%
\title{Two Stages of Curve Detection Suggest Two Styles of Visual Computation%
}
%
\author{Steven W. Zucker\thanks{Senior Fellow, Canadian Institute for Advanced 
Research.}\\
Allan Dobbins\\
Lee Iverson\\
{\small \sl Computer Vision and Robotics Laboratory,}\\
{\small \sl McGill Research Centre
for Intelligent Machines,}\\
{\small \sl McGill University, Montr\'eal, Qu\'ebec, Canada}}
%
\def\Communicate{David Mumford}
%
\markboth{Steven W. Zucker, Allan Dobbins, and Lee Iverson}
{Two Stages of Curve Detection}
%
\maketitle{zucker}

\begin{abstract}
The problem of detecting curves in visual images arises in both
computer vision and biological visual systems.  Our approach integrates
constraints from these two sources and suggests
that there are two different stages to curve detection, 
the first resulting in a local description, and the second in a global one.
Each stage involves a different
style of computation: in the first stage, hypotheses are
represented explicitly and coarsely in a fixed, preconfigured
architecture;  in the second stage, hypotheses are represented
implicitly and more finely in a dynamically-constructed architecture. 
We also show how these stages could
be related to physiology, specifying the earlier parts
in a relatively fine-grained fashion and the later ones more coarsely.
\end{abstract}

\section{Introduction}

An extensive mythology has developed around curve detection. In extrapolating
 from
orientation-selective neurons in the visual cortex (Hubel and Wiesel 1962),
it is
now widely held that curve detection is simply a matter of ``integrating''
the responses of these cells. More specifically, the mythology holds that this
integration process is global, that the initial estimates are local,
and that the relationship between them will
become clear as a more detailed understanding of cortical circuitry
is uncovered. However, this mythical process of ``integration'' has
turned out to be elusive, 
the search for it has led, instead, to a series of dilemmas, and
the quantity of physiological data is exploding. It is rarely clear how
new details of cortical circuitry relate to different components of the 
curve detection problem. 

We believe that this situation is typical of vision
in general, and amounts to ascribing too little function to the
earlier stages, and too much to the later ones. For
curve detection, virtually all of the complexity is delegated to the 
process of ``integration,'' so it is  not surprising that successful
approaches  have remained elusive. Part of the problem is that
models of integrative processes have been rich in selected
detail, but poor in abstract function. In the sense that it is often 
useful  to {\it see the forest before the trees}, we submit that solutions
will likely be found by considering both 
coarse-grained and fine-grained models, 
and that such models will suggest a partitioning of function
whose abstraction varies with granularity. To make this point
concretely, we here outline a 
coarse-grained solution to the curve detection problem from a
computational perspective, and sketch
how it could map onto physiology. The sketch is coarse enough to serve as an
organizational framework, but fine enough to suggest particular 
physiological constraints. One of these comprises our first, coarse-grain
prediction: curve detection naturally 
decomposes into two stages, the first in which a  local description
is computed, 
and the second in which a  global description is computed.  These
computations are sufficiently different that we are lead to 
hypothesize two different styles of visual computation.

\section{The Dilemma of Curve Detection}

The initial measurement of orientation information is broadly tuned, which
suggests the averaging necessary to counteract retinal (sensor) sampling,
quantization, and noise. However, the end result of curve detection is
unexpectedly precise:  
corners can be distinguished from arcs of high curvature, and nearby curves
can be
distinguished from one another to a hyperaccurate level, 
even though they might pass through the same
receptive field.

An analogous dilemma exists for computer vision systems, even with the 
spectacular numerical precision of which computers are capable: quantization 
and noise imply smoothing, but smoothing blurs corners, endpoints, and nearby
curves into confusion (Zucker 1986). 
At the foundation is a chicken-and-egg problem: if the
points through which the curve passed, together with the locations of 
discontinuities, were known, then the actual properties of the curve could
be inferred. But initially they are not known, so
any smoothing inherent in the inference process is potentially dangerous.

