Lecture Notes in Computer Science
Representation of Medial Axis from
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- Keywords: Shape
- 2 The Proposed Model
- 2.1 Model Neurons
- 2.2 Contrast Detection Stage
- 2.3 BO Detection Stage
- 2.4 MA Detection Stage
- 3 Simulation Result
- 3.1 A Single Square
Representation of Medial Axis from Synchronous Firing of Border-Ownership Selective Cells Yasuhiro Hatori and Ko Sakai Graduate School of Systems and Information Engineering, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki, 305-8573 Japan hatori@cvs.cs.tsukuba.ac.jp sakai@cs.tsukuba.ac.jp http://www.cvs.cs.tsukuba.ac.jp/
most crucial questions in brain science. Although we can perceive figure shape correctly and quickly, without any effort, the underlying cortical mechanism is largely unknown. Physiological experiment with macaque indicated the possi- bility that the brain represents a surface with Medial Axis (MA) representation. To examine whether early visual areas could provide basis for MA representa- tion, we constructed the physiologically realistic, computational model of the early visual cortex, and examined what constraint is necessary for the represen- tation of MA. Our simulation results showed that simultaneous firing of Border- Ownership (BO) selective cells at the stimulus onset is a crucial constraint for MA representation.
Segregation of figure from ground might be the first step in the cortex toward the recognition of shape and object. Recent physiological studies have shown that around 60% of neurons in cortical areas V2 and V4 are selective to Border Ownership (BO) that tells which side of a contour owns the border, or the direction of figure, even about 20% of V1 neurons also showed the BO selectivity [1]. These reports also give an insightful idea on coding of shape in early- to intermediate- level vision. The cod- ing of shape is a major question in neuroscience as well as in robot vision. Specifi- cally, it is of great interest that how the visual information in early visual areas is processed to form the representation of shape. Physiological studies in monkeys [2] suggest that shape is coded by medial axis (MA) representation in early visual areas. The MA representation is the method that codes a surface by a set of circles inscribed along the contour of the surface. An arbitrary shape would be reproduced from the centers of the circles and their diameters. We examined whether neural circuits in early- to intermediate-level visual areas could provide a basis for MA representation. We propose that the synchronized responses of BO-selective neurons could evoke the representation of MA. The physiological study [2] showed that V1 neurons re- sponded to figure shape around 40 ms after the stimulus onset, while the latency of the cells responded to MA was about 200 ms after the onset. Physiological study on
Representation of Medial Axis from Synchronous Firing of BO Selective Cells 19 BO-selective neurons reported that their latency is around 70 ms. These results give rise to the proposal that the neurons detect contours first, then the BO is determined from the local contrast detected, and finally MA representation is constructed. It should be noted that BO could be determined from local contrast surrounding the classical receptive field (CRF), thus a single neuron with surrounding modulation would be sufficient to yield BO selectivity [3]. To examine the proposal, we con- structed physiologically realistic, firing model of Border-ownership (BO) selective neuron. We assume that the onset of a stimulus with high contrast evokes simultane- ous, strong responses of the neurons. These fast responses will propagate retinotopi- cally, thus the neurons at MA (equidistance to the contour) will be activated. Our simulation results show that even relatively small facilitation from the propagated signals yield the firing of the model-cells at the MA because of the synchronization, indicating that simultaneous firing of BO-selective cells at the stimulus onset could enable the representation of MA.
The model is comprised of three stages: (1) contrast detection stage, (2) BO detection stage, and (3) MA detection stage. The first stage extracts luminance contrast as similar to V1 simple cells. The model cells in the second stage mimic BO-selective neurons, which determine the direction of BO with respect to the border at the CRF based on the modulation from surrounding contrast up to 5 deg in visual angle from the CRF. The third stage spatially pools the responses from the second stage to test the MA represen- tation from simultaneous firing of BO cells. A schematic diagram of the model is given in Figure 1. The following sections describe the functions of each stage.
