On using directional information for parameter space decomposition in ellipse detection

Abstract

Abstract--In this paper we use the parameteric polar representation to extend the application of edge directional information from circle to ellipse extraction. As a result we obtain a mapping which decomposes the parameter space required for ellipse extraction into two independent sub-spaces and one final histogram accumulator. The mapping includes the tangent of the angle of the first and second directional derivatives. These tangents are computed by considering edge direction at two border points. We show that the use of gradient information for parameter space decomposition avoids the intensive point labelling imposed by geometric constraints used by other approaches. Computer vision Ellipse detection Image segmentationFeature extraction Parameter space decomposition Hough transform l. INTRODUCTIONThe detection of geometric primitives from an image is one of the basic tasks of computer vision. A geometric primitive is represented by a mathematical expression with a number of free parameters which define instances of a shape. Hence, the extraction problem centres on obtaining an estimate of the free parameters according to the information on an image. The Hough transform (HT) is a well-known method to detect shapes by independently considering geometric data composed of edge points. The independent evidence accumulation of the HT makes it robust, providing adequate results even for overlapping or partially occluded objects.In order to gather evidence, the HT defines a mapping from the image space into a parameter space. By considering each border point in an image, the mapping adds a vote to chosen cells in an accumulator which represents the parameter space. The cells increased are those whose associated parameters define the geometric primitive which passes through the image point. Therefore, local maxima in the accumulator array correspond to the parameters of detected shapes.The HT was originally proposed to fit straight lines and then extended to analytical curves31~ Although the technique can be used directly for any parameterized curve, the generalization implies an exponential increase in computational time and space requirements. for this reason, new methods based on the original HT have been proposed (a review of the methods can be seen in Illingworth & Kittler ~2~ and Leaversta~).* Author to whom correspondence should be addressed. © Crown copyright (1996). The aim of this paper is to show how directional information can be used to derive a mapping which decomposes the parameter space required for the implementation of the HT. In addition to image points, we consider edge direction as geometric data providing an important clue for evidence gathering in the parameter space.Edge direction information has been used successfully to reduce the computational requirements of the HT in circle extraction, ca'5) The general approach is to include gradient direction information in order to constrain the parameter space. Here we show how the parametric polar representation can be used to combine edge points an...

Invariant characterisation of the Hough transform for pose estimation of arbitrary shapes

“…The advantage of this deÿnition is that it characterises a point by only one other point in the image and this point does not require a particular geometric relationship to the other two. The deÿnition of the third point is based on the pole-polar form of the shape and it has been previously used for the extraction of circles and ellipses [27,28]. In [29 -32] this relationship is exploited to deÿne indexed tables suited to invariant extraction of shapes by the HT.…”

“…By considering the second part of Eq. (21) we obtain the invariance deÿnition, Q aff ( 0 ; 2 ; 3 ) = Q aff ( (s 0 ); (s 2 ); (s 3 ))}: (27) This equation and the geometry of the distance ratio deÿne a system of three simultaneous equations. That is…”

“…(27) and (30) can be used in Eqs. (8) and (9) to develop an evidence gathering process for a ne transformations.…”

“…The deÿnition of Eqs. (27) and (30) in Eqs. (8) and (9) deÿne three mappings from the image space into three 2D accumulator spaces.…”

“…(28) to ÿnd all the points (s 0 ), (s 2 ) and (s 3 ) that satisfy Eq. (27); (3) A more e cient and accurate implementation can be achieved by using multiresolution techniques [35].…”