### 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 (that is, the smallest fraction of pixels that need to change in the image to ensure that the image satisfies 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. Prior to this work, only one tolerant testing algorithm for an image property (image partitioning) has been published. We design efficient approximation algorithms for the following fundamental questions: What fraction of pixels have to be changed in an image so that 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 a number of pixels polynomial in 1/- and independent of the size of the image. The running time of the testers for half-plane and convexity is also polynomial in 1/ϵ. Tolerant testers for these three properties were not investigated previously. For convexity and connectedness, even the existence of distance approximation algorithms with query complexity independent of the input size is not implied by previous work. (It does not follow from the VC-dimension bounds, since VC dimension of convexity and connectedness, even in two dimensions, depends on the input size. It also does not follow from the existence of non-tolerant testers.) 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. This construction might be of independent interest.

Original language | English (US) |
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Title of host publication | 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016 |

Editors | Yuval Rabani, Ioannis Chatzigiannakis, Davide Sangiorgi, Michael Mitzenmacher |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959770132 |

DOIs | |

State | Published - Aug 1 2016 |

Event | 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016 - Rome, Italy Duration: Jul 12 2016 → Jul 15 2016 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 55 |

ISSN (Print) | 1868-8969 |

### Other

Other | 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016 |
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Country | Italy |

City | Rome |

Period | 7/12/16 → 7/15/16 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Software

### Cite this

*43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016*[90] (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 55). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.ICALP.2016.90

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*43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016.*, 90, Leibniz International Proceedings in Informatics, LIPIcs, vol. 55, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, Rome, Italy, 7/12/16. https://doi.org/10.4230/LIPIcs.ICALP.2016.90

**Tolerant testers of image properties.** / Berman, Piotr; Murzabulatov, Meiram; Raskhodnikova, Sofya.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Tolerant testers of image properties

AU - Berman, Piotr

AU - Murzabulatov, Meiram

AU - Raskhodnikova, Sofya

PY - 2016/8/1

Y1 - 2016/8/1

N2 - 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 (that is, the smallest fraction of pixels that need to change in the image to ensure that the image satisfies 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. Prior to this work, only one tolerant testing algorithm for an image property (image partitioning) has been published. We design efficient approximation algorithms for the following fundamental questions: What fraction of pixels have to be changed in an image so that 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 a number of pixels polynomial in 1/- and independent of the size of the image. The running time of the testers for half-plane and convexity is also polynomial in 1/ϵ. Tolerant testers for these three properties were not investigated previously. For convexity and connectedness, even the existence of distance approximation algorithms with query complexity independent of the input size is not implied by previous work. (It does not follow from the VC-dimension bounds, since VC dimension of convexity and connectedness, even in two dimensions, depends on the input size. It also does not follow from the existence of non-tolerant testers.) 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. This construction might be of independent interest.

AB - 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 (that is, the smallest fraction of pixels that need to change in the image to ensure that the image satisfies 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. Prior to this work, only one tolerant testing algorithm for an image property (image partitioning) has been published. We design efficient approximation algorithms for the following fundamental questions: What fraction of pixels have to be changed in an image so that 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 a number of pixels polynomial in 1/- and independent of the size of the image. The running time of the testers for half-plane and convexity is also polynomial in 1/ϵ. Tolerant testers for these three properties were not investigated previously. For convexity and connectedness, even the existence of distance approximation algorithms with query complexity independent of the input size is not implied by previous work. (It does not follow from the VC-dimension bounds, since VC dimension of convexity and connectedness, even in two dimensions, depends on the input size. It also does not follow from the existence of non-tolerant testers.) 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. This construction might be of independent interest.

UR - http://www.scopus.com/inward/record.url?scp=85012909214&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85012909214&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ICALP.2016.90

DO - 10.4230/LIPIcs.ICALP.2016.90

M3 - Conference contribution

AN - SCOPUS:85012909214

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016

A2 - Rabani, Yuval

A2 - Chatzigiannakis, Ioannis

A2 - Sangiorgi, Davide

A2 - Mitzenmacher, Michael

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

ER -