Multiprimitive segmentation based on meaningful breakpoints for fitting digital planar curves with line segments and conic arcs

Abstract

This paper presents a multiprimitive segmentation method with line segments and conic arcs based on the types of breakpoints. In this method, a joint tuning procedure is proposed to merge consecutive segments and adjust their locations to achieve more accurate and stable conic arcs. No threshold is required in the multiprimitive segmentation by using the proposed scheme. And, the types of breakpoints among line segments and conic arcs are defined and they are useful and meaningful for pattern recognition and shape analysis. Besides, the computational complexity of the proposed method is O(n log n) which is lower than most other conic fitting methods. Further, the concept of types of breakpoints can be easily extended to other primitives.

Introduction

Segmentation of planar curves is one of the most important jobs in image processing since a segmented contour can be used to describe an object in a meaningful and compact form for higher level vision processing, such as pattern recognition and shape analysis. The objective is to use a minimum number of primitive pieces to approximate curves with minimum distortion. Many techniques have been proposed for this purpose in the past two decades. Polygonal approximation is the simplest approach. The line segments are almost extracted from curves based on the corner detection [1], [2], [3], [4], [5], [6] or dominant point detection [7], [8], [9], [10]. But polygonal approximation is rarely used for further shape analysis. To go from segmentation to shape analysis, one could include higher order primitives such as circular arcs, elliptic arcs, splines etc. in the segmentation. Normally, longer computation time will be involved in segmentation with such primitives since more parameters need to be solved. Hence, curve segmentation using line segments and circular arcs is selected, and it is a better representation than polygonal approximation. Thus, many methods were proposed to obtain line segments and circular arcs [11], [12], [13], [14]. The main reason is that a circular arc is easily obtained based on three parameters (center (x₀,y₀) and radius r). High order analytic curves such as conic arcs (circular, elliptic, parabolic, and hyperbolic arcs) can produce more accurate representations than only using circular arcs. Thus, the conic arcs are considered in this research because of computation time and flexibility of description. Needless to say, conic arcs in our daily life and in industry are important.

Some techniques for conic fitting were proposed [15], [16], [17], [18]. The least-squares method is often applied to fitting of conic arcs. Least-square fitting minimizes the square sum of error-of-fit in predefined measures, and two main categories (algebraic and geometric fitting) are differentiated by the respective definition of the error distance. In algebraic fitting, a conic arc is described by implicit equation, and the error distance is defined with deviation of the implicit equation from the expected value at each given point. And, in geometric fitting, the error distance is defined with the orthogonal distance from the given point to the fitted conic arc. These approaches are almost focused on accurate and efficient the result of fitting conic arcs. However, these results of conic fitting are not enough to employ on pattern recognition and shape analysis. That is, the fitting result is only flexible for curve description, but the shape analysis is not further obtained by these conic pieces (such as conic piece indexing). Hence, the multiprimitive segmentation based on conic piece indexing is very necessary to analyze the object shape.

We proposed a method for segmenting curves into line segments and circular arcs by using types of breakpoints [13]. The advantages of this technique are that it is:

• Threshold-free--No threshold within the algorithm,

• Stable--Invariant to transformations of the data (rotation, translation, and scale), and

• Extendible—Types of breakpoints can be extended to other primitives.

Hence, using the types of breakpoints for fitting curves with line segments and conic arcs is proposed in this research. The proposed scheme is a modified/extended method based on the breakpoint classification and tuning approach [13]. In the proposed scheme, the breakpoints are first categorized as five types: c-ll, c-la, c-aa, s-la, and s-aa by using AKC function and PHF [13], where c indicates a corner and s is a smooth joint. Subsequently, the procedure of line/conic segmentation and merging is employed based on the above types of breakpoints. The breakpoints are further categorized as 29 types: c-ll, c-lc, c-le, c-lp, c-lh, c-cc, c-ce, c-cp, c-ch, c-ee, c-ep, c-eh, c-pp, c-ph, c-hh, s-lc, s-le, s-lp, s-lh, s-cc, s-ce, s-cp, s-ch, s-ee, s-ep, s-eh, s-pp, s-ph, and s-hh, where c, e, p, and h indicate circular, elliptic, parabolic, and hyperbolic arcs, respectively. And, these breakpoints are defined as meaningful breakpoints in this research. These meaningful breakpoints are very useful for describing planar curves, pattern recognition, and shape analysis. Hence, the main contribution of this paper is to present the concept of meaningful breakpoints to give conic pieces indexing for shape analysis. This concept proposed in this paper is not referred in the existent methods for conic fitting.

In the remainder of this paper, the breakpoint classification is presented in Section 2 while multiprimitive segmentation based on lines and conic arcs is described in Section 3. In Section 4, the merging of conic arcs is proposed. Section 5 presents experimental examples and evaluations of the results. Finally, conclusion is made in Section 6.