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%Input Wang's changes 1-30-89
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\title{Computing Optical Flow in the Primate Visual
System} %
\author{H. Taichi Wang\\
Bimal Mathur\\
{\small \sl Science Center, Rockwell International, Thousand Oaks,
CA 91360, USA}\\
\\
Christof Koch\\
{\small \sl Computation and Neural Systems Program,}\\
{\small \sl Divisions of Biology and
Engineering and Applied Sciences, 216-76,}\\
{\small \sl California Institute of Technology,
Pasadena, CA 91125, USA}}
%
\def\Communicate{Richard Andersen}
%
\markboth{H. Taichi Wang, Bimal Mathur, Christof Koch}
{Computing Optical Flow in the Primate Visual System}
%
\maketitle{wang}

\begin{abstract}
Computing motion on the basis of the time-varying image
intensity is a difficult problem for both artificial and biological vision
systems. We show how gradient models, a well known class of motion
algorithms, can be implemented within the magnocellular pathway of the
primate's visual system.  Our cooperative algorithm computes optical
flow in two steps. In the first stage, assumed to be located in
primary visual cortex, local motion is measured while spatial
integration occurs in the second stage, assumed to be located in the middle
temporal area (MT). The final optical flow is extracted in this second stage
using population coding, such that the velocity is represented by the vector sum
of neurons coding for motion in different directions. Our theory, relating the
single-cell to the perceptual level, accounts for a number of
psychophysical and electrophysiological observations and illusions.
\end{abstract}

\section{Introduction}

In recent years, a number of theories have been advanced at
both the computational and the psychophysical level, explaining aspects of
biological motion perception (for a review see Ullman 1981; Nakayama 
1985; Hildreth and Koch 1987). One class of motion algorithms exploit
the relation between the spatial and the temporal intensity change at
a particular point (Fenneman and Thompson 1979; Horn and Schunck 
1981; Marr and Ullman 1981; Hildreth 1984). In this article we
address in detail how these algorithms can be mapped onto neurons in
striate and extrastriate primate cortex (Ballard et al.\ 1983).

Our neuronal implementation is derived from the most common version of the
gradient algorithm, proposed within the framework of machine vision (Horn
and Schunck 1981). Due to the ``aperture'' problem inherent in their
definition of optical flow, only the component of motion along the local
spatial brightness gradient can be recovered. In their formulation, optical
flow is then computed by minimizing a two-part quadratic variational 
functional. The first term forces the final optical flow to be compatible
with the locally measured motion component (``constraint line term''),
while the second term imposes the constraint that the final flow field should
be as smooth as possible. Such as ``smoothness" or ``continuity" constraint is
common to most early vision algorithms.

\section{A Neural Network Implementation}

Commensurate with this method, and in agreement with psychophysical results
(e.g.\ Welch 1989), our network extracts the optical flow in two  stages
(Fig.~1). In a preliminary stage, the time-varying image $I(i,j)$ is projected
onto the retina and relayed to cortex via two neuronal pathways providing
information as to the spatial location of image features ($S$  neurons) and
temporal changes in these features ($T$ neurons): $S(i,j)=\nabla^2G\ast I(i,j)$
and $T(i,j)=\partial(\nabla^2G\ast  I(i,j))/\partial t$, where $*$ is the
convolution operator and $G$ the 2-D Gaussian filter (Enroth-Cugell and Robson
1966; Marr and Hildreth 1980; Marr and Ullman 1981). In the first processing
stage, the local motion information is represented  using a set of $n$ ON-OFF
orientation- and direction-selective cells $U$, each with preferred direction
indicated by the unit vector  ${\bf\Theta_k}$:

