A Global Approach for Solving Edge-Matching Puzzles

We consider apictorial edge-matching puzzles, in which the goal is to arrange a collection of puzzle pieces with colored edges so that the colors match along the edges of adjacent pieces. We devise an algebraic representation for this problem and provide conditions under which it exactly characterizes a puzzle. Using the new representation, we recast the combinatorial, discrete problem of solving puzzles as a global, polynomial system of equations with continuous variables. We further propose new algorithms for generating approximate solutions to the continuous problem by solving a sequence of convex relaxations

matching, interpolation, and approximation ; a survey

In this survey we consider geometric techniques which have been used to measure the similarity or distance between shapes, as well as to approximate shapes, or interpolate between shapes. Shape is a modality which plays a key role in many disciplines, ranging from computer vision to molecular biology. We focus on algorithmic techniques based on computational geometry that have been developed for shape matching, simplification, and morphing

Deterministic Sparse Pattern Matching via the Baur-Strassen Theorem

How fast can you test whether a constellation of stars appears in the night sky? This question can be modeled as the computational problem of testing whether a set of points $P$ can be moved into (or close to) another set $Q$ under some prescribed group of transformations. Consider, as a simple representative, the following problem: Given two sets of at most $n$ integers $P,Q\subseteq[N]$, determine whether there is some shift $s$ such that $P$ shifted by $s$ is a subset of $Q$, i.e., $P+s=\{p+s:p\in P\}\subseteq Q$. This problem, to which we refer as the Constellation problem, can be solved in near-linear time $O(n\log n)$ by a Monte Carlo randomized algorithm [Cardoze, Schulman; FOCS'98] and time $O(n\log^2 N)$ by a Las Vegas randomized algorithm [Cole, Hariharan; STOC'02]. Moreover, there is a deterministic algorithm running in time $n\cdot2^{O(\sqrt{\log n\log\log N})}$ [Chan, Lewenstein; STOC'15]. An interesting question left open by these previous works is whether Constellation is in deterministic near-linear time (i.e., with only polylogarithmic overhead). We answer this question positively by giving an $n\cdot(\log N)^{O(1)}$-time deterministic algorithm for the Constellation problem. Our algorithm extends to various more complex Point Pattern Matching problems in higher dimensions, under translations and rigid motions, and possibly with mismatches, and also to a near-linear-time derandomization of the Sparse Wildcard Matching problem on strings. We find it particularly interesting how we obtain our deterministic algorithm. All previous algorithms are based on the same baseline idea, using additive hashing and the Fast Fourier Transform. In contrast, our algorithms are based on new ideas, involving a surprising blend of combinatorial and algebraic techniques. At the heart lies an innovative application of the Baur-Strassen theorem from algebraic complexity theory.Comment: Abstract shortened to fit arxiv requirement

On characteristic points and approximate decision algorithms for the minimum Hausdorff distance

We investigate {\em approximate decision algorithms} for determining whether the minimum Hausdorff distance between two points sets (or between two sets of nonintersecting line segments) is at most $\varepsilon$.\def\eg{(\varepsilon/\gamma)} An approximate decision algorithm is a standard decision algorithm that answers {\sc yes} or {\sc no} except when $\varepsilon$ is in an {\em indecision interval} where the algorithm is allowed to answer {\sc don't know}. We present algorithms with indecision interval $[\delta-\gamma,\delta+\gamma]$ where $\delta$ is the minimum Hausdorff distance and $\gamma$ can be chosen by the user. In other words, we can make our algorithm as accurate as desired by choosing an appropriate $\gamma$. For two sets of points (or two sets of nonintersecting lines) with respective cardinalities $m$ and $n$ our approximate decision algorithms run in time O(\eg^2(m+n)\log(mn)) for Hausdorff distance under translation, and in time O(\eg^2mn\log(mn)) for Hausdorff distance under Euclidean motion

Solving Jigsaw Puzzles By the Graph Connection Laplacian

We propose a novel mathematical framework to address the problem of automatically solving large jigsaw puzzles. This problem assumes a large image, which is cut into equal square pieces that are arbitrarily rotated and shuffled, and asks to recover the original image given the transformed pieces. The main contribution of this work is a method for recovering the rotations of the pieces when both shuffles and rotations are unknown. A major challenge of this procedure is estimating the graph connection Laplacian without the knowledge of shuffles. We guarantee some robustness of the latter estimate to measurement errors. A careful combination of our proposed method for estimating rotations with any existing method for estimating shuffles results in a practical solution for the jigsaw puzzle problem. Numerical experiments demonstrate the competitive accuracy of this solution, its robustness to corruption and its computational advantage for large puzzles

Partitioning de Bruijn Graphs into Fixed-Length Cycles for Robot Identification and Tracking

We propose a new camera-based method of robot identification, tracking and orientation estimation. The system utilises coloured lights mounted in a circle around each robot to create unique colour sequences that are observed by a camera. The number of robots that can be uniquely identified is limited by the number of colours available, $q$, the number of lights on each robot, $k$, and the number of consecutive lights the camera can see, $\ell$. For a given set of parameters, we would like to maximise the number of robots that we can use. We model this as a combinatorial problem and show that it is equivalent to finding the maximum number of disjoint $k$-cycles in the de Bruijn graph $\text{dB}(q,\ell)$. We provide several existence results that give the maximum number of cycles in $\text{dB}(q,\ell)$ in various cases. For example, we give an optimal solution when $k=q^{\ell-1}$. Another construction yields many cycles in larger de Bruijn graphs using cycles from smaller de Bruijn graphs: if $\text{dB}(q,\ell)$ can be partitioned into $k$-cycles, then $\text{dB}(q,\ell)$ can be partitioned into $tk$-cycles for any divisor $t$ of $k$. The methods used are based on finite field algebra and the combinatorics of words.Comment: 16 pages, 4 figures. Accepted for publication in Discrete Applied Mathematic