Neural computation of visual imaging based on Kronecker product in the primary visual cortex
© Songnian et al; licensee BioMed Central Ltd. 2010
Received: 15 June 2009
Accepted: 26 March 2010
Published: 26 March 2010
What kind of neural computation is actually performed by the primary visual cortex and how is this represented mathematically at the system level? It is an important problem in the visual information processing, but has not been well answered. In this paper, according to our understanding of retinal organization and parallel multi-channel topographical mapping between retina and primary visual cortex V1, we divide an image into orthogonal and orderly array of image primitives (or patches), in which each patch will evoke activities of simple cells in V1. From viewpoint of information processing, this activated process, essentially, involves optimal detection and optimal matching of receptive fields of simple cells with features contained in image patches. For the reconstruction of the visual image in the visual cortex V1 based on the principle of minimum mean squares error, it is natural to use the inner product expression in neural computation, which then is transformed into matrix form.
The inner product is carried out by using Kronecker product between patches and function architecture (or functional column) in localized and oriented neural computing. Compared with Fourier Transform, the mathematical description of Kronecker product is simple and intuitive, so is the algorithm more suitable for neural computation of visual cortex V1. Results of computer simulation based on two-dimensional Gabor pyramid wavelets show that the theoretical analysis and the proposed model are reasonable.
Our results are:
1. The neural computation of the retinal image in cortex V1 can be expressed to Kronecker product operation and its matrix form, this algorithm is implemented by the inner operation between retinal image primitives and primary visual cortex's column. It has simple, efficient and robust features, which is, therefore, such a neural algorithm, which can be completed by biological vision.
2. It is more suitable that the function of cortical column in cortex V1 is considered as the basic unit of visual image processing (such unit can implement basic multiplication of visual primitives, such as contour, line, and edge), rather than a set of tiled array filter. Fourier Transformation is replaced with Kronecker product, which greatly reduces the computational complexity. The neurobiological basis of this idea is that a visual image can be represented as a linear combination of orderly orthogonal primitive image containing some local feature. In the visual pathway, the image patches are topographically mapped onto cortex V1 through parallel multi-channels and then are processed independently by functional columns. Clearly, the above new perspective has some reference significance to exploring the neural mechanisms on the human visual information processing.
Human vision can be considered as a perfect image information processing device, it can easily recognize object's position, size, and orientation, pose in space, and so on. For a long time, visual scientists, computational neuroscientists, image processing experts and computer vision researchers make great effort to explore the neural mechanism of humans' remarkable visual abilities or how the retina image is represented in the primary visual cortex, which is related with the following two questions: what kind of neural computation is actually performed by the primary visual cortex, and how this is described mathematically.
It is well known that there is a one-to-one topographical mapping between retina and cortex V1, which determines projecting relations in visual space and represents some transformations from retina to cortex V1 [1–16]. Currently, it is believed that responses of neurons in cortex V1 can be simulated by a set of tiled spatio-temporal filters array. So the function of cortex V1 is to make a spatial local Fourier Transform. Theoretically, these filters involve many processes about spatial frequencies, orientations, motion and velocities (frequencies in temporal space) [17–23].
Is this notion consistent with the actual biological visual information processes? Research in neurobiology indicates that the metabolism and decay of neurons do not affect the visual function. Every neuron performs a simple ON-OFF function and transfers the information through spikes. So a dead neuron can easily be replaced by other nearby neurons [3, 24, 25]. In case of a complicated function, this replacement would be difficult. Therefore, the simplicity of algorithms not only reduces error rates to the minimum, but also guarantees repeatability and stability, i.e. robustness.
Complex computations can be carried out by parallel computations of neuronal groups with high efficiency. So we believe that the actual computations in retinal image must be simple, repeatable and robust and be performed by individual neurons at the system level and by neuronal groups with high efficiency. They are obvious requirements for neural computations in V1.
Then, how is the topographical mapping from retina to V1 be realized? Many neurobiological experiments and visual computational models show that when every primitive (edge, corner and contour) in the visual image finds matches in the receptive fields densely distributed on V1, only the neurons whose frequencies and orientations are similar to those of the primitive fire . Therefore, the patterns of the fired neurons correspond to the primitives in the visual image, which may be represented by a topographical mapping, reflecting the adjustment of the visual image to neurons in V1, and reflecting distributive and parallel visual information processing between retina and V1[27, 28]. In this paper, we discuss the mathematical representation of this information processing and use the normalized matching measure (i.e. energy function) to measure the matching extent .