To enable the simulations of precise spatiotemporal properties, we implemented sin- gle-component firing neurons and their connections through their synapses on NEU- RON simulator [4]. The cell body of the model cell is approximated by a sphere. We set the radius and the membrane resistance to 50μm and 34.5Ωcm, respectively. The model neurons calculate the membrane potential following the Hodgkin-Huxley equa- tion [5] with constant parameter as shown in table 1. We used the biophysically realis- tic spiking neuron model because we need to examine exact timing of the firing of BO-selective cells as well as the propagtation of the signals, which cannot be realized by integrate-and-fire neuron model or any other abstract models. 2.2 Contrast Detection Stage The model cells in the first stage have response properties similar to those of V1 cells, including the contrast detection, dynamic contrast normalization and static compressive nonlinearity. The model cells in this stage detect the luminance contrast from oriented Gabor filters with distnct orientations. We limited ourselves to have four orientations for the sake of simplicity; vertical (0 and 180 deg) and horizontal (90 and 270 deg). Gabor filter is defined as follows: ( ) ( ) ( ) ( ) y x y x θ σ , σ , μ , μ , y , x gaussian θ cos y θ sin x πω 2 cos y , x G × + = , (1) 20 Y. Hatori and K. Sakai
A schematic illustration of the proposed model. Luminance contrast is detected in the first stage which is then processed to determine the direction of BO by surrounding modulation. F and S represent excitatory and inhibitory regions, respectively, for the surrounding modula- tion. The last stage detects MA based on the propagations from BO model cells. where x and y represent spatial location, μ x and μ y represent central coordinate of Gauss function, σ x and σ y represent standard deviation of Gauss function, θ and ω represent orientation and spatial frequency, respectively. We take a convolution of an input image
Representation of Medial Axis from Synchronous Firing of BO Selective Cells 21 with the Gabor filters, with dynamic contrast normalization [6] including a static, com- pressive nonlinear function. For the purpose of efficient computation, the responses of the vertical pathways (0 and 180 deg) are integrated to form vertical orientation, and so do the horizontal pathways (90 and 270 deg), which will be convenient for the computa- tion of iso-orientation suppression and cross-orientation facilitation in the next stage: ( )
) ( ) y , x G * I y , x O θ θ = , (2) ( ) ( ) ( ) y , x O y , x O y , x O 180 0 1 iso + = , (3)
( ) ( ) ( ) y , x O y , x O y , x O 270 90 1 cross + = , (4)
where I represents input image, O θ (x,y)
represents the output of convolution (*), O 1 iso
(O 1 cross )
represents integrated responses of vertical (horizontal) pathway. Table 1. The constant values for the model cells used in the simulations Parameter Value Cm 1(
2 ) E Na 50(mv)
E K -77(mv) E l -54.3(mv) g Na
0.120(S/cm 2 ) g K 0.036(S/cm 2 ) g l 0.0003(S/cm 2 )
The second stage models surrounding modulation reported in early- to intermediate- level vision for the determination of BO [3]. The model cells integrate surrounding contrast information up to5 deg in visual angle from the CRF center. We modeled the surrounding region with two Gaussians, one for inhibition and the other for facilitation, that are located asymmetrically with respect to the CRF center. If a part of the contour of an object is projected onto the excitatory (or inhibitory) region, the contrast informa- tion of the contour that is detected by the first stage is transmitted via a pulse to generate EPSP (or IPSP) of the BO-selective model-cell. In other words, the projection of a fig- ure within the excitatory region facilitates the response of the BO model cell. Con- versely, if a figure is projected onto the inhibitory region, the response of the model cell is suppressed. Therefore, surrounding contrast signals from the excitatory and inhibitory regions modulate the activity of BO model cells depending on the direction of figure. In this way, we implemented BO model cells based on the surrounding modulation. Note that Jones and her colleagues reported the orientation dependency in the surrounding modulation in monkeys' V1 cells [7]. The suppression is limited to similar orientations to the preferred orientation of the CRF (iso-orientation suppression), and facilitation is dominant for other orientations (cross-orientation facilitation). We implemented this orientation dependency for surround modulation. 22 Y. Hatori and K. Sakai Taking into account the EPSP and IPSP from the surrounds, we compute the mem- brane potential of a BO-selective model cell at time t as follows: ( )
) ( )