\begin{equation}\label{wangeq:1}
U(i,j,k) ={-T(i,j)\nabla_kS(i,j)\over|\nabla
S(i,j)|^2+\epsilon}
\end{equation}
where $\epsilon$ is a constant and $\nabla_k$ is the spatial
derivative along the direction ${\bf\Theta_k}$. This derivative is
approximated by projecting the convolved image $S(i,j)$ onto a
``simple'' type receptive field, consisting of a 1 by 7 pixel positive
(ON) subfield adjacent to a 1 by 7 pixel negative (OFF) subfield.  The
cell $U$ responds optimally if a bar or grating oriented at right
angles to ${\bf\Theta_k}$ moves in direction ${\bf\Theta_k}$. Note
that $U$ is proportional to the product of a transient cell ($T$) with
a sustained simple cell with an odd-symmetric receptive field, with an
output proportional to the magnitude of local component velocity (as long as
$|\nabla_k S(i,j)| > \epsilon$). At each location $i,j$, $n$ such neurons code
for motion in $n$ directions.  Equation~(\ref{wangeq:1}) differs from the standard
gradient model in which $U= -T/\nabla_kS$, by including a gain control term,
$\epsilon$, such that $U$ does not diverge if the stimulus contrast decreases to
zero.  
$~\epsilon$ is set to a fixed fraction of the square of the maximal
magnitude of the gradient $\nabla S$ for all values of $i,j$.  Our
gradient-like scheme can be approximated for small enough values of
the local contrast (i.e.\ if $|\nabla S(i,j)|^2 < \epsilon$), by
$-T(i,j)\nabla_kS(i,j)$. Under this condition, our model can be
considered a second-order model, similar to the correlation or
spatio-temporal energy models (Hassenstein and Reichardt 1956; Poggio
and Reichardt 1973; Adelson and Bergen 1985; Watson and Ahumada 1985).  We also
require a set of $n$ ON-OFF orientation- but not direction-selective
neurons $E$, with $E(i,j,k)=|\nabla_kS(i,j)|$, where the absolute value ensures
that these neurons only respond to the magnitude of the spatial gradient,
but not to its sign.

\begin{figure}
\figspace{5.38in}{4.4in}
\caption{Computing motion in neuronal networks. (a)
Schematic representation of our model. The image $I$ is projected onto
the rectangular $64$ by $64$ retina and sent to the first processing
stage via the $S$ and $T$ channels. A set of $n=16$ ON-OFF
orientation- ($E$) and direction-selective ($U$) cells code local
motion in $16$ different directions ${\bf\Theta_k}$.  These cells are
most likely located in layers 4C$\alpha$ and 4B of V1.  Neurons with
overlapping receptive field positions $i,j$ but different preferred
directions ${\bf\Theta_k}$ are arranged here in $16$ parallel planes.
The ON subfield of one such $U$ cell is shown in Fig. 4a. The output
of both $E$ and $U$ cells is relayed to a second set of $64$ by $64$ by $16~V$
cells where the final optical flow is represented via population
coding ${\bf V(i,j)}=\sum_{k=1}^{n'}V(i,j,k){\bf\Theta_k}$, with
$n'=16$. Each cell $V(i,j,k)$ in this second stage receives input from
cells $E$ and $U$ at location $i,j$ as well as from neighboring
neurons at different spatial locations. We assume that the $V$ units
correspond to a subpopulation of MT cells. (b) Polar plot of the
median neuron (solid line) in MT of the owl monkey in response to a
field of random dots moving in different directions (Baker et al.\ 1981). The
tuning curve of one of our model $V$ cells in response to a moving bar is
superimposed (dashed line). Figure courtesy of J.~Allman and S.~Petersen.}
\end{figure}


We assume that the final optical flow field is computed in a second stage, 
using a population coding scheme such that the velocity is represented within
a set of $n'$ neurons $V$ at location $i,j$, each with preferred direction
${\bf\Theta_k}$ with ${\bf V(i,j)} = \sum_{k=1}^{n'} 
V(i,j,k){\bf\Theta_k}$. Note that the response of any individual cell 
$V(i,j,k)$ is {\bf not} the projection of the velocity field ${\bf V(i,j)}$ 
onto ${\bf\Theta_k}$. 
For any given visual stimulus, the state of the $V$ neurons is determined
by minimizing the neuronal equivalent of the above mentioned variational
functional. The first term, enforcing the constraint that the final optical
flow should be compatible with the measured data, has the form:

\begin{equation}\label{wangeq:2}
L_0 = \sum_{i,j,k}[\sum_{k'}
V(i,j,k')\cos(k'-k)-U(i,j,k)]^2 E^2(i,j,k),
\end{equation}
where $\cos(k'-k)$ is a shorthand for the $\cos$ of the angle between
${\bf\Theta_{k'}}$ and ${\bf\Theta_k}$.
The term $E^2(i,j,k)$ ensures that the local motion components $U(i,j,k)$
only have an influence when there is an appropriate oriented local pattern;
thus $E^2$ prevents velocity terms incompatible with the measured data from
contributing significantly to $L_0$. In order to sharpen the orientation
tuning of $E$, we square the output of $E$ (the orientation tuning curve of
$E$ has a half-width of about $60^o$). 

The smoothness term, minimizing the square of the first derivative 
of the optical flow (Horn and Schunck 1981) takes the following form:

\begin{eqnarray}\label{wangeq:3}
L_1&=&\sum_{i,j,k,k'}[4V(i,j,k)-V(i-1,j,k)-V(i+1,j,k)\\ \nonumber
&-&V(i,j-1,k) -V(i,j+1,k)] \times \cos(k'-k) V(i,j,k').
\end{eqnarray}
Our algorithm computes the state of the $V$ neurons that minimizes $L_0 + 
\lambda L_1$ ($\lambda$ is a free parameter, usually set at $10$). We can
always find this state by evolving $V(i,j,k)$ on the basis of the
steepest descent rule: $\partial V/\partial t=-\partial (L_0 + \lambda
L_1)/\partial V$.  Since the variational functional is quadratic in
$V$, the right hand side in the above differential equation is linear
in $V$. Conceptually, we can think of the coefficients of this linear
equation as synaptic weights, while the left hand side can be
interpreted as a capacitive term, determining the dynamics of our
model neurons. In other words, the state of the $V$ neurons evolve by
summating the synaptic contribution from $E$, $U$ and neighboring $V$
neurons and updating its state accordingly. Thus, the system
``relaxes'' into its final and unique state. To mimic neuronal
responses more accurately, the output of our model neurons $S$, $T$, $E$,
$U$, and $V$ is set to zero if the net input is negative.

\section{Physiology and Psychophysics}

Since the magnocellular pathway in primates is the one processing
motion information (Livingstone and Hubel 1988; DeYoe and Van Essen 1988) we
assume that the $U$ and $E$ neurons would be located in layers 4C$\alpha$ and 4B
of V1 (see also Hawken et al.\ 1988) and the $V$ neurons in area MT,
which contains a very high fraction of direction- and speed-tuned neurons
(Allman and Kass 1971; Maunsell and Van Essen 1983). All $2n$ neurons $U$ and
$E$ with receptive field centers at location $i$, $j$ then project to the
$n'$ MT cells $V$ in an excitatory or inhibitory (via interneurons)
manner, depending on whether the angle between the preferred direction
of motion of the pre- and post-synaptic neuron is smaller or larger
than $|90^o|$.\footnote{The appropriate weight of the synaptic connection
between $U$ and $V$ is $\cos(k-k')U(k')E^2(k')$. Various biophysical mechanisms
can implement the required multiplicative interaction as well as the
synaptic power law (Koch and Poggio 1987).} Anatomically, we then
predict that each MT cell receives input from V1 (or V2) cells located
in all different orientation-columns. The smoothness constraint of
equation~(\ref{wangeq:3}) results in massive interconnections among neighboring $V$
cells (Fig.~1a).

