Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters.
01 Jul 1985-Journal of The Optical Society of America A-optics Image Science and Vision (Optical Society of America)-Vol. 2, Iss: 7, pp 1160-1169
TL;DR: Evidence is presented that the 2D receptive-field profiles of simple cells in mammalian visual cortex are well described by members of this optimal 2D filter family, and thus such visual neurons could be said to optimize the general uncertainty relations for joint 2D-spatial-2D-spectral information resolution.
Abstract: Two-dimensional spatial linear filters are constrained by general uncertainty relations that limit their attainable information resolution for orientation, spatial frequency, and two-dimensional (2D) spatial position. The theoretical lower limit for the joint entropy, or uncertainty, of these variables is achieved by an optimal 2D filter family whose spatial weighting functions are generated by exponentiated bivariate second-order polynomials with complex coefficients, the elliptic generalization of the one-dimensional elementary functions proposed in Gabor’s famous theory of communication [ J. Inst. Electr. Eng.93, 429 ( 1946)]. The set includes filters with various orientation bandwidths, spatial-frequency bandwidths, and spatial dimensions, favoring the extraction of various kinds of information from an image. Each such filter occupies an irreducible quantal volume (corresponding to an independent datum) in a four-dimensional information hyperspace whose axes are interpretable as 2D visual space, orientation, and spatial frequency, and thus such a filter set could subserve an optimally efficient sampling of these variables. Evidence is presented that the 2D receptive-field profiles of simple cells in mammalian visual cortex are well described by members of this optimal 2D filter family, and thus such visual neurons could be said to optimize the general uncertainty relations for joint 2D-spatial–2D-spectral information resolution. The variety of their receptive-field dimensions and orientation and spatial-frequency bandwidths, and the correlations among these, reveal several underlying constraints, particularly in width/length aspect ratio and principal axis organization, suggesting a polar division of labor in occupying the quantal volumes of information hyperspace. Such an ensemble of 2D neural receptive fields in visual cortex could locally embed coarse polar mappings of the orientation–frequency plane piecewise within the global retinotopic mapping of visual space, thus efficiently representing 2D spatial visual information by localized 2D spectral signatures.
Citations
More filters
Book•
10 Mar 2005TL;DR: This unique reference work is an absolutely essential resource for all biometric security professionals, researchers, and systems administrators.
Abstract: A major new professional reference work on fingerprint security systems and technology from leading international researchers in the field Handbook provides authoritative and comprehensive coverage of all major topics, concepts, and methods for fingerprint security systems This unique reference work is an absolutely essential resource for all biometric security professionals, researchers, and systems administrators
3,821 citations
TL;DR: A method for rapid visual recognition of personal identity is described, based on the failure of a statistical test of independence, which implies a theoretical "cross-over" error rate of one in 131000 when a decision criterion is adopted that would equalize the false accept and false reject error rates.
Abstract: A method for rapid visual recognition of personal identity is described, based on the failure of a statistical test of independence. The most unique phenotypic feature visible in a person's face is the detailed texture of each eye's iris. The visible texture of a person's iris in a real-time video image is encoded into a compact sequence of multi-scale quadrature 2-D Gabor wavelet coefficients, whose most-significant bits comprise a 256-byte "iris code". Statistical decision theory generates identification decisions from Exclusive-OR comparisons of complete iris codes at the rate of 4000 per second, including calculation of decision confidence levels. The distributions observed empirically in such comparisons imply a theoretical "cross-over" error rate of one in 131000 when a decision criterion is adopted that would equalize the false accept and false reject error rates. In the typical recognition case, given the mean observed degree of iris code agreement, the decision confidence levels correspond formally to a conditional false accept probability of one in about 10/sup 31/. >
3,399 citations
TL;DR: The results obtained with six natural images suggest that the orientation and the spatial-frequency tuning of mammalian simple cells are well suited for coding the information in such images if the goal of the code is to convert higher-order redundancy into first- order redundancy.