\section{Two Stages of Curve Detection}

We have discovered a computational solution to this dilemma, which involves
decomposing the full problem into two stages, each of which has a rather
different character. In the first stage, the local properties of the
curve are computed: its trace (the set of retinotopic points through
which the curve passes), its tangent (or orientation at those points), and
its curvature. In the second stage, these properties are refined to 
create a global model of the curve.  This much --- proceeding from local to
global --- is standard; the style of the computations is not. The key to the 
first stage is to infer the local properties coarsely --- not in fine 
detail --- but without sacrificing reliability or robustness. 
Coarseness is here related to quantization, which must
limit error propagation without blurring over corners. Observe that this is
precisely what is lacking in the
standard myth, where errors (e.g., about placing discontinuities)
can have far reaching consequences. The result is
a style of computation in which the different (quantized)
possibilities are made explicit,
and arranged in a fixed, preconfigured computational 
architecture that imposes no {\it a
priori\/} ordering over them. Each distinct hypothesis, say rough 
orientation and
curvature at every position, 
forms a unit in a fixed network that strongly resembles
neural-network-style models. Reliability and robustness are then maintained
by the network; hence the local description is not computed
locally! A mapping onto orientation hypercolumns will
be discussed shortly.

The second stage embodies a rather different style of computation. Now the
possibilities no longer need be general, but are constrained to be in the
range dictated by the first stage. Thus the architecture can be tailored to
each problem---that is, constructed adaptively rather than preconfigured---and
variables can be represented implicitly. With these highly focused
resources, the key limitation on precision is implementation, and
it need not be hampered by uncontrolled error propagation. From the outside,
this constructive style of computation holds certain key properties in common
with later visual areas, such as V4 and IT, where receptive field structure has
been shown to vary with problem  constraints (e.g., Maunsell and
Newsome 1987; Moran and Desimone 1985).

\section{The Model of Curve Detection}

In physiological terms, neurons are said to be orientation selective
if they respond differentially to stimulus (edge or line)
orientation. We take this operational statement one step further by defining
orientation selection to be the inference of a 
local description of the curve everywhere along it, and postulate orientation
selection as the goal of our first stage. In the second stage, global curves
are inferred through this local description. The various stages of our process
are shown in figure~1, and expanded below.

\subsection{Stage 1: Inferring the Tangent Field}

Formally orientation selection amounts
to inferring the trace of the curve, or the set of points (in the image) 
through which the curve passes, its (approximate) tangent and curvature
at those points, and their discontinuities (Zucker 1986). 
We refer to such information as the
{\it tangent field}, 
and note that, since the initial measurements are discrete,
this will impose constraints on the (inferred) tangents, curvatures, and
discontinuities (Parent and Zucker 1985).

This first stage of orientation  selection is in turn 
modeled as a two step process:
\begin{description}
\item[Step 1.1.] {\it Initial Measurement\/} of the local fit at each point to 
estimate orientation and curvature. These estimates derive from  a model 
of simple cell
receptive fields instantiated at multiple scales and orientations
at each image position. 
However, 
these local measurements are inherently inaccurate (e.g., broadly
tuned), so we require:
\item[Step 1.2.] {\it Interpretation\/} into an explicit distributed
representation of tangent and curvature by  establishing consistency
between the local measurements.  This is accomplished by
modifying them according to their geometric relationships with
nearby estimates.
\end{description}

\subsection{Stage 2: Inferring a Covering of the Curve}

Since the tangent is the first derivative of a curve
(with respect to arc length), the global curve can be recovered as an integral
through the tangent field. Such a view typically leads to sequential
recovery   algorithms (e.g., Kass and Witkin 1987). But these
algorithms require global parameters, starting points,
and some amount of topological structure 
(i.e., which tangent point follows which); in short, they are biologically
implausible.
In contrast, we propose a novel approach in which a collection of short,
dynamically modifiable curves (``snakes'' in computer vision; see
Montanari 1971; Kass et al.\ 1988) move in
parallel.

