Curve matching for open 2D curves

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Abstract

We present a curve matching framework for planar open curves under similarity transform1 based on a new scale invariant signature. The signature is derived from the concept of integral of unsigned curvatures. If one input curve as a whole can be aligned with some part in the second curve then the algorithm will find the requisite starting and end positions and will estimate the similarity transform in O(Nlog(N)) time. We extend our frame work to a more general case where some part of the first input curve can be aligned with some part of the second input curve. This is a more difficult problem that we solve in O(N3) time. The contributions of the paper are the new signature as well as faster algorithms for matching open 2D curves. We present examples from diverse application set to show that our algorithm can work across several domains.

Introduction

Given two curves as input, we seek to find what part of the first matches the best with a part or the whole of the second curve.2 This type of query is useful in many applications involving shape comparison. Example applications areas are computer vision, geospatial analysis and registration of images, computer aided geometric design, manufacturing, etc.

We want to motivate the discussion about curve matching through some examples illustrated in Fig. 1, which is organized as follows.

The first and second columns represent the input curves while the last column represents the two curves aligned in the same coordinate space using the algorithms proposed by this paper.

Application 1 (first row): The first curve is a handwritten stroke and the second curve is a feature, whose presence or absence in the first curve is what we are interested in. The second curve can be completely embedded in the first. We call it a whole-to-part curve matching problem.

Application 2 (second row): The two input curves are contours extracted from aerial images. They represent the same area of interest. The goal here is to find two sub-curves that differ only by a similarity transform,which enables image registration and change detection. We call it a part-to-part curve matching problem. Note that whole-to-part is a special case of part-to-part and we will present a specialized, faster solution.

Our motivation is a well known technique that solves the part-to-part problem invariant to Euclidean transform3 (Wolfson, 1990). It makes use of curvature with respect to arclength as a signature invariant to Euclidean transform, see Fig. 3a for an example. However, (Wolfson, 1990) cannot be extended to curves under similarity transform: if the potentially matched parts have a scale difference, then they do not span the same length on the curvature plot, and thus cannot be compared directly. See Fig. 2 for an example. In contrast, our signature is scale invariant and can deal with similarity transform.

In this paper we propose algorithms that perform whole-to-part matching in O(Nlog(N)) and part-to-part matching in O(N3), where the input curves can differ by a similarity transform. This is a significant improvement over the state of the art methods which are O(N4). Hash-based methods (Wolfson and Lamdan, 1998, Tomas Pajdla and Luc Van Gool, 1995) can do the comparison in O(N), but they require pre-processing steps that are of O(N4) asymptotic complexity.

The main contributions of this paper are:

  • 1.

    We outline the theoretical framework and develop a new curve signature based on the concept of integral of curvatures of a curve. This is also a novel way to parameterize curves that is scale invariant. We use this for solving the general problem of matching open curves.

  • 2.

    We present an O(Nlog(N)) algorithm for whole-to-part matching of a pair of open curves under similarity transform. We extend this to address part-to-part matching under similarity transform and the solution is an O(N3) algorithm.

The rest of the paper is organized as follows: Section 2 covers existing algorithms for curve matching. Section 3 describes the scale invariant signature proposed. Section 4 covers the topic of curve matching using the proposed signature and is followed by sections on experiments and discussion of results (Section 5) which includes a comparison to other methods. The paper concludes with a brief summary (Section 6).

Section snippets

Related work

We focus on work that serves to place this paper relative to the state of the art. See (Remco Veltkamp, 2001) for a more general review of the area.

Methods like (Mokhtarian et al., 1992, Arbter et al., 1990, Liao and Pawlak, 1996) belong to whole-to-whole matching, use global features and only return a distance estimation. Methods like (Kaygin and Mete Bulut, 2002, Tsay and Tsai, 1993) belong to whole-to-whole methods, use global features and return correspondence as well as similarity.

Scale invariant signature based on integral of curvature

The input to our algorithm can be conceptually characterized as follows. First, we only deal with the problem of matching open 2D (planar) curves such as in Fig. 1. Second, the curves may come from any source i.e. vectorized from a digital source or a Computer Aided Design (CAD) system, as long as the curves have a parametric representation (for ease of computation), they are twice differentiable (for robust curvature estimation) and other filters or algorithms have pre-processed the input

Whole-to-part matching

The domain of our signature is scale invariant but the range still has a scaling factor. We adopt normalized cross correlation (Zitova and Flusser, 2003), which is not influenced by the scale, to evaluate similarity between two matched curves.

For whole-to-part match, we know one curve is potentially embedded in the other one, therefore, we only need to find the position where the first curve aligns to in the second curve. Since we are using the curvature signature parameterized by the integral

Experiments and discussion

To test our algorithm, we have drawn some test cases from different applications domains. These are handwriting strokes, aerial images and Kimia image database (Kimia shape database).

We implemented a naive arclength based search algorithm to compare with ours. This algorithm is of complexity O(N4). It works directly in arclength domain instead of the integral of curvature domain. It has four loops over all possible start and end points of matched parts on two curves. Within the most inner loop

Conclusions

In this paper we derived a framework with a new curve signature and a parametrization that is invariant under similarity transform. Using this signature, we solve curve matching problems for open 2D curves. Our novel approach is a significant improvement over previous methods. In the future we would like to extend the signature to include affine invariant signature and deal with curve matching under affine transform. Another extension would be to use this for 3D/space curves.

Acknowledgement

This work was funded in part by the National Geospatial Agency (NGA), Grant # HM1582-05-1-2004), and National Science Foundation (NSF), Grant # IIS-0612269.

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    1

    Similarity transform is defined as a 2D transform that is limited to translation, rotation and uniform scaling.

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