视觉系统初级信息加工的一种数学模型——<Ⅰ>模型的空间性质

本文用广义Gabor(EQ)函数作为视觉系统进行初级信息加工的核函数.所谓广义Gabor函数,就是把原始Gabor函数中的一个独立变量扩展到三个(即空间变量从一维推广到二维,再加时间变量),并相应地引入一些参数.本文中EG取二大类形式:各向同性系统中表达式和各向异性系统表达式.当EG取某一特定形式,而且其中参数取某些特定值时,本模型可分别定性描述电生理实验中发现的几种主要感受野类型(Kuffler,Hubel和Wiesel型等).在空间频率域中引入对数变换假设后,EG经付里叶变换后可很好地合成人眼调制传递函数(MTF)     

Are V1 Simple Cells Optimized for Visual Occlusions? A Comparative Study

Simple cells in primary visual cortex were famously found to respond to low-level image components such as edges. Sparse coding and independent component analysis (ICA) emerged as the standard computational models for simple cell coding because they linked their receptive fields to the statistics of visual stimuli. However, a salient feature of image statistics, occlusions of image components, is not considered by these models. Here we ask if occlusions have an effect on the predicted shapes of simple cell receptive fields. We use a comparative approach to answer this question and investigate two models for simple cells: a standard linear model and an occlusive model. For both models we simultaneously estimate optimal receptive fields, sparsity and stimulus noise. The two models are identical except for their component superposition assumption. We find the image encoding and receptive fields predicted by the models to differ significantly. While both models predict many Gabor-like fields, the occlusive model predicts a much sparser encoding and high percentages of ‘globular’ receptive fields. This relatively new center-surround type of simple cell response is observed since reverse correlation is used in experimental studies. While high percentages of ‘globular’ fields can be obtained using specific choices of sparsity and overcompleteness in linear sparse coding, no or only low proportions are reported in the vast majority of studies on linear models (including all ICA models). Likewise, for the here investigated linear model and optimal sparsity, only low proportions of ‘globular’ fields are observed. In comparison, the occlusive model robustly infers high proportions and can match the experimentally observed high proportions of ‘globular’ fields well. Our computational study, therefore, suggests that ‘globular’ fields may be evidence for an optimal encoding of visual occlusions in primary visual cortex.     

Are receptive fields of the visual cortex detectors or spatial frequency filters?

Besides its principal maximum, the spatial frequency characteristic curve of the complex visual cortical receptive field of curarized cats also has additional maxima and also negative regions, as predicted by the theory of piecewise Fourier analysis. Comparison of responses of the complex receptive field to sinusoidal gratings completely and incompletely contained in the field and comparison of responses to sinusoidal and square-wave gratings indicate that the receptive field, as a spatial frequency filter, has linear properties. The response of the complex receptive field rises with an increase in the number of periods of the sinusoidal grating. Several periods of optimal frequency match the complex field. Receptive fields tuned to a broad band of spatial frequencies were found in neuron columns. The results confirm the view that complex receptive fields are spatial frequency filters and not detectors.I. P. Pavlov Institute of Physiology, Academy of Sciences of the USSR, Leningrad. Translated from Neirofiziologiya, Vol. 11, No. 5, pp. 403–411, September–October, 1979.     

Spatiotemporal organization of simple-cell receptive fields in area 18 of cat’s cortex

Spatiotemporal structures of receptive-fields (RF) have been studied for simple cells in area 18 of eat by measuring the temporal transfer function (TTF) over different locations (subregions) within the RF. The temporal characteristics of different subregions differed from each other in the absolute phase shift (APS) to visual stimuli. Two types of relationships can be seen: (i)The APS varied continuously from one subregion to the next: (ii) A 180°-phase jump was seen as the stimulus position changed somewhere within the receptive field. Spatiotemporal receptive field profiles have been determined by applying reverse Fourier analysis to responses in the frequency domain. For the continuous type, spatial and temporal characteristics cannot be dissociated (space time inseparable) and the spatiotemporal structure is oriented. On the contrary, the spatial and temporal characteristics for the jumping type can be dissociated (space-time separable) and the structure is not oriented in the space-time plane. Based on the APSs measured at different subregions, the optimal direction of motion and optimal spatial frequency of neurons can be predicted.     