The key idea behind our approach is to recover the
global curve by computing a {\it covering\/} of it; i.e., a set of
objects whose union is equivalent to the original curve.
The elements of the covering
are unit-length dynamic splines,
initially equivalent to the elements of the tangent field,
but which then evolve according to a potential
distribution constructed from the tangent field. The evolution takes
two forms: (i) a migration in position to achieve smooth coverings;
and (ii) a ``growth'' to triple their initial length. Furthermore, since
the splines are initially independent, 
it is not known which should be grouped into the covering of each 
distinct global curve. 
For graphical purposes we represent
this by creating each one with a different ``color,'' and include
a second process  which converts overlapping splines to the
same color. In the end, then, the cover is given by a collection of
overlapping splines, or short ``snakes,'' each of which is the same color.

Again, there are two conceptually distinct steps to Stage 2 of the algorithm
(David and Zucker 1989):

\begin{description}
\item[Step 2.1.] {\it Constructing the Potential Distribution\/} from the
discrete tangent field.  Each entry in the tangent field actually represents a 
discretization  of the many possible curves in the world that could
project onto that particular (tangent, curvature) hypothesis.
Now  these pieces must be put together,  so 
consider a measure (or envelope) over all of these possible curves.
Assuming the curves are continuous but not necessarily differentiable
everywhere, each contribution to the potential can be modeled as a
Gaussian (the Wiener measure)
oriented in the direction of the tangent field entry. The full
potential distribution is their pointwise sum; see figure~3.
\item [Step 2.2.] {\it Spline Dynamics\/}
The discrete entities in the tangent field are converted into
unit splines initialized in the valleys of the potential distribution.
They evolve according to
a variational scheme that depends on spline properties (tension and rigidity)
as well as the  global potential. 
\end{description}


\begin{figure}
\vspace{6in}
\caption{An illustration of the different stages of curve detection. In (a) we
show a section of a fingerprint image; note the smooth curves and discontinuities
  around the ``Y'' in the center. (b) Graphical illustration of the initial 
  information, or those orientation/curvature hypotheses resulting from 
  convolutions above the noise level. (c) The discrete tangent field
  resulting from the relaxation process
  after 2 iterations; note that most of the spurious initial responses
  have been 
  eliminated.  (d) Final snake positions, or coverings of the global
  curves. (e) The potential distribution constructed from the entries in
  the tangent field.}
\end{figure}

\section{Implementing the Model}

Each stage of the model has different implementation requirements. To
differentiate between smooth curves, curves with
corners, crossing curves and branching curves, it is necessary to represent
each possible tangent (orientation) and  curvature value at every possible
position. Smooth curves are then represented as a single (tangent, curvature)
hypothesis at each (retinotopic) trace point, corners as multiple tangents
at a single point, and bifurcations as a single tangent but multiple curvatures
at a single point. Orientation hypercolumns in the visual cortex are thus
a natural representational substrate, with explicit representation of
each possible orientation and curvature at each position. This leads to
a new observation regarding discontinuities: explicit neurons to represent
them are unnecessary, and leads to our first physiological
prediction: 

\begin{quote}
Prediction 1. 
Crossings, corners, and bifurcations  are represented at the early processing
stages by multiple neurons firing within a ``hypercolumn.''
\end{quote}

\subsection{Stage 1, Step 1: Intra-Columnar Initial Measurements}

We first seek a physiologically plausible mechanism for measuring orientation
and curvature. Observe that
an orientation-selective cortical neuron carries information about the
tangent to curves as they pass through its receptive field, 
and an ensemble of such cells of different size 
carries information about how orientation is changing over it. Such
differences are related to curvature (or deviation
from straightness), and adding appropriate rectification leads to
a model of endstopped neurons (Dobbins et al.\ 1987; cf.\
Hubel and Wiesel 1965).  This model
exhibits curvature-selective response at the preferred orientation, as
do endstopped neurons. Thus 
\begin{quote}
Prediction 2. 
Endstopped neurons carry the quantized representation of orientation
and (non-zero) curvature at each position.
\end{quote}

\noindent
By  varying the components one obtains cells
selective for different ranges and signs of curvature.

