Duality of log-polar image representations in the space and spatial-frequency domains

Abstract

Abstract-In this paper, we study the result of applying a lowpass variant filtering using scaling-rotating kernels to both the spatial and spatial-frequency representations of a two-dimensional (2-D) signal (image). It is shown that if we apply this transformation to a Fourier pair, the two resulting signals can also form a Fourier pair when the filters used in each domain maintain a dual relationship. For a large class of "self-dual" filters, a perfect symmetry exists, so that the lowpass scaling-rotating variant filtering (SRVF) is the same in both domains, thus commuting with the Fourier transform operator. The lowpass SRVF of an image is often referred to as a "foveated" image, whereas its Fourier pair (the lowpass SRVF of its spectrum) can be realized as a local spectrum estimation around the point of attention. This lowpass SRVF is equivalent to a log-polar warping of the image representation followed by a lowpass invariant filtering and the corresponding inverse warping. The use of the log-polar warped representation allows us to extend the one-dimensional (1-D) scale transform to higher dimensions, in particular to images, for which we have defined a scale-rotation invariant representation. We also present an efficient implementation using steerable filters to compute both the foveated image and the local spectrum.

Approximating filtered scale-variant signals

“…While short-duration linear degradation functions might often be encountered in practice, and so (21) might be small, the problem that arises is expressed well by (22): the duration of h is controlled by the reciprocal of G. Low-pass blur functions that completely or nearly eradicate high frequencies will have large durations, hence (10) will grow quite large. This is a new interpretation of the main limitation of inverse filters: excessive and unpredictable amplification of high signal frequencies, especially when noise is present.…”

“…Prior work on scale-variant signal processing has consisted mainly in applications to modeling biological vision systems [15]- [17] including fast implementation of foveation filtering [18][19]. Scale-variant filtering has also been touched upon in the context of steerable filters [20][21], and log-polar representation of images [22]. Equations…”

Efficient shift-variant image restoration using deformable filtering (Part I)

“…However, it was not applied to perform SV filtering, but rather to obtain continuous tunable filters from a (reduced) set of discrete filters. Later on, several authors (including a coauthor of this study, see [16]) used it for SV filtering, although no one, to the best of our knowledge, for the purpose of image restoration.…”

“…Perona's main motivation was to obtain a set of tunable kernels for mimicking early vision. However, another direct application of this approach is SV filtering (see, e.g., [16]) in image processing: the local integration of a signal with a locally varying linear combination of some reference kernels can be expressed as a locally varying linear combination of the convolutions of the original signal with those kernels (as explained in detail in Section 3). From a different, analysis-driven point of view, PCA-based optimal basis functions have been recently used to describe with increased accuracy the response of astronomical instruments [17][18][19][20] in a more compact and numerically-stable way than other recent techniques.…”

Rotation invariant feature lines transform for image matching

“…In the spatial domain, logpolar schemes have been used to model the strongly inhomogeneous sampling of the retinal image by the HVS. 19 Polar transform has two principal advantages: rotation invariance and scale invariance. Lines through the center of the image will be mapped into horizontal lines in the polar image and the concentric circle into vertical lines.…”