Abstract: The relative efficiency of any particular image-coding scheme should be defined only in relation to the class of images that the code is likely to encounter. To understand the representation of images by the mammalian visual system, it might therefore be useful to consider the statistics of images from the natural environment (i.e., images with trees, rocks, bushes, etc). In this study, various coding schemes are compared in relation to how they represent the information in such natural images. The coefficients of such codes are represented by arrays of mechanisms that respond to local regions of space, spatial frequency, and orientation (Gabor-like transforms). For many classes of image, such codes will not be an efficient means of representing information. However, the results obtained with six natural images suggest that the orientation and the spatial-frequency tuning of mammalian simple cells are well suited for coding the information in such images if the goal of the code is to convert higher-order redundancy (e.g., correlation between the intensities of neighboring pixels) into first-order redundancy (i.e., the response distribution of the coefficients). Such coding produces a relatively high signal-to-noise ratio and permits information to be transmitted with only a subset of the total number of cells. These results support Barlow's theory that the goal of natural vision is to represent the information in the natural environment with minimal redundancy.
3,077 citations
Additional excerpts
...G(f) = expi-[log(f/f 0 )] 2 /2[log(oi/f0 )] 2 I, (12)...
[...]
TL;DR: Algorithms developed by the author for recognizing persons by their iris patterns have now been tested in many field and laboratory trials, producing no false matches in several million comparison tests.
Abstract: Algorithms developed by the author for recognizing persons by their iris patterns have now been tested in many field and laboratory trials, producing no false matches in several million comparison tests. The recognition principle is the failure of a test of statistical independence on iris phase structure encoded by multi-scale quadrature wavelets. The combinatorial complexity of this phase information across different persons spans about 249 degrees of freedom and generates a discrimination entropy of about 3.2 b/mm/sup 2/ over the iris, enabling real-time decisions about personal identity with extremely high confidence. The high confidence levels are important because they allow very large databases to be searched exhaustively (one-to-many "identification mode") without making false matches, despite so many chances. Biometrics that lack this property can only survive one-to-one ("verification") or few comparisons. The paper explains the iris recognition algorithms and presents results of 9.1 million comparisons among eye images from trials in Britain, the USA, Japan, and Korea.
2,829 citations
Cites methods from "Uncertainty relation for resolution..."
...Each isolated iris pattern is then demodulated to extract its phase information using quadrature 2D Gabor wavelets ( Daugman 1985, 1988, 1994 )....
[...]
10 Dec 2002
TL;DR: Algorithms developed by the author for recognizing persons by their iris patterns have now been tested in many field and laboratory trials, producing no false matches in several million comparison tests.
Abstract: The principle that underlies the recognition of persons by their iris patterns is the failure of a test of statistical independence on texture phase structure as encoded by multiscale quadrature wavelets. The combinatorial complexity of this phase information across different persons spans about 249 degrees of freedom and generates a discrimination entropy of about 3.2 bits/mm/sup 2/ over the iris, enabling real-time decisions about personal identity with extremely high confidence. Algorithms first described by the author in 1993 have now been tested in several independent field trials and are becoming widely licensed. This presentation reviews how the algorithms work and presents the results of 9.1 million comparisons among different eye images acquired in trials in Britain, the USA, Korea, and Japan.
2,437 citations
Cites background from "Uncertainty relation for resolution..."
...D integral; is the raw iris image in a dimensionless polar coordinate system that is size- and translation-invariant and which corrects for pupil dilation as explained in a later section; and are the multiscale 2-...
[...]
References
More filters
TL;DR: This method is used to examine receptive fields of a more complex type and to make additional observations on binocular interaction and this approach is necessary in order to understand the behaviour of individual cells, but it fails to deal with the problem of the relationship of one cell to its neighbours.