Visual analysis in unspecialized receptive fields as an orthogonal series expansion

It was postulated on the basis of results obtained by investigating retinal receptive fields of cats that experimental data reflecting the pattern of sensitivity of the receptive field coincide with certain orthogonal polynomials. These special functions have certain properties which can be regarded as spatial characteristics of the receptive fields whose differential sensitivity they describe. Within the framework of the suggested hypothesis visual analysis in the retina from the mathematical point of view can be regarded as a process of expansion of the curve of spatial distribution of brightness within the receptive fields in an orthogonal series. Physiologically speaking the retinal receptive fields can be interpreted as filters distinguishing an elementary orthogonal harmonic from the signal. This method of analysis results in maximal economy of the coding of visual information in the retina. The concepts put forward provide a fresh explanation of a number of physiological facts and they provide precise quantitative evaluations of the response of the receptive field to an arbitrary stimulus.     

Organization of receptive fields of motion-senstitive neurons in the cat pulvinar

The distribution of 70 visually sensitive neurons in the cat pulvinar sensitive to motion in the receptive fields was studied. The experimental results showed that components with directional characteristics are present in the structure of these fields of both direction-selective and unselective neurons. In the receptive fields of direction-selective neurons the directional elements of the substructure have identical preferred directions, which coincide with the preferred directions of response to stimulus movement over the entire receptive field. The organization of receptive fields of direction-selective neurons of the visual association structure thus does not differ significantly from that of analogous fields of visual projection neurons. Directional elements of the receptive fields of direction-unselective neurons were found to have different preferred directions, thereby providing a basis for organization of the nondirectional response of the neuron to a stimulus moving across the entire receptive field.L. A. Orbeli Institute of Physiology, Academy of Sciences of the Armenian SSR, Erevan. Translated from Neirofiziologiya, Vol. 14, No. 4, pp. 339–346, July–August, 1982.     

A model for size- and rotation-invariant pattern processing in the visual system

The mapping of retinal space onto the striate cortex of some mammals can be approximated by a log-polar function. It has been proposed that this mapping is of functional importance for scale-and rotation-invariant pattern recognition in the visual system. An exact log-polar transform converts centered scaling and rotation into translations. A subsequent translation-invariant transform, such as the absolute value of the Fourier transform, thus generates overall size-and rotation-invariance. In our model, the translation-invariance is realized via the R-transform. This transform can be executed by simple neural networks, and it does not require the complex computations of the Fourier transform, used in Mellin-transform size-invariance models. The logarithmic space distortion and differentiation in the first processing stage of the model is realized via Mexican hat filters whose diameter increases linearly with eccentricity, similar to the characteristics of the receptive fields of retinal ganglion cells. Except for some special cases, the model can explain object recognition independent of size, orientation and position. Some general problems of Mellin-type size-invariance models-that also apply to our model-are discussed.     

Spatiotemporal organization of simple-cell receptive fields in area 18 of cat''s cortex

Spatiotemporal structures of receptive-fields (RF) have been studied for simple cells in area 18 of eat by measuring the temporal transfer function (TTF) over different locations (subregions) within the RF. The temporal characteristics of different subregions differed from each other in the absolute phase shift (APS) to visual stimuli. Two types of relationships can be seen: (i)The APS varied continuously from one subregion to the next: (ii) A 180°-phase jump was seen as the stimulus position changed somewhere within the receptive field. Spatiotemporal receptive field profiles have been determined by applying reverse Fourier analysis to responses in the frequency domain. For the continuous type, spatial and temporal characteristics cannot be dissociated (space time inseparable) and the spatiotemporal structure is oriented. On the contrary, the spatial and temporal characteristics for the jumping type can be dissociated (space-time separable) and the structure is not oriented in the space-time plane. Based on the APSs measured at different subregions, the optimal direction of motion and optimal spatial frequency of neurons can be predicted. Project supported by the National Natural Science Foundation of China (Grant Nos. 39570206, 39330110) and the Laboratory of Visual Information Processing, Chinese Academy of Sciences.     

Emergence of spatiotemporal receptive fields and its application to motion detection

A model of motion sensitivity as observed in some cells of area V1 of the visual cortex is proposed. Motion sensitivity is achieved by a combination of different spatiotemporal receptive fields, in particular, spatial and temporal differentiators. The receptive fields emerge if a Hebbian learning rule is applied to the network. Similar to a Linsker model the network has a spatially convergent, linear feedforward structure. Additionally, however, delays omnipresent in the brain are incorporated in the model. The emerging spatiotemporal receptive fields are derived explicitly by extending the approach of MacKay and Miller. The response characteristic of the network is calculated in frequency space and shows that the network can be considered as a spacetime filter for motion in one direction. The emergence of different types of receptive field requires certain structural constraints regarding the spatial and temporal arborisation. These requirements can be derived from the theoretical analysis and might be compared with neuroanatomical data. In this way an explicit link between structure and function of the network is established.     

Can the data of Campbell and Robson be explained without assuming fourier analysis?

