICAM, Linz 4 / 29
Detecting Aligned Points Suppose we want to detect aligned points in a given picture. Represent a straight line as y = ax + b ( not the best representation!). Consider points in the picture (practically pixels , small cells ). M. Torrente AVP and Applications to HT RICAM, Linz 4 / 29
Detecting Aligned Points Suppose we want to detect aligned points in a given picture. Represent a straight line as y = ax + b ( not the best representation!). Consider points in the picture (practically pixels , small cells ). Let p 1 = ( x 1 , y 1 ) be a point of the picture; a straight line containing it satisfies y 1 = ax 1 + b . M. Torrente AVP and Applications to HT RICAM, Linz 4 / 29
Detecting Aligned Points Suppose we want to detect aligned points in a given picture. Represent a straight line as y = ax + b ( not the best representation!). Consider points in the picture (practically pixels , small cells ). Let p 1 = ( x 1 , y 1 ) be a point of the picture; a straight line containing it satisfies y 1 = ax 1 + b . MAIN IDEA: move to the parameter space , so y 1 = ax 1 + b is a straight line in this space. Repeat this process for every point p 2 , p 3 , . . . in the picture. Let ( A , B ) the intersection point of many lines, it means that the corresponding points in the picture lie on y = Ax + B !!! M. Torrente AVP and Applications to HT RICAM, Linz 4 / 29
Aligned Points Image space y ✻ ❅ p 1 = ( x 1 , y 1 ) r ❅ ❅ ❅ ✲ O x M. Torrente AVP and Applications to HT RICAM, Linz 5 / 29
Aligned Points Image space Parameter space y ✻ ✻ b ❅ p 1 = ( x 1 , y 1 ) ❍❍❍❍❍❍ r ❅ ❅ ❅ b = − x 1 a + y 1 ✲ ✲ O x O a M. Torrente AVP and Applications to HT RICAM, Linz 5 / 29
Aligned Points Image space Parameter space ✻ b y ✻ ❅ p 1 = ( x 1 , y 1 ) ❍❍❍❍❍❍ r ❅ p 2 r ❅ p 3 r ❅ b = − x 1 a + y 1 ✲ ✲ O a O x M. Torrente AVP and Applications to HT RICAM, Linz 6 / 29
Aligned Points Image space Parameter space y ✻ ✻ b b = − x 3 a + y 3 � � ✟✟✟✟✟✟ ❅ p 1 = ( x 1 , y 1 ) ❍❍❍❍❍❍ � r ❅ b = − x 2 a + y 2 p 2 B � r r ❅ p 3 � r ❅ � b = − x 1 a + y 1 ✲ ✲ O x O a A M. Torrente AVP and Applications to HT RICAM, Linz 7 / 29
Detecting Aligned Points Suppose we want to detect aligned points in a given picture. Represent a straight line as y = ax + b ( not the best representation!). Consider points in the picture (that is pixels , small cells ). Let p 1 = ( x 1 , y 1 ) be a point of the picture; a straight line containing it satisfies y 1 = ax 1 + b . MAIN IDEA: move to the parameter space, so y 1 = ax 1 + b 1 is a straight line in this space. Repeat this process for every point p 2 , p 3 , . . . in the picture. Let ( A , B ) be the intersection point of many lines, it means that the corresponding points in the picture lie on y = Ax + B !!! M. Torrente AVP and Applications to HT RICAM, Linz 8 / 29
Aligned Points Image space Parameter space y ✻ ✻ b b = − x 3 a + y 3 � � ✟✟✟✟✟✟ ❅ p 1 ❍❍❍❍❍❍ � r ❅ b = − x 2 a + y 2 p 2 B � r r ❅ p 3 � r ❅ � b = − x 1 a + y 1 y = Ax + B ✲ ✲ O x O a A M. Torrente AVP and Applications to HT RICAM, Linz 9 / 29
The Hough Transform (Formalism) Our interest is to detect more complicated curves. M. Torrente AVP and Applications to HT RICAM, Linz 10 / 29