Thus the initial measurements can  be 
built up by intra-columnar local circuits, with the match to each (quantized)
orientation and curvature represented explicitly as, say, firing rate in
endstopped neurons.  However,  these  measurements of
orientation and curvature  are broadly
tuned; nearby curves are blurred together and
multiple possibilities arise at many positions. 
Introducing
further non-linearities into the initial measurements eliminates some spurious
responses (Zucker et al.\ 1988), 
but the broadly-tuned smearing remains.
We thus seek an abstract principle by which these broadly tuned responses
can be
refined into crisper distributions.

\subsection{Stage 1, Step 2: Inter-Columnar Iterative Refinement}

Again curvature enters the model, but
now as a way of expressing the relationship
between nearby tangent (orientation) hypotheses. 
Consider an arc of a curve, and observe that
tangents to this arc must conform to certain position and orientation 
constraints for a given amount of curvature;
we refer to such constraints geometrically as co-circularity
(Fig.~2a). Discretizing all continuous curves in the world that project 
into the columnar space of coarse (orientation, curvature) hypotheses
partitions these curves into equivalence classes, 
examples of which are shown in figure~2b (Parent and 
Zucker 1985; Zucker et al.\ 1988). Interpreting the (orientation, curvature)
hypotheses as endstopped neurons,  
such co-circularly-consistent relationships are what is to be expected
of the firing pattern between endstopped neurons in nearby orientation
hypercolumns given such a curve as stimulus. 
Turning this around, when such intercolumnar patterns
arise from the initial measurements, a curve from one of the
equivalence classes is to be expected.

Such inter-columnar interactions can be viewed physiologically as 
excitatory and inhibitory projections
between endstopped cells at nearby positions (adjacent hypercolumns), and
can be used as follows. Since curvature is a relationship between tangents
at nearby positions, two tangents  should 
support one another if and only if they agree under a curvature hypothesis, and
co-circularity provides the measure of such support. In addition, two tangents
that disagree with the curvature estimate should detract support from one
another. Relaxation labeling provides a formal mechanism for defining such
support, and for specifying how to use it (Hummel and Zucker 1983). 
Mathematically it amounts to
gradient descent; physiologically it can be viewed as a mechanism for
specifying how the response of neighboring neurons will interact. In summary:

\begin{quote}
Prediction 3. 
Inter-columnar interactions exist between curvature consistent (co-circular)
tangent hypotheses. 
\end{quote}

\begin{figure}
%\figspace{4.25in}{4.25in}
\vspace{6in}
\caption{(a) The geometric relationships necessary for defining the
   compatibilities between two label pairs at points $i$ and $j$.
(b) Compatibilities between coarse (orientation, curvature) hypotheses
at nearby positions. 8 distinct orientations and 7 curvatures were represented,
and 3 examples are shown.
   (top) The labels which give positive (left) and negative (right) support
   for a diagonal orientation that curves slightly left; (middle) positive and
   negative support for a straight curvature class; (bottom) positive and
   negative support for the maximum curvature class. The magnitude of the
interactions varies as well, roughly as a Gaussian superimposed on these
diagrams. The values were obtained by numerically solving  a 6-dimensional
closest point problem (Zucker et al.\ 1988). Physiologically these
projective fields represent inter-columnar interactions.
Multiplied by the
   original tangent receptive fields, they represent the units 
 for building the potential distribution that guides Stage 2.
  }
\end{figure}

Given interaction, the next question relates to  precision.
Earlier we hypothesized that this first stage was coarse.
Both computational experiments (Zucker et al.\ 1988), psychophysics
(Link and Zucker 1988), 
and the range of receptive field sizes in
striate cortex (Dobbins et al.\ 1988)
provide independent evidence about the quantization of
curvature:

\begin{quote}
Prediction 4. 
The initial representation of curvature in the visual cortex is 
quantized into $5\pm2$ distinct classes; namely,
straight, curved to the left a small amount, curved to the left a
large amount, and similarly to the right.
\end{quote}