Certain experiments on the detection of low-contrast gratings, occasionally cited as evidence of Fourier analysis within the visual system, are interpreted without the assumption of Fourier analysis. Theoretical curves are obtained and compared with the published experimental points, showing mostly satisfactory agreement. The computations utilize Gaussian receptive fields (on-center and off-center) for the retinal ganglion cells, spatial summation, center-surround antagonism, quasilinear response at low contrasts (X-cells), and the assumption that the first significant convergence is primarily between cells of like response type and like receptive field geometry.     

The central performance drop in texture segmentation: a simulation based on a spatial filter model

 This article presents a computational model of early visual information processing that attempts to account for the central performance drop (CPD) in texture segmentation. CPD is the finding that detection performance on short stimulus displays of line textures using orientation differences to set off the target is not maximal at the foveal center but in parafoveal areas. A comparison between a simulation and psychophysical experimental data supported the assumption that the CPD may be explained by properties of spatial frequency channels whose band-pass filter characteristics are not constant over the retina but differ with eccentricity in a defined manner. The model provided satisfactory predictions of experimental data based on densely or widely spaced line elements in texture fields. It is concluded that preattentive texture analysis might be performed by a relatively small number of simple spatial filters. Received: 14 November 1996 / Accepted in revised form: 3 June 1997     

Receptive field properties of near neighbor orientation selective neurons in the visual cortex: a modeling study

The primary visual cortex is organized into clusters of cells having similar receptive fields (RFs). A purely feedforward model has been shown to produce realistic simple cell receptive fields. The modeled cells capture a wide range of receptive field properties of orientation selective cortical cells. We have analyzed the responses of 78 nearby cell pairs to study which RF properties are clustered. Orientation preference shows strongest clustering. Orientation tuning width (hwhh) and tuning height (spikes/sec) at the preferred orientation are not as tightly clustered. Spatial frequency is also not as tightly clustered and RF phase has the least clustering. Clustering property of orientation preference, orientation tuning height and width depend on the location of cells in the orientation map. No such location dependence is observed for spatial frequency and RF phase. Our results agree well with experimental data.     

A computational theory of visual receptive fields

A receptive field constitutes a region in the visual field where a visual cell or a visual operator responds to visual stimuli. This paper presents a theory for what types of receptive field profiles can be regarded as natural for an idealized vision system, given a set of structural requirements on the first stages of visual processing that reflect symmetry properties of the surrounding world. These symmetry properties include (i) covariance properties under scale changes, affine image deformations, and Galilean transformations of space–time as occur for real-world image data as well as specific requirements of (ii) temporal causality implying that the future cannot be accessed and (iii) a time-recursive updating mechanism of a limited temporal buffer of the past as is necessary for a genuine real-time system. Fundamental structural requirements are also imposed to ensure (iv) mutual consistency and a proper handling of internal representations at different spatial and temporal scales. It is shown how a set of families of idealized receptive field profiles can be derived by necessity regarding spatial, spatio-chromatic, and spatio-temporal receptive fields in terms of Gaussian kernels, Gaussian derivatives, or closely related operators. Such image filters have been successfully used as a basis for expressing a large number of visual operations in computer vision, regarding feature detection, feature classification, motion estimation, object recognition, spatio-temporal recognition, and shape estimation. Hence, the associated so-called scale-space theory constitutes a both theoretically well-founded and general framework for expressing visual operations. There are very close similarities between receptive field profiles predicted from this scale-space theory and receptive field profiles found by cell recordings in biological vision. Among the family of receptive field profiles derived by necessity from the assumptions, idealized models with very good qualitative agreement are obtained for (i) spatial on-center/off-surround and off-center/on-surround receptive fields in the fovea and the LGN, (ii) simple cells with spatial directional preference in V1, (iii) spatio-chromatic double-opponent neurons in V1, (iv) space–time separable spatio-temporal receptive fields in the LGN and V1, and (v) non-separable space–time tilted receptive fields in V1, all within the same unified theory. In addition, the paper presents a more general framework for relating and interpreting these receptive fields conceptually and possibly predicting new receptive field profiles as well as for pre-wiring covariance under scaling, affine, and Galilean transformations into the representations of visual stimuli. This paper describes the basic structure of the necessity results concerning receptive field profiles regarding the mathematical foundation of the theory and outlines how the proposed theory could be used in further studies and modelling of biological vision. It is also shown how receptive field responses can be interpreted physically, as the superposition of relative variations of surface structure and illumination variations, given a logarithmic brightness scale, and how receptive field measurements will be invariant under multiplicative illumination variations and exposure control mechanisms.     

