An Efficient Approximation Algorithm for Point Pattern Matching Under Noise

Point pattern matching problems are of fundamental importance in various areas including computer vision and structural bioinformatics. In this paper, we study one of the more general problems, known as LCP (largest common point set problem): Let \PP and \QQ be two point sets in $\mathbb{R}^3$, and let $\epsilon \geq 0$ be a tolerance parameter, the problem is to find a rigid motion $\mu$ that maximizes the cardinality of subset \II of $Q$, such that the Hausdorff distance \distance(\PP,\mu(\II)) \leq \epsilon. We denote the size of the optimal solution to the above problem by \LCP(P,Q). The problem is called exact-LCP for $\epsilon=0$, and \tolerant-LCP when $\epsilon>0$ and the minimum interpoint distance is greater than $2\epsilon$. A $\beta$-distance-approximation algorithm for tolerant-LCP finds a subset I \subseteq \QQ such that |I|\geq \LCP(P,Q) and \distance(\PP,\mu(\II)) \leq \beta \epsilon for some $\beta \ge 1$. This paper has three main contributions. (1) We introduce a new algorithm, called {\DA}, which gives the fastest known deterministic 4-distance-approximation algorithm for \tolerant-LCP. (2) For the exact-LCP, when the matched set is required to be large, we give a simple sampling strategy that improves the running times of all known deterministic algorithms, yielding the fastest known deterministic algorithm for this problem. (3) We use expander graphs to speed-up the \DA algorithm for \tolerant-LCP when the size of the matched set is required to be large, at the expense of approximation in the matched set size. Our algorithms also work when the transformation $\mu$ is allowed to be scaling transformation.Comment: 18 page

Tight Approximation of Image Matching

In this work we consider the {\em image matching} problem for two grayscale $n \times n$ images, $M_1$ and $M_2$ (where pixel values range from 0 to 1). Our goal is to find an affine transformation $T$ that maps pixels from $M_1$ to pixels in $M_2$ so that the differences over pixels $p$ between $M_1(p)$ and $M_2(T(p))$ is minimized. Our focus here is on sublinear algorithms that give an approximate result for this problem, that is, we wish to perform this task while querying as few pixels from both images as possible, and give a transformation that comes close to minimizing the difference. We give an algorithm for the image matching problem that returns a transformation $T$ which minimizes the sum of differences (normalized by $n^2$) up to an additive error of $\epsilon$ and performs $\tilde{O}(n/\epsilon^2)$ queries. We give a corresponding lower bound of $\Omega(n)$ queries showing that this is the best possible result in the general case (with respect to $n$ and up to low order terms). In addition, we give a significantly better algorithm for a natural family of images, namely, smooth images. We consider an image smooth when the total difference between neighboring pixels is O(n). For such images we provide an approximation of the distance between the images to within an additive error of $\epsilon$ using a number of queries depending polynomially on $1/\epsilon$ and not on $n$. To do this we first consider the image matching problem for 2 and 3-dimensional {\em binary} images, and then reduce the grayscale image matching problem to the 3-dimensional binary case

Combinatorial and experimental methods for approximate point pattern matching

Point pattern matching is an important problem in computational geometry, with applications in areas like computer vision, object recognition, molecular modelling, and image registration. Traditionally, it has been studied in an exact formulation, where the input point sets are given with arbitrary precision. This leads to algorithms that typically have running times of the order of high degree polynomials, and require robust calculations of intersection points of high degree surfaces. We study approximate point pattern matching, with the goal of developing algorithms that are more efficient and more practical than exact algorithms. Our work is motivated by the observation that in practice, data sets that form instances of pattern matching problems are noisy, and so approximate formulations are more appropriate. We present new and efficient algorithms for approximate point pattern matching in two and three dimensions, based on approximate combinatorial distance bounds on sets of points, and via the use of methods from combinatorial pattern matching. We also present an average case analysis and a detailed empirical study of our methods