Relaxation processes can be realized iteratively, and computational experiments
suggest that about 3 interations suffice (Zucker et al.\ 1988). At this
time we can only speculate how these iterations relate to physiology, but 
perhaps the first iteration is carried out by a recurrent network within V1,
and the subsequent iterations through the feed-forward and\penalty-100000 {-back}
projections to extrastriate cortex (e.g., V2 or V4 in monkey). There is no
doubt, however, that interactions beyond the classical receptive field abound
(Allman et al.\ 1985).

The advantage of this style of ``coarse modeling'' is that 
a number of testable physiological hypotheses do emerge, and we are now
beginning to explore them. The requirement of initial curvature estimates
led to the connection with endstopping, and the current model suggests 
roles for inter-columnar interactions. In particular, we predict that they 
should be a function of position and orientation, a prediction for which
some support exists (e.g.\ Nelson and Frost 1985)
in the zero-curvature case; 
experiments with curved
stimuli remain to be done.

\subsection{Stage 2: Potential Distributions and Evolving Spline Covers}

The tangent field serves as a coarse model for the curve, represented locally.
The next task is to infer a smooth, global curve running through it. We perform
this inference in a rather different kind of architecture, one that involves
potential distributions  constructed specifically for each instance. It proceeds
as follows. 
 
The potential distribution is created by adding together contributions from
each element in the tangent field; see figure~3.
Changing the representation from the tangent field to the
potential distribution changes what is explicit and what is implicit in the 
representation. In Stage 1 there were discrete coarse entities; now there are
smooth 
valleys that surround each of the global curves, with a separation
between them. 
The ``jaggies'' imposed by
the initial image sampling have been eliminated, and interpolation to
sub-pixel resolution is viable.

\begin{figure}
\figspace{4.25in}{4.02in}
\caption{Illustration of how a potential distribution is constructed from
tangent
   field entries. (a) A small number of tangents, showing the
individual contributions from each one. (b) As more tangents are included,
long ``valleys'' begin to form when the individual entries are added 
together. (c) The complete tangent field and potential distribution as
shown in figure~1. Physiologically one might think of such potentials as being
mapped onto neuronal membranes. Not shown is the possible effect of attention
in gating the tangent field contributions, the smallest unit for which could
correspond to a tangent field entry.
   }
\end{figure}

To recover the curves through the valleys, 
imagine creating, at each tangent field entry,
a small spline of unit length 
oriented according to the tangent and curvature estimates (Fig.~4).
By construction, we know that this spline will be born in a valley of
the tangent field potential distribution, so they are then
permitted to migrate to both smooth out the curve and to find the
true local minima. But the inference of a cover for the global curves requires
that the splines overlap, so that each point on every curve is covered by
at least one spline. 
We therefore let the splines extend in length while they migrate in position,
until they each reach a prescribed length.
The covering is thus composed of these  extensible splines 
which have grown in the valleys of the tangent field potential.
Their specific dynamics and properties are described more fully in
(David and Zucker 1989).

It is difficult to interpret these ideas physiologically within the
classical view of neurons, in which  inputs are summed and transformed
into an output train of action potentials.  Dendrites simply support passive
diffusion of depolarization.
Recently, however, a richer view of neuronal processing has
emerged, with a variety of evidence pointing to active
dendritic computation and dendro-dendritic interaction
(Schmitt and Worden 1979). Active conductances
in dendrites functionally modify the geometry, and
dendro-dendritic interactions suggest that the
output transformation is not uniquely mediated by the axon.
Taken together,
these facts imply that patterns of activity can be sustained in the
dendritic arbor, and that this membrane could be the substrate of the
above potential distribution computations. For this to be feasible, however,
we require
\begin{quote}
Prediction 5. 
The mapping of the potential distribution onto the neuronal membrane
implies that the retinotopic coordinates are similarly mapped (at least
in open neighborhoods) onto the membrane.
\end{quote}