This paper proposes a model of neural information processing based on the topographical map in place of Fourier Transform. In this model, the functional columns are not considered as sets of tiled filters, but basis elements of the visual information, including orientation selectivity and feature matching [2, 30–32]. The visual image carried by spike trains is processed by Kronecker product with functional columns in V1. So synchronously parallel computations on the whole image can be performed by receptive field-to-receptive field rather than by pixel-to-pixel, and be represented by Kronecker product between matrixes. Its complexity is greatly reduced as compared with Fourier transform and other matrix computations [33, 34]. What is more, this algorithm can simulate stimulations of the elements in the visual image to cortical neurons as the embodiment of simple neural functions. The aggregative computation based on simple functions is one of plausible approaches of the visual cortex in realizing topographical mapping.
Numerical simulations are carried out to justify the above notions. In our experiments, receptive fields of neurons in V1 are simulated by hierarchical Gabor functions [35–40]. Visual image is the feature image of Lenna processed by pre-processing (filtered) in front of the pathway. Results of our algorithm are consistent with theoretical expectations.
The visual pathway ('what' pathway ) from retina across LGN to V1 is modelled. The following discussions are focussed on: 1. the optimal detection in V1 of retinal image R(x, y); 2. the optimal matching between R(x, y) and firing pattern of neuronal groups in V1; 3. Kronecker product obtained by optimal detection and matching; 4. determination of kernel function G(x, y); 5. realization of Kronecker product; 6. numerical experiments and discussions.
Image division and cortical response
Figure 1. Visual image R(x, y) is divided into M × N local patches according to a ganglion cell's receptive field.
This division involves two aspects. First, every patch ri,j(a) contains local features (such as shapes of receptive fields and orientations) at (i, j) with area (Δx × Δy), while a pixel at (i, j) only contains intensity information. Second, in the parallel multi-channel vision system, every channel only deals with a local patch. Obviously this division is consistent with parallel and multi-channel properties of a vision system, and in a topographical mapping any patch can be located in the retina, so the neural processing based on functional columns can be realized and the corresponding mathematical description is possible. A patch ri,j(a) will activate the corresponding ganglion cell and output a coded firing spike train. This is transferred across LGN into V1 as a topographical mapping. Then, the firing spike train is decoded and the image represented by ri,j(a) is restored. A Kronecker product of the restored image with functional columns Bk,l(s) in V1 [40, 41, 33] leads to firing of neurons in receptive fields having similar orientation and bandwidth (that is so-called firing under a preferential stimulus). We denote the firing pattern of one singular neuron (or simple cell) as ϕi,j(b), where b denotes the area of the receptive field of the neuron. Since, A, the retinal image area, will be enlarged in the cortex V1 , for simplicity, with a magnification factor z = 2 h (h = 0,1,2,⋯); if the area of image on the cortex V1 is denoted by B, we have B = zA = 2 h A, b = za = 2 h a. In this way, the spatial sum of all signals ϕi,j(b) in an orderly manner will form the overall firing pattern, that is , which represents a reconstruction of the retinal image R(A) on V1.
Optimum detection in V1
where is a ratio coefficient.
We can see ai,jreaches its optimum value when image patch ri,j(a) coincides with receptive fields feature gi,j(b), i.e. ai,jri,j(2 h a) = gi,j(b). So, can be taken as the measure for matching extent between neuronal receptive field gi, j(b) and patch ri,j(a) in image R(A).
Figure 2. Selective matching between ri,j(a) (a horizontal edge) and different receptive fields in functional columns. The receptive field gi,j(b) with horizontal orientation responses strongly.
It is worthy noting that the multi-scale processing function of a visual pathway is to guarantee a clear image at a proper resolution in V1. So, the optimal matching is related with some resolution. When the scale or resolution changes, the optimal matching may concentrate on different extents of details, or may include or exclude some details, as is determined by the circumstances when the vision system is "observing" the world.
The whole matching between the retinal image and receptive field patterns in V1
It can be seen later, R(x, y) and G(x, y) can be expressed as matrix form, for this reason, the formula (8) is essentially an inner product operation, it is not only more elegant on the mathematical form, but also more clear on the neurobiological significance.
Additionally, we can see the normalized matching coefficient ρ rg is equivalent to in the previous section.
Determination of integral kernel function
where γ is the ratio of the length in the major axis direction to that of in minor axis direction, usually set to a constant 0.5; σ is derivative of Gauss, determining the size of receptive fields; φ is the phase, when φ = 0; π; G(x, y) λ,σ,θ,φ,γ is symmetric about the origin; when φ = -(π/2); (π/2), G(x, y) λ,σ,θ,φ,γ is anti-symmetric about the origin; Θ is the optimal orientation, and λ is the wavelength. These arguments should be determined by experimental results from morphology and biophysics, but the exact data are not available so far . One plausible way is to set the arguments according to input image features in an input-driven topological mapping . This will be explained in the last section.
