Walsh functions provide a viable and interesting tool in the treatment of problems of optical imagery. A reformulation of Walsh functions in polar coordinates facilitates the analysis which is presented, along with a restatement of the basic properties of these functions pertinent to the purpose. Some numerical results of aberrational diffraction images are cited to illustrate the compatibility of the proposed approach. © 1977 Taylor & Francis Group, LLC.

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University of Calcutta (informally known as Calcutta University or CU) is a collegiate public state university located in Kolkata, West Bengal, India.&nbsp;Within India it is recognized as a &quot;Five-Star University&quot; and accredited &quot;A&quot; Grade by National Assessment and Accreditation Council.

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The University of Calcutta has secured the Fifth position among the Universities of India in the prestigious &quot;Indian University Ranking 2019&quot; list, released by the NIRF of the Ministry of Human Resource Development, Government of India.

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It was established on 24 January 1857, and was one of the first institutions in Asia to be established as a multidisciplinary and Western-style university. Its alumni and faculty include several heads of state, heads of government, social reformers, prominent artists, only academy award winner and Dirac medal winner in India, many Fellows of the Royal Society and five Nobel laureates as of 2019 which is highest in South Asia.

University Of Calcutta

Walsh Functions in Problems of Optical Imagery

Optica Acta

With a brief description of the origin, basic characteristics and major practical applications of Walsh functions, different forms of Walsh functions in one and two dimensions are systematically developed from the orthogonality considerations. Approximation of a continuous function over a given domain by a finite set of Walsh functions provides a piecewise constant approximation of the function. The latter has important consequences in tackling practical problems by way of opening up new techniques for analysis and synthesis. Highlights of these aspects are provided along with a brief description of Walsh Block functions and Hadamard matrices. © 2018, The Author(s).

SpringerBriefs in Applied Sciences and Technology

Walsh Functions

The set of orthogonal Walsh filters consists of groups, members of which are self-similar. The corresponding axial and transverse intensity distributions in the farfield diffraction patterns of these filters possess self-similarity. Results of our investigations on the same are presented. © 2015 Copyright SPIE.

Proceedings of SPIE - The International Society for Optical Engineering

Self-similarity in Walsh filters and wave-optical engineering

Azimuthal Walsh Functions can be utilized to obtain an orthogonal set of Azimuthal Walsh Filters where the transmission values of +1 and -1 are achieved by using 0 and pi phase respectively. The inherent asymmetry in the azimuthal Walsh filters leads to asymmetrical Point Spread Function (PSF) in the far field diffraction pattern. This report presents results of our preliminary investigation on the effects of phase asymmetry in the filters on the PSF in the Far-field patterns. © 2015 IEEE.

2015 2nd International Conference on Opto-Electronics and Applied Optics: Advances in Optical Sciences and Engineering II, IEM OPTRONIX 2015

Asymmetrical PSF by Azimuthal Walsh filters

Pupil plane filtering by radial Walsh filters is a convenient technique for tailoring the axial intensity distribution near the focal plane of a rotationally symmetric imaging system. Radial Walsh filters, derived from radial Walsh functions, form a set of orthogonal phase filters that take on values either 0 or p phase, corresponding to 1 or -1 values of the radial Walsh functions over prespecified annular regions of the circular filter. Order of these filters is given by the number of zero-crossings, or equivalently phase transitions within the domain over which the set is defined. In general, radial Walsh filters are binary phase zone plates, each of them demonstrating distinct focusing characteristics. The set of radial Walsh filters can be classified into distinct groups, where the members of each group possess self-similar structures. Self-similarity can also be observed in the corresponding axial intensity distributions. These observations provide valuable clues in tackling the inverse problem of synthesis of phase filter in accordance with prespecified axial intensity distributions. This paper reports our observations on self-similarity in radial Walsh filters of various orders and corresponding axial intensity distributions. © 2014 Optical Society of America.

Journal of the Optical Society of America A: Optics and Image Science, and Vision

Self-similarity in radial Walsh filters and axial intensity distributions in the far-field diffraction pattern

Annular Walsh filters are derived from the rotationally symmetric annular Walsh functions which form a complete set of orthogonal functions that take on values either +1 or -1 over the domain specified by the inner and outer radii of the annulus. The value of any annular Walsh function is taken as zero from the centre of the circular aperture to the inner radius of the annulus. The three values 0, +1, and -1 in an annular Walsh function can be realized in a corresponding annular Walsh filter by using transmission values of zero amplitude (i.e., an obscuration), unity amplitude and zero phase, and unity amplitude and π phase, respectively. Not only the order of the Walsh filter but also the size of the inner radius of the annulus provides an additional degree of freedom in tailoring of point spread function by using these filters for pupil plane filtering in imaging systems. In this report, we present the far-field amplitude characteristics of some of these filters to underscore their potential for effective use in several demanding applications like high-resolution microscopy, optical data storage, microlithography, optical encryption, and optical micromanipulation. © 2013 Pubali Mukherjee and Lakshminarayan Hazra.

Fulltext

Advances in Optical Technologies

Far-field diffraction properties of Annular Walsh filters

Prefaced with a brief survey of the state of the art in restoration techniques for atmospherically degraded imagery, this paper presents an analytical approach pertinent to the studies on the effects of wavefront distortion and compensation of the latter through a finite number of corrector elements. Our method is essentially based on an analysis in terms of Walsh functions which form a closed set of normal orthogonal functions and are quite appropriate to the problem. Some numerical results on phase compensation through a finite number of corrector elements are given. The analytical technique and the results point towards the feasibility of a new restoration scheme enabling one to replace the usually conceived scheme of transverse segmentation by one with a finite array of correctors placed longitudinally. The scheme is briefly outlined. © 1977 Taylor & Francis Group, LLC.

Journal	Optica Acta
Publisher	-
ISSN	0030-3909