NEURAL DYNAMICS OF 3-D SURFACE PERCEPTION:
FIGURE-GROUND SEPARATION AND LIGHTNESS PERCEPTION

Frank Kelly and Stephen Grossberg

Perception and Psychophysics, in press

APPENDIX AND PARAMETER TABLE

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A.1 General Introduction

This section describes the FACADE model's BCS and FCS equations. These equations are similar to those in Grossberg and McLoughlin (1997) and Gove, Grossberg and Mingolla (1995) but there are several refinements: LGN, simple cell and complex stages better sharpen the responses of cells to boundaries. The boundary grouping process incorporates inhibitory feedback from bipoles at other positions and orientations that helps to break T-junctions. The remaining stages are unchanged, except again for minor parameter differences. See Grossberg and McLoughlin (1997) for a more complete discussion of the basic equations.

A.2 LGN ON and OFF Channels

The model LGN discounts the illuminant and computes Weber-law modulated and normalized estimates of image contrasts above an adaptation level. The LGN ON activities tex2html_wrap_inline2074 and OFF activities tex2html_wrap_inline2076 are described by on-center off-surround and off-center on-surround networks, respectively; that obey membrane, or shunting, equations (Grossberg, 1973, 1983; Hodgkin, 1964):

equation 1

and

equation 2

where the decay constant tex2html_wrap_inline2078 , the upper and lower activity bounds are tex2html_wrap_inline2080 and the center tex2html_wrap_inline2082 and surround tex2html_wrap_inline2084 are defined by Gaussian kernels:

equation 3

equation 4

where

equation 5

and

equation 6

To give the ON and OFF signals the same total strength, we use tex2html_wrap_inline2086 where the width of center and surround are described by tex2html_wrap_inline2088 . At equilibrium:

equation106

and

equation123

The difference of these ON and OFF activities is computed to generate opponent output signals:

equation140

equation163

The output is rectified using tex2html_wrap_inline2090

A.3 Simple Cells

Model simple cells respond to oriented contrasts in the image. They respond to a prescribed contrast polarity. Even-symmetric and odd-symmetric simple cell receptive fields centered on the two-dimensional location (i,j) and of orientation k were defined using even and odd Gabor kernels:

equation193

and

equation208

where

equation223

and

equation238

For vertical cells at Scale 1 : tex2html_wrap_inline2096 . For horizontal cells at Scale 1 : tex2html_wrap_inline2098 . For vertical cells at the larger Scale 2 : tex2html_wrap_inline2100 . For horizontal cells at Scale 2 : tex2html_wrap_inline2102 . These parameters are slightly smaller than those in the Grossberg and McLoughlin (1997) implementation to make edges slightly sharper. The extent of the receptive fields of the cells at each scale are as follows: For Scale 1 : tex2html_wrap_inline2104 , and for Scale 2 : tex2html_wrap_inline2106 .

A.4 Complex Cells

Model complex cells pool signals from like-oriented simple cells that are sensitive to opposite contraxt polarities. They also pool left and right eye input to compute binocular disparity. Different cell sizes, or scales, can compute different disparity ranges. The two scales are repeated at the complex cell stage. Scale 1 contains two pools of disparity-sensitive cells (near-zero disparity and a disparity of 3 pixels). Scale 2 contains three pools of such cells (near-zero disparity, a disparity of 3 pixels, and a disparity of 7 pixels). Two fields of monocular complex cells are used to represent the left and right images. The dynamics of the complex cell stage are defined mathematically as follows:

equation266

where: tex2html_wrap_inline2108 The inhibitory signal function

equation283

where, tex2html_wrap_inline2110 for all scales and disparities. tex2html_wrap_inline2112 is the excitatory input formed by the binocular filter:

eqnarray295

The binocular complex cell receptive fields are as follows: For scale 1, L, R = 20 ; for scale 2, L, R = 28 . Kernels satisfy:

equation348

and

equation358

The inhibitory connections between complex cells tuned to different disparities (possibly at different scales) obey:

equation368

for tex2html_wrap_inline2118 . The strength of the inhibitory connections between complex cells depends on the scale and disparity of the pool of cells in question. For cells at Scale 1 whose disparity is d : tex2html_wrap_inline2122 tex2html_wrap_inline2124 For cells at Scale 2 with disparity d : tex2html_wrap_inline2128 tex2html_wrap_inline2130 . In (A19), tex2html_wrap_inline2132 determines the shift in the center of the Gaussians between disparities d and e. Also

equation393

where for Scale 1, tex2html_wrap_inline2138 whereas for Scale 2: tex2html_wrap_inline2138