It is also the inner product, because the formula can be also expressed as matrix form. The formulae (8) and (10), as the inner product, which shows such an important neurobiological fact, that is, in the visual pathway topographical mapping indicates accurate positioning of retinal image in the visual cortex, therefore, these primitives can only respectively activate cells which are in the corresponding locations in visual cortex. Since it is a one to one excitation, scanning and convolution is no longer needed.
Comparison of inner product with convolution
We know that convolution and cross-correlation operations are essentially filtering operations in the frequency domain, which is not needed for V1, because such a filtering operation would lead to loss of high-and low-frequency information from the retinal picture. The second reason is that the scan process in such operations (convolution and cross-correlation) is a calculation with a high cost (see the section of discussion in this paper, for detail), in which, G(x, y) λ,σ,θ,φ,γ should be taken as the template to scan the whole image R(x, y) from top to bottom and from left to right. Obviously, it is not an effective method, because this scanning will cause too many responses of corresponding cells and the energy cost is too great.
Figure 3. Convolution operation for horizontal lines (A) and vertical lines (B) in image patches in R(x, y).
The use of the inner product reveals that neuron firing caused by a visual stimulus is in fact a simple function. This is a simpler neural computation than the cross-correlation and convolution operations, because it needs only multiplication between corresponding pixels of Φ(x, y) λ,σ,θ,φ,γ and R(x k , y k ). It is worth pointing out that the product R(x k , y k )Φ(x k , y l ) λ,σ,θ,φ,γ means that the retinal image R(x k , y k ) excites all cortical cells and forms a global activity pattern Φ (x, y) λ,σ,θ,φ,γ in V1. Different visual stimuli will excite and form different activity patterns corresponding to that stimulus; the differences in activity patterns occur only in a topographically connected weight coefficient of the pixels of R(x k , y k ) with the corresponding pixels of Φ(x, y) λ,σ,θ,φ,γ in a fully mapping neural computation. Generally, the weight coefficients corresponding to detailed image information are much smaller than those corresponding to contour and edge information. The intensity of the spike firing of simple cells excited by the details of the stimulus is also weaker than the intensity corresponding to contours and edges.
The inner product in equation (8) reveals the collective calculation of a simple neuronal "on" or "off" function. From this, we can see that the calculation of the inner product is very well suitable to the visual system in that it satisfies the prerequisites of efficiency, simplicity, and robustness and also provides an optimal means of detection under the condition of least-mean-square-error reconstruction.
In fact, formula (9) reflects a specific wavelet transform on retinal image R(x, y) by basis function G(x, y) λ,σ,θ,φ,γ . This formula reflects the neural firing stimulated by the retinal image at the system level. Next we will discuss how to process visual images according to this formula. Two important problems will be discussed, that is, how to divide visual image R(x, y) according to structures and functions of the visual pathway and how to express the orientation selectivity of functional columns in V1 by two-dimensional wavelet function G(x, y) λ,σ,θ,φ,γ .
Kronecker product in V1
According to neurophysiology and neuroanatomy , cortical modules are densely distributed in V1, with approximately 103 modules; the area of each module is approximately 1.8 mm × 1.8 mm, containing two function columns for both left and right eyes. Thus, the area related with every function column Bk,l(s) is 0.9 mm × 0.9 mm. At the system level, before adequate neurophysiological and neuroanatomical knowledge may be available, these function columns are assumed to have the same function and be composed of many receptive fields with different orientations and frequencies .
Figure 5. Optimal matching between a patch (upper right corner of the hat) and receptive fields of specific orientations in cortical modules [Bk,l(s)]K × L.
When all the M × N patches in retinal image [Ri,j(a)]M × Nsimultaneously (in a parallel manner) activate topographically corresponding neurons, response pattern [Φi,j(b)]M × Nis formed in V1. At the system level, this process can be described by Kronecker Product.
In formula (17), [Φi,j(b)]M × Nis the representation of retinal image [Ri,j(2 h a)]M × Nin V1, which involves an essential difference with the traditional coding.
Figure 6. Orthogonal division of a visual image.
Therefore, every patch ri,j(a) can be processed independently by functional columns. This is consistent with the neural mechanism of cortical information processing, and reduces computational complexity as well.
Currently, it is widely believed that simple cells densely distributed in V1 function similarly as a tiled set of selective spatio-temporal filters, while V1 carries out operations similar to the local complex Fourier transform. Theoretically, various kinds of neural processing about frequency, orientation, motion and other spatio-temporal operations can thus be performed [49, 50].
That is to say, G(x, y) λ,σ,θφ,γ is taken as a template to scan the whole image R(x, y) from above to bottom and from left to right. For example, if G(x, y) λ,σ,θ,φ,γ is a horizontal orientation receptive field, it will match to many edges with a similar orientation in R(x, y), so many cells in V1 are activated. The activated pattern Φ(x, y) λ,σ,θ,φ,γ is shown in Figure 3. A similar activated pattern corresponding to a vertical edge is shown in Figure 3(b). This is not an effective method for it stimulates too many responses of relative cells and costs a large amount of energy .
