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Nearness of Associated Rough Sets

Case Study in Image Analysis

  • Chapter
Book cover Rough Sets: Selected Methods and Applications in Management and Engineering

Part of the book series: Advanced Information and Knowledge Processing ((AI&KP))

Abstract

The focus of this chapter is on the nearness of associated sets and the answer to the question Is a pair of sets sufficiently near to be considered similar? In answering this question, we consider associated sets of a set. In particular, associated rough sets as well as associated near rough sets are introduced. A rough set X is associated with another set Y, provided that X is sufficiently near Y. In general, nonempty sets are sufficiently near, provided that the Čech distance between the sets is less than some number ε in the interval (0,∞]. An application of the proposed approach is given in the context of image analysis with emphasis on detecting patterns in visual rough sets.

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Notes

  1. 1.

    See, e.g., [30].

  2. 2.

    Using this approach, one can introduce parallel computation as a means of finding the distance between large tolerance rough sets commonly found in digital images (see, e.g., [35]).

  3. 3.

    It is seldom possible to find distinct objects in the physical world that have identical descriptions. Hence, there is interest in discovering sets derived from the physical world and that are sufficiently near each other.

  4. 4.

    The Caltech image database contains large collections of images of front and rear views of automobiles, side view of motorcycles and aircraft, front view of faces, top view of leaves as well as 550 images of various scenes, gadgets, toys, offices, and Pasadena houses in 101 categories (available at http://www.vision.caltech.edu/html-files/archive.html).

  5. 5.

    In an RGB colour space where individual pixel colours are a mixture of varying amounts of the R (red), green (G) and blue (B) primary colours. A pixel greyscale intensity i is computed using \(i = \frac{R+G+B}{3}\).

  6. 6.

    A common way to measure pixel gradient is to pick a pixel g in location (x,y) in a small n×n subimage in a greyscale image (usually, a 3×3 subimage). Then estimate the centre pixel gradient g x in the x-direction and g y in the y-direction and use

    $$\alpha(x,y)=\tan^{-1}\left[\frac{g_y}{g_x}\right].$$

    For a detailed explanation of how to determine g x , g y , see, e.g., [51, Sect. 3.6.4].

  7. 7.

    The id letters KGB in Fig 9.4 have been whimsically inverted by the aircraft owner (see, also, Fig. 10.4).

  8. 8.

    APm is available at http://wren.ece.umanitoba.ca. Caution: In attempting to reproduce results shown in Table 2, the actual values will vary from the reported nearness measurements, depending on an image region that is selected.

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Acknowledgements

Many thanks to S. Tiwari, S.A. Naimpally, M. Borkowski, A. Skowron and P. Wasilewski for their insights concerning a number of topics in this paper. This research has been supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) research grants 185986 and 194376.

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Authors and Affiliations

  1. Department of Applied Computer Science, University of Winnipeg, Winnipeg, MB, Canada

    Sheela Ramanna

  2. Computational Intelligence Laboratory, Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, MB, Canada

    Sheela Ramanna & James F. Peters

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  1. Sheela Ramanna
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  2. James F. Peters
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Correspondence to Sheela Ramanna .

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Editors and Affiliations

  1. Munich University of Applied Sciences, Lothstrasse 34, Munich, 80335, Germany

    Georg Peters

  2. Dpt. of Mathematics and Computer Science, St. Mary's University, Halifax, B3H 3C3, Nova Scotia, Canada

    Pawan Lingras

  3. University of Warsaw, Banacha 2, Warsaw, 02-097, Poland

    Dominik Ślęzak

  4. Department of Computer Science, University of Regina, Regina, S4S0A2, Saskatchewan, Canada

    Yiyu Yao

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Ramanna, S., Peters, J.F. (2012). Nearness of Associated Rough Sets. In: Peters, G., Lingras, P., Ślęzak, D., Yao, Y. (eds) Rough Sets: Selected Methods and Applications in Management and Engineering. Advanced Information and Knowledge Processing. Springer, London. https://doi.org/10.1007/978-1-4471-2760-4_11

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  • DOI: https://doi.org/10.1007/978-1-4471-2760-4_11

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