The large constructed
potential distributions may bear some resemblance to the large receptive fields
observed in areas V4 and 
IT (Maunsell and Newsome 1987). While any such relationship
is clearly speculative at this time, it should be noted that they have
two key similarities: (i) extremely large receptive fields (potential 
distributions)  have
been created, but they maintain about the same orientation selectivity
as in 
V1 (Desimone et al.\ 1985); (ii) their structure can change. We have stressed
how structure is controlled by upward flowing information, but it should 
be modifiable
by ``top-down'' attentional influences as well
(Maunsell and Newsome 1987; Moran and Desimone 1985). 
Attention could easily ``gate'' the
tangent field entries at the creation of the potential, which leads to:

\begin{quote}
Prediction 6. 
There exists a smallest scale of attentional control, and it
corresponds (in size) to the scale of the unit potential contributions.
\end{quote}

\begin{figure}
\figspace{4.25in}{4.25in}
\caption{Illustration of the splines in motion. Initially, each spline is born
at a  tangent field location, with unit length. Then, according to the
potential distribution shown in figure~1e, the splines migrate in position
(to find minima in the distribution) and in length, so that they overlap and
fill in short gaps. At convergence, the length of each spline has tripled.
Not shown is the fact that each spline is born with a different ``color,''
and that, as they overlap, the colors equilibrate to a unique value for
the entire covering of each global curve. Also, those splines that migrate
to positions unsupported by the potential distribution are eliminated at
convergence. (a) Initial distribution; (b) and (c) intermediate iterations;
(d) final convergence. Physiologically one might think of the spline
computations as being supported by localized
dendric or dendro-dendritic interactions.}
\end{figure}


\section{Conclusions}

This paper is both constructive and
speculative. 
On the constructive side, we have outlined a computational solution to
the curve detection problem that fills the
wide gulf between initial broad measurements of orientation and
precise final descriptions of global curve structure. Much of 
the mythology that has developed around curve detection is due, we believe,
to ascribing  too little 
function  to the first (measurement) stage, and too much function to
the second (integration) stage. Our solution was to interpose a stable 
description---the tangent field---between the stages, to
represent the local 
properties of curves (and their discontinuities). Three points emerged:
(i)  represent the
local structure coarsely, not in fine detail; so that (ii) the different
possibilities can be represented explicitly
and (iii)
do not  assume that local properties must be computed purely locally. 
Once the tangent field was in place, the task for the second, global stage
could then be posed, and led to
the introduction of the mathematical
notion of a cover to suggest parallel (and hence at least not 
biologically implausible) mechanisms for recovering global information.
Finally, we introduced the notion of a potential distribution as the
representation for mediating the local to global transition between the two
stages.

The paper has also been speculative. Problems in vision are complex, and
computational modeling can certainly help in understanding them. But in our
view computational modeling cannot proceed without direct constraints from
the biology, and modeling --- like curve detection --- should 
involve both coarse-grained and finer-grained theories.
We attempted to illustrate how such constraints could be abstracted 
by speculating how our model could map onto physiology.
While much clearly remains to be done, the role for curvature at several levels 
now seems 
evident. That such roles for curvature would have emerged from more traditional
neural network modeling seems doubtful.

Two different styles of computation emerged in the two stages of curve 
detection. Although we stressed their differences in the paper, in closing we
should like to stress their similarities. Both stages
enjoy formulations as variational problems, and recognizing the hierarchy of
visual processing, we cannot help but
postulate that the second, fine stage of curve detection may well be
the first, coarse stage of shape description. The fine splines then
become the coarse units of shape.

\section*{Acknowledgments}

This research was sponsored by NSERC grant A4470. 
We thank R.~Milson and especially C.~David for their contributions to
the second stage of this project.


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{\flushleft
\rule{2in}{.1mm}\hfill\break
\footnotesize Received 14 October; accepted 23 October 1988.
\normalsize}
\endpaper{zucker}