While in our case, in order to reconstruct retinal image [Ri,j(a)]M × N, we only calculate activated pattern ϕi,j(b) = ri,j(2 h a)Bk,l(s) of the receptive field stimulated by every patch ri,j(a) according to formula (15), and then the location of every patch is determined according to the topological mapping to V1 according to formulas (17) and (18). Finally, the whole activated pattern [Φi,j(b)]M × Nstimulated by image [Ri,j(a)]M × Nis obtained. Obviously, the related computation is much less complicated, which thus is more consistent with the multi-channel parallel processing mechanism in biology vision.
Every element ϕ(i, j) in array [Φi,j(b)]M × Ninvolves M × N times of calculations, making the total calculations for all elements as M2 × N2.
While in our case, the main computation is ϕi,j(b) = ri,j(2 h a)Bk,l(s), so the total number of calculations are M × N × K × L (K <<M, L <<N). So the computation of Kronecker product is much less complicated than that of convolution.
We already noticed that a number of other researchers have developed linear-nonlinear models based on response properties of visual neurons [52, 53], or on optimal nonlinear transformation . In essence, they are a combination of linear filtering and divisive inhibition model; all of the models have been used to model the nonlinear responses of visual neurons and primary visual cortex. In terms of our proposed model, as already pointed out that theoretical analysis and simulated results show that at the system level, the inner product operator reflects the nature of the excitation of neurons in the cortex V1 by local characteristics of the external stimuli. This is also a plausible assumption for neural computation in the cortex V1. Therefore, it may have some reference value in investigations of neural mechanisms in visual information processing.
It is understood that the retinal image must be in one-to-one correspondence with cortex V1, for all subsequent processing will extract information from V1 and the information kept in V1 is vital. Only in this way, the brain can perceive a vision through the retinal image with high fidelity. Neurophysiology shows that when a retinal image topographically projects to the visual cortex, corresponding neurons will be activated. The whole activated pattern is a copy of the retinal image with high fidelity. In view of signal processing, product ri,j(2 h a)Bk,l(s) means that receptive fields gi,j(b) in V1 are activated when stimulated by retinal image ri,j(a). Therefore, this operator is consistent with this neurobiology mechanism. It involves both the simple function of a single neuron and the population function of neuronal groups. ϕi,j(b) is the local activated pattern corresponding to patch ri,j(a). Different stimuli produce different activated patterns of neuronal groups. The signals activated by details in visual stimuli are much weaker than those activated by contours. According to our understanding of the precise reconstruction of retinal images in the visual process, and based on multi-channel parallel processing features of the visual pathway, a visual image is divided into basic image units (patches) or primitives, which are topographically mapped onto the visual cortex by a one-to-one correspondence, by means of multiplication computing, features contained in image's primitives can be extracted by thousands of visual cortical modules in parallel and synchronously, where only an inner product (or Kronecker product)is needed, then an image will be formed in the primary visual cortex. This algorithm is simple, efficient and in line with the current knowledge about the neural mechanism of visual information processing, the mathematical description is also appropriate to the visual neural computation.
Visual information processing that is actually carried out in V1 is very important, but so far our knowledge of it at the system level remains inadequate [55–57] apart from Hubel and Wiesel's discovery  in the 1960s and Field and Olshausen's sparse coding theory  in the 1990s. Therefore, the neural computation model based on available knowledge about structure and function of V1 [59–65], presented in this paper may throw some light towards that direction, of course, will require further proof in neurobiology.
The next step of the studies on the visual information processing will be focussed on functional modules in cortex V1.
Figure 7. Three representations of eight types of receptive fields in function columns in V1 calculated by Gabor function, in which orientations 0°, 10°, 20°, ⋯, 180° in turn.
Figure 8. Image reconstruction by topological mapping and Kronecker product (A)Source image Lenna; (B)Retinal image [Ri,j(2 h a)]M × N; (C) Receptive fields array Bk,l(s) of functional columns in V1; (D) The whole activated pattern of receptive field image [Φi,j(b)]M × N; (E), (F) and (G) A part of activated pattern [Φi,j(b)]M × Nin V1 calculated by formulas 14-16 (upper right corner of the hat).
Figure 9. Result (only of the upper right part of the hat shown) corresponding to the damaged function columns at i = 5, j = 9; i = 7, j = 2; i = 11, j = 15.
The authors appreciated help from Li Weige and Wang Keren for their assistance in English editing for this manuscript and would like to thank the anonymous reviewers for their detailed comments and helpful suggestions which improved the quality of our manuscript. This work was supported by the NSFC of China (Grant no. 60902058, 60371045, 60905063 & 60931003). Authors declare they have no conflict of interest.
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