The atoms of vision: Cartesian or polar?

Zetzsche, Christoph; Krieger, Gerhard; Wegmann, Bernhard

doi:10.1364/josaa.16.001554

Cited by 38 publications

(32 citation statements)

References 42 publications

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“…This type of cooperative coding will give rise to a certain form of statistical dependency among coefficients-namely, a circularly symmetric, non-factorial distribution. Such distributions were first described by Zetzsche et al, 10 who observed that the responses of pairs of Gabor functions in quadrature phase have a circularly symmetric, yet sparse (non-Gaussian), joint distribution. This structure suggests that pairs of such filter outputs are better described in polar coordinates-i.e., in terms of amplitude and phase-rather than cartesian coordinates (the responses of individual filers).…”

Section: Stable Representation Via Amplitude and Phase Factorizationmentioning

confidence: 88%

Learning real and complex overcomplete representations from the statistics of natural images

2009

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We show how an overcomplete dictionary may be adapted to the statistics of natural images so as to provide a sparse representation of image content. When the degree of overcompleteness is low, the basis functions that emerge resemble those of Gabor wavelet transforms. As the degree of overcompleteness is increased, new families of basis functions emerge, including multiscale blobs, ridge-like functions, and gratings. When the basis functions and coefficients are allowed to be complex, they provide a description of image content in terms of local amplitude (contrast) and phase (position) of features. These complex, overcomplete transforms may be adapted to the statistics of natural movies by imposing both sparseness and temporal smoothness on the amplitudes. The basis functions that emerge form Hilbert pairs such that shifting the phase of the coefficient shifts the phase of the corresponding basis function. This type of representation is advantageous because it makes explicit the structural and dynamic content of images, which in turn allows later stages of processing to discover higher-order properties indicative of image content. We demonstrate this point by showing that it is possible to learn the higher-order structure of dynamic phase-i.e., motion-from the statistics of natural image sequences.

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Section: Stable Representation Via Amplitude and Phase Factorizationmentioning

confidence: 88%

Learning real and complex overcomplete representations from the statistics of natural images

2009

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“…This was first noted with regard to pairwise joint statistics of Gabor filters of differing phase (Wegmann & Zetzsche, 1990), and later extended to filters at nearby positions, orientations and scales (Zetzsche & Krieger, 1999;Wainwright & Simoncelli, 2000). As a result, many recent models of local image statistics are members of the elliptically symmetric family (Zetzsche & Krieger, 1999;Huang & Mumford, 1999;Wainwright & Simoncelli, 2000;Hyvärinen et al, 2001;Parra et al, 2001;Srivastava et al, 2002;Sendur & Selesnick, 2002;Portilla et al, 2003;Teh et al, 2003;Gehler & Welling, 2006). This suggests that radial gaussianization may be an appropriate methodology for eliminating statistical dependencies in local image regions.…”

Section: Local Image Statisticsmentioning

confidence: 89%

“…We introduce an alternative nonlinear procedure, which we call radial gaussianization (RG), whereby the norms of whitened signal vectors are nonlinearly adjusted to ensure that the resulting output density is a spherical gaussian, whose components are thus statistically independent. We apply our methodology to natural images, whose local statistics have been modeled by a variety of different ESDs (Zetzsche & Krieger, 1999;Wainwright & Simoncelli, 2000;Huang & Mumford, 1999;Parra, Spence, & Sajda, 2001;Hyvärinen, Hoyer, & Inki, 2001;Srivastava, Liu, & Grenander, 2002;Sendur & Selesnick, 2002;Portilla, Strela, Wainwright, & Simoncelli, 2003;Teh, Welling, & Osindero, 2003;Gehler & Welling, 2006). We show that RG produces much more substantial reductions in dependency, as measured with multi-information of pairs or blocks of nearby bandpass filter responses, than does ICA.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Extraction of Independent Components of Natural Images Using Radial Gaussianization

2009

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We consider the problem of efficiently encoding a signal by transforming it to a new representation whose components are statistically independent. A widely studied linear solution, known as independent component analysis (ICA), exists for the case when the signal is generated as a linear transformation of independent nongaussian sources. Here, we examine a complementary case, in which the source is nongaussian and elliptically symmetric. In this case, no invertible linear transform suffices to decompose the signal into independent components, but we show that a simple nonlinear transformation, which we call radial gaussianization (RG), is able to remove all dependencies. We then examine this methodology in the context of natural image statistics. We first show that distributions of spatially proximal bandpass filter responses are better described as elliptical than as linearly transformed independent sources. Consistent with this, we demonstrate that the reduction in dependency achieved by applying RG to either nearby pairs or blocks of bandpass filter responses is significantly greater than that achieved by ICA. Finally, we show that the RG transformation may be closely approximated by divisive normalization, which has been used to model the nonlinear response properties of visual neurons.

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“…These observations at the neurobiological level are supported by human psychophysics. The sensitivity of human observers to quadrature pair Gabor stimuli is aligned with a polar decomposition and not a Cartesian decomposition (Zetzsche et al, 1999). In sum, these observations strongly advocate the use of angular decompositions, such as those that can be obtained with complex variables, in models of natural images.…”

Section: First Layer: Sparse and Temporally Persistent Representationmentioning

confidence: 99%

“…This step is motivated by a number of findings from natural scene statistics, neurophysiology, and human psychophysics and draws heavily from the work of Christoph Zetzsche described in Zetzsche, Krieger, & Wegmann (1999). In particular, we are motivated by two observations: The first is that although the prior over the coefficients in sparse coding models is typically factorial, the actual joint distribution of coefficients, even after learning, exhibits strong statistical dependencies in response to natural images.…”

Section: First Layer: Sparse and Temporally Persistent Representationmentioning

confidence: 99%

Learning Intermediate-Level Representations of Form and Motion from Natural Movies

Cadieu

Olshausen

2012

Neural Computation

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We present a model of intermediate-level visual representation that is based on learning invariances from movies of the natural environment.The model is composed of two stages of processing: an early feature representation layer and a second layer in which invariances are explicitly represented. Invariances are learned as the result of factoring apart the temporally stable and dynamic components embedded in the early feature representation. The structure contained in these components is made explicit in the activities of second-layer units that capture invariances in both form and motion. When trained on natural movies, the first layer produces a factorization, or separation, of image content into a temporally persistent part representing local edge structure and a dynamic part representing local motion structure, consistent with known response properties in early visual cortex (area V1). This factorization linearizes statistical dependencies among the first-layer units, making them learnable by the second layer. The second-layer units are split into two populations according to the factorization in the first layer. The form-selective units receive their input from the temporally persistent part (local edge structure) and after training result in a diverse set of higher-order shape features consisting of extended contours, multiscale edges, textures, and texture boundaries. The motion-selective units receive their input from the dynamic part (local motion structure) and after training result in a representation of image translation over different spatial scales and directions, in addition to more complex deformations. These representations provide a rich description of dynamic natural images and testable hypotheses regarding intermediate-level representation in visual cortex.

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The atoms of vision: Cartesian or polar?

Cited by 38 publications

References 42 publications

Learning real and complex overcomplete representations from the statistics of natural images

Learning real and complex overcomplete representations from the statistics of natural images

Nonlinear Extraction of Independent Components of Natural Images Using Radial Gaussianization

Learning Intermediate-Level Representations of Form and Motion from Natural Movies

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