### Abstract

Original language | English |
---|---|

Pages (from-to) | 1748-1765 |

Number of pages | 18 |

Journal | Vision Research |

Volume | 50 |

Issue number | 17 |

Early online date | 2 Jun 2010 |

DOIs | |

Publication status | Published - 6 Aug 2010 |

### Fingerprint

### Keywords

- Fourier analysis
- humans
- biological models
- orientation
- psychophysics
- visual perception

### Cite this

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*Vision Research*, vol. 50, no. 17, pp. 1748-1765. https://doi.org/10.1016/j.visres.2010.05.031

**The Riesz transform and simultaneous representations of phase, energy and orientation in spatial vision.** / Langley, Keith; Anderson, Stephen J.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The Riesz transform and simultaneous representations of phase, energy and orientation in spatial vision

AU - Langley, Keith

AU - Anderson, Stephen J

PY - 2010/8/6

Y1 - 2010/8/6

N2 - To represent the local orientation and energy of a 1-D image signal, many models of early visual processing employ bandpass quadrature filters, formed by combining the original signal with its Hilbert transform. However, representations capable of estimating an image signal's 2-D phase have been largely ignored. Here, we consider 2-D phase representations using a method based upon the Riesz transform. For spatial images there exist two Riesz transformed signals and one original signal from which orientation, phase and energy may be represented as a vector in 3-D signal space. We show that these image properties may be represented by a Singular Value Decomposition (SVD) of the higher-order derivatives of the original and the Riesz transformed signals. We further show that the expected responses of even and odd symmetric filters from the Riesz transform may be represented by a single signal autocorrelation function, which is beneficial in simplifying Bayesian computations for spatial orientation. Importantly, the Riesz transform allows one to weight linearly across orientation using both symmetric and asymmetric filters to account for some perceptual phase distortions observed in image signals - notably one's perception of edge structure within plaid patterns whose component gratings are either equal or unequal in contrast. Finally, exploiting the benefits that arise from the Riesz definition of local energy as a scalar quantity, we demonstrate the utility of Riesz signal representations in estimating the spatial orientation of second-order image signals. We conclude that the Riesz transform may be employed as a general tool for 2-D visual pattern recognition by its virtue of representing phase, orientation and energy as orthogonal signal quantities.

AB - To represent the local orientation and energy of a 1-D image signal, many models of early visual processing employ bandpass quadrature filters, formed by combining the original signal with its Hilbert transform. However, representations capable of estimating an image signal's 2-D phase have been largely ignored. Here, we consider 2-D phase representations using a method based upon the Riesz transform. For spatial images there exist two Riesz transformed signals and one original signal from which orientation, phase and energy may be represented as a vector in 3-D signal space. We show that these image properties may be represented by a Singular Value Decomposition (SVD) of the higher-order derivatives of the original and the Riesz transformed signals. We further show that the expected responses of even and odd symmetric filters from the Riesz transform may be represented by a single signal autocorrelation function, which is beneficial in simplifying Bayesian computations for spatial orientation. Importantly, the Riesz transform allows one to weight linearly across orientation using both symmetric and asymmetric filters to account for some perceptual phase distortions observed in image signals - notably one's perception of edge structure within plaid patterns whose component gratings are either equal or unequal in contrast. Finally, exploiting the benefits that arise from the Riesz definition of local energy as a scalar quantity, we demonstrate the utility of Riesz signal representations in estimating the spatial orientation of second-order image signals. We conclude that the Riesz transform may be employed as a general tool for 2-D visual pattern recognition by its virtue of representing phase, orientation and energy as orthogonal signal quantities.

KW - Fourier analysis

KW - humans

KW - biological models

KW - orientation

KW - psychophysics

KW - visual perception

UR - http://www.scopus.com/inward/record.url?scp=77955093302&partnerID=8YFLogxK

U2 - 10.1016/j.visres.2010.05.031

DO - 10.1016/j.visres.2010.05.031

M3 - Article

C2 - 20685326

VL - 50

SP - 1748

EP - 1765

JO - Vision Research

JF - Vision Research

SN - 0042-6989

IS - 17

ER -