Invariance of visual operations at the level of receptive fields

The brain is able to maintain a stable perception although the visual stimuli vary substantially on the retina due to geometric transformations and lighting variations in the environment. This paper presents a theory for achieving basic invariance properties already at the level of receptive fields. Specifically, the presented framework comprises (i) local scaling transformations caused by objects of different size and at different distances to the observer, (ii) locally linearized image deformations caused by variations in the viewing direction in relation to the object, (iii) locally linearized relative motions between the object and the observer and (iv) local multiplicative intensity transformations caused by illumination variations. The receptive field model can be derived by necessity from symmetry properties of the environment and leads to predictions about receptive field profiles in good agreement with receptive field profiles measured by cell recordings in mammalian vision. Indeed, the receptive field profiles in the retina, LGN and V1 are close to ideal to what is motivated by the idealized requirements. By complementing receptive field measurements with selection mechanisms over the parameters in the receptive field families, it is shown how true invariance of receptive field responses can be obtained under scaling transformations, affine transformations and Galilean transformations. Thereby, the framework provides a mathematically well-founded and biologically plausible model for how basic invariance properties can be achieved already at the level of receptive fields and support invariant recognition of objects and events under variations in viewpoint, retinal size, object motion and illumination. The theory can explain the different shapes of receptive field profiles found in biological vision, which are tuned to different sizes and orientations in the image domain as well as to different image velocities in space-time, from a requirement that the visual system should be invariant to the natural types of image transformations that occur in its environment.     

Initial processing of visual information within the retina and the LGN

The initial stage of information processing by the visual system reduces the information contained in the continuous image on the retina into a discrete set of responses which are carried from the lateral geniculate nucleus (LGN) to the visual cortex.-1. The optimal sampling of the light intensity distribution in the visual environment is achieved only if each channel in the visual pathways carries undistorted information corresponding to an image element. The visual system approaches as closely as possible the scheme of optimal spatial sampling, retaining the full information on the low spatial frequency content of the object light intensity. The ideal receptive field of a sustained LGN cell is then of the form J ₁ (Kr)/Kr.-2. The experimentally determined receptive fields of sustained LGN cells (and to some extent retinal ganglion cells as well) in cat closely resemble the functional form J ₁ (Kr)/Kr. The centre-surround organization of the receptive fields is therefore understood as a scheme which leads to a maximal information flow through the visual pathways.-3. The optimal sampling scheme cannot be realized by the retina alone, because of restrictions on the size of neural networks. It is therefore constructed in two stages, ending at the LGN level. A recombination of ganglion cell signals into optimal receptive fields is a major role of the LGN.     

The Effects of Context and Attention on Spiking Activity in Human Early Visual Cortex

Here we report the first quantitative analysis of spiking activity in human early visual cortex. We recorded multi-unit activity from two electrodes in area V2/V3 of a human patient implanted with depth electrodes as part of her treatment for epilepsy. We observed well-localized multi-unit receptive fields with tunings for contrast, orientation, spatial frequency, and size, similar to those reported in the macaque. We also observed pronounced gamma oscillations in the local-field potential that could be used to estimate the underlying spiking response properties. Spiking responses were modulated by visual context and attention. We observed orientation-tuned surround suppression: responses were suppressed by image regions with a uniform orientation and enhanced by orientation contrast. Additionally, responses were enhanced on regions that perceptually segregated from the background, indicating that neurons in the human visual cortex are sensitive to figure-ground structure. Spiking responses were also modulated by object-based attention. When the patient mentally traced a curve through the neurons’ receptive fields, the accompanying shift of attention enhanced neuronal activity. These results demonstrate that the tuning properties of cells in the human early visual cortex are similar to those in the macaque and that responses can be modulated by both contextual factors and behavioral relevance. Our results, therefore, imply that the macaque visual system is an excellent model for the human visual cortex.     

Neural network model of visual cortex for determining surface curvature from images of shaded surfaces

The visual system can extract information about shape from the pattern of light and dark surface shading on an object. Very little is known about how this is accomplished. We have used a learning algorithm to construct a neural network model that computes the principal curvatures and orientation of elliptic paraboloids independently of the illumination direction. Our chief finding is that receptive fields developed by units of such model network are surprisingly similar to some found in the visual cortex. It appears that neurons that can make use of the continuous gradations of shading have receptive fields similar to those previously interpreted as dealing with contours (i.e. 'bar' detectors or 'edge' detectors). This study illustrates the difficulty of deducing neuronal function within a network solely from receptive fields. It is also important to consider the pattern of connections a neuron makes with subsequent stages, which we call the 'projective field'.