FIHT2 AlGORITHM : A FAST INCREMENTAL HOUGH TRANSFORM
MVAY9O IAPR Workshop on Machine Vision Applications Nov. 28-30,1990, Tokyo
FIHT2 AlGORITHM : A FAST INCREMENTAL HOUGH TRANSFORM Hiroyasu KOSHIMIZU SCCS, Chukyo University
t : School of Computer
+
Munetoshi NUMADA* Lossev Technology Corp.
and Cognitive Sciences 101 Tokodate, Kaizu-cho, Toyota, 470-03 Japan
* : Fukuno-chou,Toyama,
939-15 Japan
ABSTRACT
+
+
FIHT2 algorithm defined by p = x .cos 0 y .sin 0 (a/(21()) . x . s i n e at 0 5 6 < a / 2 and at p = x . cose y . sin 0 (aJ(2It')) . y . cos 0 at a / 2 5 0 < a is a Hough transform which requires nothing of the trigonometric and functional operations to generate the Hough distributions. It is demonstrated in this paper that the FIHT2 is a complete alternative of the usual Hough transform(HT) defined by p = x.cos O+y.sin 0 in the sense that the both transforms could work perfectly as a line detector. It is easy to show that the Hough curves of the FIIJT2 can be generated in a incremental way where addition operation is exclusively needed. It is also investigated that the difference between HT and FIHT2 could be estimated to be neglected.
+
+
2. INCREMENTAL HOUGH TRANSFORM
FIHT: A N APPROXIMATED HT When let n and K be the sequence number and the number of the division of the 0 - axis of the parameter space, respectively, the H T for a point(x,y) can be expressed by eq.(2) and it is easily certified that the difference equations given by eq.(3) are approximately equivalent to the modified version of the Hough transform defined by eq.(4). A new Hough transform defined by eq.(3) is called as FIHT in this paper.
1. INTRODUCTION It is important to reduce the computing cost of Hough transform t o enforce its validity to apply t o the real world of the computer vision. In order t o reduce the cost, several researches have been reported on the basis of the reduction of the 'frequency cost' both of the numbers of the edge points and the divisions of the parameter space. On the other hand, it is still expectative to reduce the 'core cost' by direct reducing the Hough transform calculation. If it becomes possible t o reduce the core cost, the total cost reduction of the Hough transform would b e drastically improved. This paper proposes a method, called FIHT2 algorithm, to realize the reduction of the 'core cost' of the Hough transform H T defined by eq.(l). FIHT2(Fast Incremental Hough Transform-2) algorithm is a method t o generate the Hough curve by the incremental way of voting of the Hough curves to t h e parameter space.
Firstly in Chapter 2, it is introduced that H T defined by eq.(l) can be approximately repalaced by the difference equations called FIHT. In chapter 3, it is proved that if a little modifications are given to eq.(l), the expression of the difference equation becomes the exact expression of the modified Hough transform. In chapter 4, from the view point of the error evaluation, the relations among FIHT2, FIHT and H T are investigated t o show the validity and efficiency of FIHT2 algorithm.
+
(2)
+ Pn
(3)
y . sin 0, p, = x . cos 0, n = 0'1'2 ,...,K - 1 0, = n . a / I < ~n+l=
~n t
Pn+l = Pn where po = x,
E .
- & ' Pn+l r
pb= y
pn = x . cos 0, +(a/2I()
pi
=
and
E
= a/I(
+ y . sin 8,
(4)
. x . sine, -x . cos 0, + y . sin On -(a/2K) . y . sin 0,
O
FIHT2 AlGORITHM : A FAST INCREMENTAL HOUGH TRANSFORM Hiroyasu KOSHIMIZU SCCS, Chukyo University
t : School of Computer
+
Munetoshi NUMADA* Lossev Technology Corp.
and Cognitive Sciences 101 Tokodate, Kaizu-cho, Toyota, 470-03 Japan
* : Fukuno-chou,Toyama,
939-15 Japan
ABSTRACT
+
+
FIHT2 algorithm defined by p = x .cos 0 y .sin 0 (a/(21()) . x . s i n e at 0 5 6 < a / 2 and at p = x . cose y . sin 0 (aJ(2It')) . y . cos 0 at a / 2 5 0 < a is a Hough transform which requires nothing of the trigonometric and functional operations to generate the Hough distributions. It is demonstrated in this paper that the FIHT2 is a complete alternative of the usual Hough transform(HT) defined by p = x.cos O+y.sin 0 in the sense that the both transforms could work perfectly as a line detector. It is easy to show that the Hough curves of the FIIJT2 can be generated in a incremental way where addition operation is exclusively needed. It is also investigated that the difference between HT and FIHT2 could be estimated to be neglected.
+
+
2. INCREMENTAL HOUGH TRANSFORM
FIHT: A N APPROXIMATED HT When let n and K be the sequence number and the number of the division of the 0 - axis of the parameter space, respectively, the H T for a point(x,y) can be expressed by eq.(2) and it is easily certified that the difference equations given by eq.(3) are approximately equivalent to the modified version of the Hough transform defined by eq.(4). A new Hough transform defined by eq.(3) is called as FIHT in this paper.
1. INTRODUCTION It is important to reduce the computing cost of Hough transform t o enforce its validity to apply t o the real world of the computer vision. In order t o reduce the cost, several researches have been reported on the basis of the reduction of the 'frequency cost' both of the numbers of the edge points and the divisions of the parameter space. On the other hand, it is still expectative to reduce the 'core cost' by direct reducing the Hough transform calculation. If it becomes possible t o reduce the core cost, the total cost reduction of the Hough transform would b e drastically improved. This paper proposes a method, called FIHT2 algorithm, to realize the reduction of the 'core cost' of the Hough transform H T defined by eq.(l). FIHT2(Fast Incremental Hough Transform-2) algorithm is a method t o generate the Hough curve by the incremental way of voting of the Hough curves to t h e parameter space.
Firstly in Chapter 2, it is introduced that H T defined by eq.(l) can be approximately repalaced by the difference equations called FIHT. In chapter 3, it is proved that if a little modifications are given to eq.(l), the expression of the difference equation becomes the exact expression of the modified Hough transform. In chapter 4, from the view point of the error evaluation, the relations among FIHT2, FIHT and H T are investigated t o show the validity and efficiency of FIHT2 algorithm.
+
(2)
+ Pn
(3)
y . sin 0, p, = x . cos 0, n = 0'1'2 ,...,K - 1 0, = n . a / I < ~n+l=
~n t
Pn+l = Pn where po = x,
E .
- & ' Pn+l r
pb= y
pn = x . cos 0, +(a/2I()
pi
=
and
E
= a/I(
+ y . sin 8,
(4)
. x . sine, -x . cos 0, + y . sin On -(a/2K) . y . sin 0,
O