Our model can explain a number of perceptual phenomena. When two
identical square gratings, oriented at a fixed angle to each other,
are moved perpendicular to their orientation (Fig.~2a), human
observers see the resulting plaid pattern move coherently in the
direction given by the intersection of their local constraint lines
(``velocity constraint combination rule''; in the case of two gratings
moving at right angle at the same velocity, the resultant is the
diagonal; Adelson and Movshon 1982). The response of our network to such an
experiment is illustrated in figure~2: the $U$ cells only respond to the local
component of motion (component selectivity; Fig.~2b), while the $V$
cells respond to the global motion (Fig.~2c), as can be seen by computing the
vector sum over all $V$ cells at every location (pattern selectivity;
Fig.~2d). About $30\%$ of all MT cells do respond in this manner,
signaling the motion of the coherently moving plaid pattern (Movshon et al.\ 
1985). In fact, under the conditions of rigid motion in the plane observed in
Adelson and Movshon's experiments, both their ``velocity space combination 
rule'' and the ``smoothness'' constraint converge to the solution perceived 
by human observers (for more results see Wang et al.\ 1989). Given
the way the response of the $U$ neurons vary with visual contrast
(equation~\ref{wangeq:1}), our model predicts that if the two gratings making up
the plaid pattern differ in contrast, the final pattern velocity will be 
biased in the direction of the component velocity of the grating with the
larger contrast. Recent psychophysical experiments support this conjecture 
(Stone et al.\ 1988). It should be noted that the optical flow field is not
represented explicitly within neurons in the second stage, but only implicitly,
via vector addition.

\begin{figure}
\figspace{4.75in}{4.75in}
\caption{(a) Two gratings moving towards the lower right (one at
$-26^o$ and one at $-64^o$), the first moving at twice the speed of the 
latter. The amplitude of the composite is the sum of the amplitude of
the individual bars. The neuronal responses of a 12 by 12 pixel
patch (outlined in (a)) is shown in the next three sub-panels. (b)
Response of the $U$ neurons to this
stimulus. The half wave rectified output of all $16$ cells at each location is
plotted in a radial coordinate system at each location as long as the response is
significantly different from zero. (c) The output of the $V$
cells using the same needle diagram representation after the network converged.
(d) The resulting optical flow field, extracted from (c) via population coding,
corresponding to a coherent plaid moving towards the right, similar to the
perception of human observers (Adelson and Movshon 1982) as well as to the
response of a subset of MT neurons in the macaque (Movshon et al.\ 1985). The
final optical flow is within $5\%$ of the correct flow field.}
\end{figure}

Our algorithm reproduces both ``motion capture'' (Ramachandran and
Anstis 1983; see Wang et al.\ 1989) and ``motion coherence''
(Williams and Sekuler 1984), as illustrated in figure~3. As
demonstrated previously, these phenomena can be explained, at least
qualitatively, by a smoothness or local rigidity constraint (Yuille
and Grzywacz 1988; B\"ulthoff et al.\ 1989). Finally,
$\gamma$ motion, a visual illusion first reported by the Gestalt
psychologists (Lindemann 1922; Kofka 1931; for a related illusion in
man and fly see B\"ulthoff and G\"otz 1979), is also mimicked by our
algorithm. This illusion arises from the initial velocity measurement
stage and does not rely on the smoothness constraint.

\begin{figure}
\figspace{4.75in}{4.75in}
\caption{(a) In ``motion coherence,'' a cloud of dots is
perceived to move in the direction of their common motion component.
In this sequence, all dots have an upward velocity
component, while their horizontal velocity component is random. (b) The
final velocity field only shows the motion component common to all dots. Humans
observe the same phenomena (Williams and Sekuler 1984). (c) In $\gamma$-motion,
a dark stimulus flashed onto a homogenously lit background appears to expand
(Lindemann 1922; Kofka, 1931). It will disappear with a motion of contraction.
(d) Our algorithm perceives a similar expansion when a disk appears. The contour
of the  stimulus is projected onto the final optical flow.}
\end{figure}
 
Cells in area MT respond well not only to motion of a bar or grating
but also to a moving random dot pattern (Albright 1984; Allman et al.\ 1985).
Similarly, our algorithm detects a random-dot figure moving over a stationary
random-dot background, as long as the spatial displacement between two
consecutive frames is not too large (Fig.~4).  An interesting distinction
arises between direction-selective cells in V1 and MT. While the optimal
orientation in V1 cells is always perpendicular to their optimal direction, this
is only true for about 60\% of MT cells (type I cells; Albright 1984; Rodman and
Albright 1988).  30\% of MT cells respond strongly to flashed bars oriented
parallel to the cells' preferred direction of motion (type II cells). If we
identify our $V$ cells with this MT subpopulation, we predict that type II cells
should respond to an extended bar (or grating) moving parallel to its edge
(Fig.~4).