Abstract: What chiefly distinguishes cerebral cortex from other parts of the central nervous system is the great diversity of its cell types and interconnexions. It would be astonishing if such a structure did not profoundly modify the response patterns of fibres coming into it. In the cat's visual cortex, the receptive field arrangements of single cells suggest that there is indeed a degree of complexity far exceeding anything yet seen at lower levels in the visual system. In a previous paper we described receptive fields of single cortical cells, observing responses to spots of light shone on one or both retinas (Hubel & Wiesel, 1959). In the present work this method is used to examine receptive fields of a more complex type (Part I) and to make additional observations on binocular interaction (Part II). This approach is necessary in order to understand the behaviour of individual cells, but it fails to deal with the problem of the relationship of one cell to its neighbours. In the past, the technique of recording evoked slow waves has been used with great success in studies of functional anatomy. It was employed by Talbot & Marshall (1941) and by Thompson, Woolsey & Talbot (1950) for mapping out the visual cortex in the rabbit, cat, and monkey. Daniel & Whitteiidge (1959) have recently extended this work in the primate. Most of our present knowledge of retinotopic projections, binocular overlap, and the second visual area is based on these investigations. Yet the method of evoked potentials is valuable mainly for detecting behaviour common to large populations of neighbouring cells; it cannot differentiate functionally between areas of cortex smaller than about 1 mm2. To overcome this difficulty a method has in recent years been developed for studying cells separately or in small groups during long micro-electrode penetrations through nervous tissue. Responses are correlated with cell location by reconstructing the electrode tracks from histological material. These techniques have been applied to
12,923 citations
"Uncertainty relation for resolution..." refers background in this paper
...It should be noted that 2D Gabor functions (1) have the same functional form in both 2D domains, as can be verified by applying the similarity theorem and the shift/modulation theorem to Eqs....
[...]
...(1)], expresses the theoretical limit of joint 2D resolution in the two 2D domains....
[...]
...(1) that if (Ax)/(Ay) = A, then also (Av)/Au) = A....
[...]
TL;DR: The contrast thresholds of a variety of grating patterns have been measured over a wide range of spatial frequencies and the results show clear patterns of uniformity in the response to grating noise.
Abstract: 1. The contrast thresholds of a variety of grating patterns have been measured over a wide range of spatial frequencies.2. Contrast thresholds for the detection of gratings whose luminance profiles are sine, square, rectangular or saw-tooth waves can be simply related using Fourier theory.3. Over a wide range of spatial frequencies the contrast threshold of a grating is determined only by the amplitude of the fundamental Fourier component of its wave form.4. Gratings of complex wave form cannot be distinguished from sine-wave gratings until their contrast has been raised to a level at which the higher harmonic components reach their independent threshold.5. These findings can be explained by the existence within the nervous system of linearly operating independent mechanisms selectively sensitive to limited ranges of spatial frequencies.
3,073 citations
"Uncertainty relation for resolution..." refers background in this paper
...Equation (4) expresses explicitly the fact that for a fixed spatial aspect ratio A, there is a positive correlation between spatial-frequency bandwidth and orientation bandwidth....
[...]
...(4) predicts for this 2D Gabor filter that A01/2 = 190, and their empirical regression line predicts 170; when Aw = 1....
[...]
...A01/2 = arcsin[X (2"- - 1)| (2A- + 1)1 (4)...
[...]
TL;DR: Among other things, it is shown that many stirate cells have quite narrow spatial bandwidths and at a given retinal eccentricity, the distribution of peak frequency covers a wide range of frequencies; these findings support the basic multiple channel notion.
1,437 citations
Book•
01 Jan 1971
TL;DR: A second edition was begun in 1970, the aim was to retain the original format, but to expand the content, especially in the areas of digital communications and com puter techniques for speech signal processing.
Abstract: The first edition of this book has enjoyed a gratifying existence. 1s sued in 1965, it found its intended place as a research reference and as a graduate-Ievel text. Research laboratories and universities reported broad use. Published reviews-some twenty-five in number-were universally kind. Subsequently the book was translated and published in Russian (Svyaz; Moscow, 1968) and Spanish (Gredos, S.A.; Madrid, 1972). Copies of the first edition have been exhausted for several years, but demand for the material continues. At the behest of the publisher, and with the encouragement of numerous colleagues, a second edition was begun in 1970. The aim was to retain the original format, but to expand the content, especially in the areas of digital communications and com puter techniques for speech signal processing. As before, the intended audience is the graduate-Ievel engineer and physicist, but the psycho physicist, phonetician, speech scientist and linguist should find material of interest."
1,386 citations