Tolerant Testers of Image Properties

IF 0.9 3区 计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS ACM Transactions on Algorithms Pub Date : 2022-10-10 DOI:https://dl.acm.org/doi/10.1145/3531527
Piotr Berman, Meiram Murzabulatov, Sofya Raskhodnikova
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Abstract

We initiate a systematic study of tolerant testers of image properties or, equivalently, algorithms that approximate the distance from a given image to the desired property. Image processing is a particularly compelling area of applications for sublinear-time algorithms and, specifically, property testing. However, for testing algorithms to reach their full potential in image processing, they have to be tolerant, which allows them to be resilient to noise.

We design efficient approximation algorithms for the following fundamental questions: What fraction of pixels have to be changed in an image so it becomes a half-plane? A representation of a convex object? A representation of a connected object? More precisely, our algorithms approximate the distance to three basic properties (being a half-plane, convexity, and connectedness) within a small additive error ε, after reading poly(1/ε) pixels, independent of the image size. We also design an efficient agnostic proper PAC learner of convex sets (continuous and discrete) in two dimensions under the uniform distribution.

Our algorithms require very simple access to the input: uniform random samples for the half-plane property and convexity, and samples from uniformly random blocks for connectedness. However, the analysis of the algorithms, especially for convexity, requires many geometric and combinatorial insights. For example, in the analysis of the algorithm for convexity, we define a set of reference polygons Pε such that (1) every convex image has a nearby polygon in Pε and (2) one can use dynamic programming to quickly compute the smallest empirical distance to a polygon in Pε.

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图像特性公差测试仪
我们开始对图像属性的容忍度测试进行系统研究,或者等效地,对从给定图像到期望属性的距离进行近似的算法进行系统研究。图像处理是亚线性时间算法的一个特别引人注目的应用领域,特别是属性测试。然而,为了测试算法在图像处理中充分发挥其潜力,它们必须具有容忍度,这使它们能够适应噪声。我们为以下基本问题设计了有效的近似算法:图像中需要改变多少像素才能使其成为半平面?一个凸面物体的表示?一个连接对象的表示?更准确地说,我们的算法在读取多边形(1/ε)像素后,在一个小的附加误差ε内近似地逼近到三个基本属性(半平面、凸性和连通性)的距离,与图像大小无关。我们还设计了一个有效的二维均匀分布凸集(连续和离散)的不可知论适当PAC学习器。我们的算法需要非常简单的输入:均匀随机样本用于半平面性质和凸性,均匀随机样本用于连通性。然而,对算法的分析,特别是对凸性的分析,需要许多几何和组合的见解。例如,在分析凸性算法时,我们定义了一组参考多边形Pε,使得(1)每个凸图像在Pε中都有一个邻近的多边形,(2)可以使用动态规划快速计算到Pε中多边形的最小经验距离。
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来源期刊
ACM Transactions on Algorithms
ACM Transactions on Algorithms COMPUTER SCIENCE, THEORY & METHODS-MATHEMATICS, APPLIED
CiteScore
3.30
自引率
0.00%
发文量
50
审稿时长
6-12 weeks
期刊介绍: ACM Transactions on Algorithms welcomes submissions of original research of the highest quality dealing with algorithms that are inherently discrete and finite, and having mathematical content in a natural way, either in the objective or in the analysis. Most welcome are new algorithms and data structures, new and improved analyses, and complexity results. Specific areas of computation covered by the journal include combinatorial searches and objects; counting; discrete optimization and approximation; randomization and quantum computation; parallel and distributed computation; algorithms for graphs, geometry, arithmetic, number theory, strings; on-line analysis; cryptography; coding; data compression; learning algorithms; methods of algorithmic analysis; discrete algorithms for application areas such as biology, economics, game theory, communication, computer systems and architecture, hardware design, scientific computing
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