\begin{figure}
\figspace{4.75in}{4.75in}
\caption{(a) A dark bar (outlined) is moved parallel to its
orientation towards the right. Due to the aperture problem, those $U$
neurons whose receptive field only ``see'' the straight elongated
edges of the bar --- and not the leading or trailing edges --- will fail
to respond to this moving stimulus, since motion remains invisible on
the basis of purely local information. The ON subfield of the
receptive field of a vertically oriented $U$ cell is superimposed onto
the image. (b) It is only after information has been
integrated, following the smoothing process inherent in the second
stage of our algorithm, that the $V$ neurons respond to this motion.
The type II cells of Albright (1984) in MT should respond to this
stimulus. (c) Response of our algorithm to a random-dot
figure-ground stimulus. The central 10 by 10 pixel square moved by 1
pixel toward the left. (d) Final optical flow after smoothing.
The $V$ cells detect the figure, similar to cells in MT. The contour of the 
translated square is projected onto the final optical flow.}
\end{figure}

\section{Discontinuities in the Optical Flow}
The major drawback of all motion algorithms is the degree of
smoothness required, smearing out any discontinuities in the optical
flow field, such as those arising along occluding objects or along a
figure-ground boundary.  It has been shown previously how this can be
dealt with by introducing the concept of line processes which
explicitly code for the presence of discontinuities in the motion
field (Hutchinson et al.\ 1988; see also Poggio et al.\ 1988). If the spatial
gradient of the optical flow between two neighboring points is larger than some
threshold,  the flow field ``is broken''; that is the process or ``neuron''
coding for a  motion discontinuity at that location is switched on and no
smoothing occurs. If little spatial variation exists, the discontinuity remains
off. The performance of the original version of the Horn and Schunck version is
greatly improved using this idea (Hutchinson et al.\ 1988).
Perceptually, it is known that the visual system uses motion to segment
different parts of visual scenes (Baker and Braddick 1982; van Doorn
and Koenderink 1983; Hildreth 1984). But what about the possible
cellular correlate of line processes? Allman and colleagues (Allman
et al.\ 1985) first described cells in area MT in the owl monkey
whose ``true'' receptive field extended well beyond the classical
receptive field as mapped with bar or spot stimuli. About half of all
MT cells have an antagonistic direction-selective surround, such that
the response of the cell to motion of a random dot display or an edge
within the center of the receptive field can be modified by moving a
stimulus within a very large surrounding region. The response depends
on the difference in speed and direction of motion between the center
and the surround, and is maximal if the surround moves at the same
speed as the stimulus in the center but in the opposite direction.
Thus, tantalizing hints exist as to the possible neuronal basis of
motion discontinuities.

Recently, two variations  of Horn and Schunck's (1981) original
algorithm have been proposed, based on computational considerations
(Uras et al.\ 1988; Yuille and Grzywacz 1988).  Both algorithms can be mapped
onto the neuronal network we have proposed, with minimal changes either in the
$U$ stage (Uras et al.\ 1988) or by increasing the connectivity among
more distant $V$ cells (Yuille and Grzywacz 1988).  It remains an open
challenge to provide both psychophysical and electrophysiological
evidence to evaluate the validity of these and similar schemes (e.g.\
Nakayama and Silverman 1988). We are currently trying to
extend our model to account for the intriguing phenomena of ``motion
transparency,'' such as when two fields of random dots moving in
opposite directions are perceived to form the 3-D motion field
associated with a transparent and rotating cylinder (Siegel and
Andersen 1988).

\section*{Acknowledgments}
 We thank John Allman, David Van Essen, and Alan Yuille
for many fruitful discussions and Andrew Hsu for computing the figure-ground
response. Support for this research came from NSF grant EET-8700064, ONR
Young Investigator Award and NSF Presidential Young Investigator Award to C.K.

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\end{thebibliography}
{\flushleft
\rule{2in}{.1mm}\hfill\break
\footnotesize Received 28 October; accepted 6 December 1988.
\normalsize}
\endpaper{wang}