The Hough Transform (Formalism) Our interest is to detect more complicated curves. Let: n , t be positive integers; K = R ; x = ( x 1 , . . . , x n ) and a = ( a 1 , . . . , a t ); M. Torrente AVP and Applications to HT RICAM, Linz 10 / 29
The Hough Transform (Formalism) Our interest is to detect more complicated curves. Let: n , t be positive integers; K = R ; x = ( x 1 , . . . , x n ) and a = ( a 1 , . . . , a t ); (Beltrametti, Robbiano 2012) F ( x , a ) ∈ K [ x 1 , . . . , x n , a 1 , . . . , a t ] = K [ x , a ] such that for each α = ( α 1 , . . . , α t ) ∈ A t ( K ) (parameter space) and for each p = ( p 1 , . . . , p n ) ∈ A n ( K ) (image space) we have: H α : F ( x , α ) := f α ( x ) = 0 irreduc. hypersurface degree d Γ p : F ( p , a ) := Γ p ( a ) = 0 Hough transform of the point p M. Torrente AVP and Applications to HT RICAM, Linz 10 / 29
The Hough Transform (Formalism) Our interest is to detect more complicated curves. Let: n , t be positive integers; K = R ; x = ( x 1 , . . . , x n ) and a = ( a 1 , . . . , a t ); (Beltrametti, Robbiano 2012) F ( x , a ) ∈ K [ x 1 , . . . , x n , a 1 , . . . , a t ] = K [ x , a ] such that for each α = ( α 1 , . . . , α t ) ∈ A t ( K ) (parameter space) and for each p = ( p 1 , . . . , p n ) ∈ A n ( K ) (image space) we have: H α : F ( x , α ) := f α ( x ) = 0 irreduc. hypersurface degree d Γ p : F ( p , a ) := Γ p ( a ) = 0 Hough transform of the point p Proposition (Regularity Property): the following conditions are equivalent: a) for any H α , H α ′ , we have H α = H α ′ ⇒ α = α ′ ; b) for any H α , we have � p ∈H α Γ p ( a ) = { α } . M. Torrente AVP and Applications to HT RICAM, Linz 10 / 29
Detection procedure Consider the case n = 2 (detection of curves in images). Let F = {H α } be a suitable (irreducible, with fixed degree...) family of curves. Assume that Regularity Property (RP) holds. M. Torrente AVP and Applications to HT RICAM, Linz 11 / 29
Detection procedure Consider the case n = 2 (detection of curves in images). Let F = {H α } be a suitable (irreducible, with fixed degree...) family of curves. Assume that Regularity Property (RP) holds. Recognition Algorithm 1 Choose a set X = { p 1 , . . . , p ν } of points of interest in A 2 ( R ). 2 In A t ( R ) find the (unique) intersection of the HT corresponding to the points p i , that is compute { α } = � i =1 ,...,ν Γ p i ( a ). 3 Return the parameter α , and the curve H α uniquely determined by α . M. Torrente AVP and Applications to HT RICAM, Linz 11 / 29
Detection procedure Consider the case n = 2 (detection of curves in images). Let F = {H α } be a suitable (irreducible, with fixed degree...) family of curves. Assume that Regularity Property (RP) holds. Recognition Algorithm 1 Choose a set X = { p 1 , . . . , p ν } of points of interest in A 2 ( R ). 2 In A t ( R ) find the (unique) intersection of the HT corresponding to the points p i , that is compute { α } = � i =1 ,...,ν Γ p i ( a ). 3 Return the parameter α , and the curve H α uniquely determined by α . Note that, in practice, { α } = � i =1 ,...,ν Γ p i ( a ) is NOT CORRECT since: RP applies to infinite intersection; M. Torrente AVP and Applications to HT RICAM, Linz 11 / 29
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