Mellin - Riesz transforms and linear summation across scale and orientation

Keith Langley, Stephen J. Anderson

Research output: Contribution to journalConference abstract

Abstract

How the visual system combines the responses of its orientation and spatial frequency tuned filters is a fundamental issue. Starting with the 3-D Newtonian potential, defined by the original image coordinates and a 'dummy' scale variable, we Taylor expand this 3-D holomorphic function using cascades of first-order constraints along the direction of the maximum local gradient. In collapsing this 3-D signal representation using a 2-D Fourier transform, the resulting operator delivers the Riesz transform and also provides rules by which filters tuned to different scale and orientation may be combined. Unlike the visual system, filters derived from a Taylor series expansion are neither distributed nor efficient. We show how Taylor's expansion maybe embedded in a polar decomposition of the image signal from which both distributed and efficient signal representations can be encoded across scale and orientation by 'steering'. Radial (scale)computations are, however, largely unconstrained in our approach, which allows us to introduce a radial Mellin-like transform via complex log-exponential filters to facilitate scale-invariant computations. We implement the Mellin - Riesz transform using a distributed signal representation, after which we linearly compress across scale and orientation. Image features are then detected using Bayesian methods for model selection. We show: (i) that the inclusion of an image signal's mean reduces noise when identifying image features, and (ii) that higher-order image features (eg saddle points) can also be identified. We predict that the power invariance afforded by the compressed Mellin - Riesz transform may help to explain fundamental mechanisms that drive visual adaptation.
Original languageEnglish
Article numberT22
Pages (from-to)111
Number of pages1
JournalPerception
Volume40
Issue number1
Publication statusPublished - 1 Jan 2011
Event1st French–British Conference on Visual Perception - Paris, France
Duration: 17 Dec 201018 Dec 2010

Fingerprint Dive into the research topics of 'Mellin - Riesz transforms and linear summation across scale and orientation'. Together they form a unique fingerprint.

  • Research Output

    • 3 Meeting abstract

    Retinal inhomogeneity and the witch's hat: contrast sensitivity declines as a bi-linear function of eccentricity in each direction

    Baldwin, A., Meese, T. S. & Baker, D., 1 Jan 2011, In : Perception. 40, 1, p. 112 1 p., P3.

    Research output: Contribution to journalConference abstract

  • The scale dependence of binocular fusion, suppression and diplopia

    Wallis, S. & Georgeson, M. A., 1 Jan 2011, In : Perception. 40, 1, p. 115-116 2 p., P12.

    Research output: Contribution to journalConference abstract

  • Varying extrinsic uncertainty affects the slope and position of the psychometric function for contrast detection and contrast discrimination

    Baker, D. & Meese, T., 2011, In : Perception. 40, 1, p. 113 1 p., P4.

    Research output: Contribution to journalConference abstract

  • Cite this