( ) ( ) ( ) { } ∑ , 1 , 1 1 , 1 1 1 1 1 2 , - , , , - , , , , , y x y x cross y x iso y x d t y x E y x d t y x E c y x input t y x O + + = (5)
,where x 1 and y 1 represent the spatial position of the BO-selective cell, input(x 1 ,y 1 ) represents the output of the first stage that is O 1 iso or O 1 cross , c represents a weight of synaptic connection, E
by the pulse which is generated at t-d x1,y1. E cross (x,y,t-d x1,y1 (x,y)) is the same except for input orientation. And, d x1,y1 (x,y) shows time delay in proportion to the distance be- tween the BO-selective cell whose coordinate is (x 1 ,y 1 ) and the connected cell whose coordinate is (x,y). We defined E iso (x,y,t-dx 1 ,y 1 (x,y)) and d x1,y1 (x,y) as: ( ) ( ) ( ) ( ) ( ) ( ) ( ) e - v exp σ , σ , μ , μ , y , x gaussian y , x d - t , y , x E τ y , x d - t - y x y x y , x iso 1 y , 1 x 1 1 1 1 1 1 × × = , (6)
( ) ( ) ( ) 2 1 2 1 time y , x y - y x - x c y , x d 1 1 + = , (7) where τ represents time constant, v represents membrane potential, e represents rever- sal potential, c
is a constant that converts distance to time. We set c, τ, e and c time to
0.6 (or 1.0), 10ms, 0mv and 0.2ms/μm, respectively. And E cross (x,y,t - dx 1 ,y 1 (x,y)) is also calculated similarly to eq.6. 2.4 MA Detection Stage The third stage integrates BO information from the second stage to detect the MA. A MA model cell has a single excitatory surrounding region that is represented by a Gaussian. The membrane potential of a MA model cell is given by:
(
( ) ( ) ( ) ( ) ( ) ∑ ∈ × =
1 2 2 2 2 2 2 y , x y , x y , x 2 y x y x medial 2 2 3 y , x d - t , y , x O σ , σ , μ , μ , y , x gaussian c t , y , x O , (8) where x
and y 2 represent the spatial position of the MA model cell, μ x2 and μ y2 repre-
sent the center of gauss function, c medial is a constant that we set to 1.8 (or 10.0), d x2,y2 (x,y) is calculated similarly to eq.7. Note that MA model cells receive EPSP only from BO model cells. When a BO model cell is activated, the model cell transmits a pulse to MA model cells, with its magnitude and time delay depending on the distance between the two. If a MA model cell was located equidistance to some parts of the contours, the pulses from the BO model cells on the contours reach the MA cell at the same time to evoke strong EPST that will generate a spike. On the other hand, MA model cells that are located not equidistance to the contours will never evoke a pulse. Therefore, the model cells that are located equidistance to the contours are activated based on simultaneous activation from BO model cells, the neural population of which will represent the medial axis of the object. 3 Simulation Result We carried out the simulations of the model to test whether the model shows the rep- resentation of MA. As typical examples, the results for thee types of stimuli are
Representation of Medial Axis from Synchronous Firing of BO Selective Cells 23 shown here, a square, a C-shaped figure, and a natural image of an eagle [8], as shown in Fig 1. Note that the model cells described in the previous sections are dis- tributed retinotopically to form each layer with 138 × 138 cells.
(A) (B) (C) Fig. 2. Three examples of stimuli used for the simulations. A square (A), a C-shaped figure (B), and a natural image of an eagle from Berkeley Segmentation Dataset [8] (C). 3.1 A Single Square First, we tested the model with a single square similar to that used in corresponding physiological experiment [2], as shown in Fig.1(A). Although we carried out the simula- tions retinotopically in 2D, for the purpose of graphical presentation, the responses of a horizontal cross section of the stimulus indicated by Fig. 3(A) are shown here in Fig. 3(B). The figure exhibits the firing rates for two types of the model cells, BO model cells responding to contours (solid lines at the horizontal positions of -1 and 1) and MA model cells (dotted lines at the horizontal position of 0). We observe a clear peak corre- sponding to MA at the center, as similar to the results of physiological experiments by Lee, et al. [2]. Although we tested the model along the horizontal cross section, the response of a vertical is identical. This result suggests that simultaneous firing of BO cells is capable of generating MA without any other particular constraints.
0
4 6 8 10 12 14 -1 0 1 Horizontal position sp ike
s/ 40 0m se c
(A) (B) Fig. 3. The simulation results for a square. (A) We show the responses of the cells located along the dashed line (the number of neurons is 138). Horizontal positions -1 and 1 represent the places on the vertical edges of the square. Zero represents the center of the square. (B) The responses of the model cells in firing rate along the cross-section as a function of the horizontal location. A clear peak at the center corresponding to MA is observed. |
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