A.4.1 Horizontal Complex Cells

Cells sensitive to horizontal boundaries are spatially sharpened using an on-center off-surround network:

equation412

where tex2html_wrap_inline2142 and tex2html_wrap_inline2144 . The Gaussian kernels are:

equation431

and

equation437

where tex2html_wrap_inline2146 . For Scale 1, tex2html_wrap_inline2148 . For Scale 2 : tex2html_wrap_inline2150 . At equilibrium, after rectification to generate output signals, we find:

equation449

A.5 Hypercomplex Cells : Spatial Competition

Hypercomplex cells carry out spatial and orientational competition in response to complex cell inputs. The spatial competition models the neural process of endstopping. The spatial competition uses an on-center off-surround network that includes both excitatory and inhibitory shunting feedback from the bipole cells, rather than just the additive excitatory feedback used by Grossberg and McLoughlin (1997):

equation467

At equilibrium:

equation481

where tex2html_wrap_inline2152 T= 0.0000189 , and the feedback terms tex2html_wrap_inline2156 are defined as follows:

equation502

and

equation506

where tex2html_wrap_inline2158 for both scales. Term tex2html_wrap_inline2160 provides positive feedback from like-oriented bipole cells (see Section A6) to help complete boundaries, whereas tex2html_wrap_inline2162 sums bipole cell outputs across orientations and space to provide negative feedback with which to supress weaker boundaries. This orientational competition works across space to break the stems of T-junctions from their tops. The feedback signal functions f and g are linear, with gains that depend upon scale. Thus for Scale 1, tex2html_wrap_inline2168 , and tex2html_wrap_inline2170 , whereas for Scale 2, tex2html_wrap_inline2172 , and tex2html_wrap_inline2174 . The feedforward excitatory term is:

equation521

with Gaussian kernels

equation527

The inhibitory term is:

equation543

where

equation549

For Scale 1 and Scale 2, tex2html_wrap_inline2176. The size of the kernels is tex2html_wrap_inline2178 for both Scales 1 and 2. The output signals from this stage are tex2html_wrap_inline2180 .

A.5.1. Hypercomplex Cells : Orientational Competition

The hypercomplex competition between orientations obeys:

equation571

where tex2html_wrap_inline2182 and F(x) = 20x . This inhibitory feedback term F(x) comes from those monocular FIDO cells which represent nearer depths (i.e., smaller disparities) as defined by equation (A64). The excitatory input is

equation590

where the amount of inhibition between orientations tex2html_wrap_inline2188 is

equation597

The inhibitory input is

equation609

with

equation615

For both Scales: tex2html_wrap_inline2190 and tex2html_wrap_inline2192 . Solving at equilibrium and rectifying yields the output signals

equation629

A.6 Cooperative Bipole Cells

The bipole cells initiate boundary grouping by collecting oriented signals from two oriented branches of their receptive fields. Bipole cell activity satisfies:

equation649

where tex2html_wrap_inline2194 bounds each branch's activity:

equation657

and D = 0.1. The output threshold, tex2html_wrap_inline2198 , in h helps ensure that both lobes are active before a bipole cell fires :

equation666

Here tex2html_wrap_inline2202 . Terms tex2html_wrap_inline2204 and tex2html_wrap_inline2206 in (A39) correspond to two oriented branches of the bipole cell receptive field:

equation675

and

equation683

where orientation R is perpendicular to orientation r, a = 1.0, b = 2.0 . The subfraction of mutually perpendicular input prevents colinear grouping from crossing regions that contain other, non-colinear contrasts. For Scale 1 and Scale 2: tex2html_wrap_inline2214 . The bipole cell's receptive fields are implemented as in Gove et al. (1995). In particular, the branches of the bipole cell receptive field obey a Gaussian weighting operator:

equation696

Term tex2html_wrap_inline2216 modulates filter values based on their distance from the bipole's center, where tex2html_wrap_inline2218 is the optimal distance from the center:

equation704

In (A45), we set tex2html_wrap_inline2220 and tex2html_wrap_inline2222 . In the current implementation the input images were 164 x 204 pixels in size and tex2html_wrap_inline2224 was chosen to make the bipoles 40 pixels in length. Term tex2html_wrap_inline2226 favors tangent values closer to the orientation of the bipole's main axis:

equation718

where tex2html_wrap_inline2228 . Term tex2html_wrap_inline2230 measures the similarity of the orientation of a point tex2html_wrap_inline2232 and the angle formed by the tangent at that point. The tangent defines the optimal orientation for that point and filter element orientations closer to this optimal value will have greater strength than those at larger angular separations:

equation728

with tex2html_wrap_inline2234 .

Bipole Cell Feedback to Hypercomplex Cells : Orientational Competition

Unlike in Gove et al. (1995), the bipole-to-hypercomplex cell feedback is calculated directly from the bipole cell output, as in equations (A25) - (A28).

A.7 Monocular Filling-in and Monocular FIDOs

The monocular FIDO ON cell activities tex2html_wrap_inline2238 and OFF cell activities tex2html_wrap_inline2240 diffuse the FCS outputs tex2html_wrap_inline2242 and tex2html_wrap_inline2244, respectively, from the monocular preprocessing stage. Boundary outputs create resistive barriers to the diffusion process. Filling-in obeys the following equations (Grossberg and Todorovic, 1988):

equation754

and

equation770

where N consists of the four nearest neighbors to a cell and where the boundary-dependent diffusion coefficient obeys

equation787

where tex2html_wrap_inline2248 and tex2html_wrap_inline2250. The boundary term

equation795

Thus any large boundary value at the nearest neighbor positions reduces the diffusion coefficient and thereby blocks filling-in. At equilibrium:

equation800

and

equation813

A.7.1 Output from Monocular FIDOs

Outputs from the monocular FIDOs generate both boundary pruning and surface pruning signals. In order to generate such signals at the contours of filled-in regions, the filled-in activities are processed by a contrast-sensitive on-center off-surround network as follows:

equation827

and

equation843

where tex2html_wrap_inline2252 and tex2html_wrap_inline2254 . The excitatory input

equation862

has the on-center kernel

equation868

where tex2html_wrap_inline2256. The inhibitory input

equation879

has the off-surround kernel

equation885

where S = 0.181 and tex2html_wrap_inline2260 . At equilibrium:

equation897

and

equation916

These contrast-sensitive signals were then subtracted and rectified to generate double-opponent output signals:

equation935

and

equation944

A.8 Boundary Pruning Signals to Hypercomplex Cells

In order to transform unoriented FIDO activities into boundary pruning signals at oriented hypercomplex cell activities in equation (A33), they are processed by oriented filters:

equation956

where

equation967

and

equation982

with odd-symmetric kernels

equation997

and even-symmetric kernels

equation1013

All parameters for these FCS simple cells are the same as for the BCS simple cells in Section A3.

A.9 Surface Pruning Signals to the Binocular FIDOs

The binocular FIDOs receive inputs from both the left and right eye monocular FIDOs and the monocular preprocessing stage. Inputs from the monocular preprocessing stage are excitatory and are binocularly matched at the binocular FIDOs. Inputs from the monocular FIDOs are inhibitory surface pruning signals. Both excitatory and inhibitory signals are combined binocularly via the following equations:

equation1028

and

equation1046

where tex2html_wrap_inline2262 and tex2html_wrap_inline2264 . The excitatory term tex2html_wrap_inline2266 matches left and right monocular preprocessing signals:

equation1068

and

equation1076

The Binocular FIDOs also receive inhibitory surface pruning inputs from monocular FIDO cells that represent smaller disparities:

equation1084

and

equation1091

where tex2html_wrap_inline2268 and tex2html_wrap_inline2270 are the monocular FIDO outputs in (A62) and (A63). The values of the saturation terms tex2html_wrap_inline2272 and tex2html_wrap_inline2274 are chosen to enable the monocular FIDO outputs to inhibit monocular preprocessing signals and thereby prevent filling-in of occluded regions at the Binocular FCS. Solving at steady state and rectifying yields:

equation1102

and

equation1112

The diffusive spread of binocular FIDO activity is defined by the following equations:

equation1126

and

equation1142

where N is the set of nearest neighbors, M = 0.1, and the boundary gating term is defined by

equation1160

where tex2html_wrap_inline2280 and tex2html_wrap_inline2282 In (A79), the boundary signals tex2html_wrap_inline2284 are enriched by adding the boundaries Z of nearer objects to the boundaries of farther objects, thereby preventing occluded regions of the binocular FIDO from filling-in and giving a percept of transparency where none exists. Thus:

equation1172

where tex2html_wrap_inline2288 is defined by (A51). Solving at equilibrium yields:

equation1179

and

equation1194

These equations were solved at equilibrium using the Y12M package (Zlatev, Wasniewski & Schaumburg, 1981) because solving these equations using 4th-order Runge-Kutta (step size 0.0000025) or adaptive step Runge-Kutta was computationally intractable. The Y12M package uses an approximation algorithm to calculate final FCS values based on filling-in. Unfortunately one of the problems with this approximation is that it allows boundaries to leak color. This problem was solved by increasing the strength of boundaries by setting tex2html_wrap_inline2290 in (A79) to 100,000. In the Munker-White simulation, because of the sparsity of filling-in signals at the far depth binocular FCS (Figure 15f) relative to the area these signals must fill in, we set tex2html_wrap_inline2292 and tex2html_wrap_inline2294

Equilibrium opponent ON - OFF and OFF - ON values are calculated as follows:

equation1214

and

equation1226

These are the binocular FIDO activities that are plotted in the simulations.


The parameters used in the simulations are listed below. Note: Older browsers may substitute for the Greek letters. Please download the postscript or PDF versions for complete accuracy.


LGN ON and OFF Channels

LGN ON and OFF Channels

a1

100

U1

50

L1

50

A1

1.0

A2

1.03361

sc

0.5

ss

1.5

Simple Cells

Scale 1

Vertical Cells

Horizontal Cells

spk

1.0

0.75

sqk

0.75

1.0

Scale 2

Vertical Cells

Horizontal Cells

spk

1.25

1.0

sqk

1.0

1.25

Complex Cells

a2

0.01

b

15.0

U2

1.0

L2

1.0

Ge

0.2

 

Scale 1

Scale 2

Add

2.5

2.5

Ade

1.0

1.0

Ad,mon

0.25

0.15

mdd

0.05

0.05

mde

0.05

0.05

md,mon

0.075

0.075

mcdd

0.015

0.015

msdd

0.5

0.5

Horizontal Complex Cells

a3

0.1

U3

1.0

L3

1.0

A1

1.0

A2

1.0

 

Scale 1

Scale 2

mc

1.5

0.5

ms

0.06

0.06

Hypercomplex Cells: Spatial Competition

a4

0.01

U4

1.0

L4

1.0

T

0.0000189

 

Scale 1

Scale 2

sc

1.0

1.0

ss

2.0

2.0

 

 

Hypercomplex cells: Orientational Competition

a 5

1.0

U5

1.0

L5

1.0

 

Scale 1

Scale 2

C

1.0

1.0

S

1.5

1.5

sc

0.5

0.5

ss

0.75

0.75

Bipole Cells

D

0.1

G

0.1

A

1.0

B

2.0

Z

1.0

r

0.0

sg

7.0

sk

0.15

sr

0.15

Monocular FIDO

Mm

0.1

d

100,000.0

k

1.0

e

1000.0

Output from Monocular FIDOs

a8

0.01

L8

1.0

U8

1.0

C

0.0398

sc

2.0

S

0.181

ss

3.0

Binocular FIDO

a9

0.01

U9

0.5

L9

50.0

Mb

0.1

d

100,000.0

k

1.0

e

1000.0

 

TABLE 1. Model Parameters.

Diana Meyers

Tue March 7 14:29:09 EST 2000