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Marr
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LEGENDS
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ε3Legends to the figuresε0

1.  Examples of pairs of perspective line drawings presented to the
subjects of Shepard & Metzler's (1971) experiments on mental rotation.
(A) A "same" pair, which differs by an 80 degree rotation in
the picture plane; (B) a "same" pair which differs by an 80 degree
rotation in depth; (C) a "different" pair, which cannot be brought into
congruence by any rotation. The time taken to decide whether a pair
is the "same" varies linearly with the (3-D) angle by which one must
be rotated to be brought into correspondence with the other.
(reconstructed from figure 1 of Shepard & Metzler, 1971).

2.  This diagram summarises our overall view of the visual
recognition problem, and it embodies several points that our approach
takes as assumptions.  The first is that the recognition process
decomposes to a set of modules that are to a first approximation
independent.  The simplified subdivision shown here consists of four
main stages, each of which may contain several modules. (1) The
translation of the image into a primitive description called the
ε1primal sketchε0 (Marr 1976b); (2) The division of the primal sketch
into regions or forms, through the action of various grouping processes
ranging in scope from the very local to global predicates like a rough
type of connectedness;  (3) The assignment of an axis-based description
to each form (see figure 4); and (4) The construction of a 3-D model for
the viewed shape, based initially on the axes delivered by (3).  The
relation between the 3-D model representation of a shape and the image
of that shape is found and maintained with the help of the image-space
processor. Finally, the representation of the geometry of a shape is
separate from the representation of the shape's use or purpose
(Warrington & Taylor 1973).

3. 3a shows the image of a toy bear, printed in a font with 16 grey levels.
In 3b, the intensity at each point is represented along the z-axis. 3c illustrates
the spatial component of the raw primal sketch as obtained from this image.
Associated with each line segment are measures of contrast, type and extent of
the intensity change, position and orientation. This image is so simple
that purely local grouping processes suffice to extract the major forms
from the primal sketch. These forms are exhibited in 3d, e & f.

4. The geometry of constraints on the computation of binocular disparity.
4a illustrates the constraints for the case of a one-dimensional image.
Lx and Ly represent the positions of descriptive elements from the left
and right views, and the horizontal and vertical lines indicate the range
of disparity values that can be assigned to left-eye and right-eye elements.
The uniqueness condition states that only one disparity value may be assigned
to each descriptive element. That is, only one disparity value may be "on"
along each horizontal or vertical line. The continuity condition states that
we seek solutions in which disparity values vary smoothly almost everywhere.
That is, solutions tend to spread along the dotted diagonals, which are lines
of constant disparity, and between adjacent diagonals.
4b shows how this geometry appears at each intersection
point. The constraints may be implemented by a network with positive and negative
interactions that obey this geometry, because the stable states of such a network
are precisely the states that satisfy the constraints on the computation.
4c shows the constraint geometry for a 2-dimensional image. The negative
interactions remain essentially unchanged, but the positive ones now extend
over a small 2-dimensional neighbourhood.  A network with this geometry was
used to perform the computation exhibited in figure 8.

5.  Examples of 3-D models, and their arrangement into the 3-D
model representation of a human shape.  A 3-D model consists of a model
axis (a) and component axes (b) that consist of a principal axis (the
torso) and several auxiliary axes (the head and limbs) whose positions
are described relative to the principal axis.  The complete human 3-D
model is enclosed in a rectangle (c).  The 3-D model representation is
obtained by concatenating 3-D models for different parts at different
levels of detail.  This is achieved by allowing a component axis of one
3-D model to be the model axis of another.  Here, for example, the arm
auxiliary axis in the human 3-D model acts as the model axis for the arm
3-D model, which itself has two component axes, the upper and lower
arms.  The figure shows how this scheme extends downwards as far as the
fingers.

6. Analysis of a contour from Vatan and Marr (1976). The outline
(a) was obtained by applying local grouping operations to a primal
sketch, as in figure 4.  It is then smoothed, and divided into convex and
concave components (b).  The outline is searched for deeply concave
points or components, which correspond to strong segmentation points.  One
such point is marked with an open circle in (c).  There are usually
several possible matching points for each strong segmentation point, and
the candidates for the marked point are shown here by filled circles
(c).  The correct mates for each segmentation point can usually be found
by eliminating relatively poor candidates. The result of doing this here
is the segmentation shown in (d).  Once these segments have been
defined, their corresponding axes (thick lines) are easy to obtain (e).
They do not usually connect, but may be related to one another by
intermediate lines which are called ε1embedding relationsε0 (thin lines
in f).  According to the 3-D representation theory, the resulting stick figure (f)
is the deep structure on which interpretation of this image is based.

7.  (a) and (b) show two views of a water-pail.  Warrington &
Taylor's (1973) patients are impaired on (b), but not on (a).  This is
consistent with the 3-D model representation, for reasons that are clear from (c)
and (d).  The outlines of the original figures are shown as thin lines,
and the axis is shown as a thick one.  This axis is directly recoverable
from image (a), but not from (b) where it is severely foreshortened.
Since the 3-D model representation relies on an explicit representation
of this axis, the successful recognition of views like (b) requires
considerable extra computation.

8. Graphical representation (a) of the decomposition of a "simple" nonlinear,
n-input "algorithm"into a sum of interactions of various order. The
functional representation
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S{ . . x (t) . .⎇  =  L   +  ε2Sε0   L   {x  (t)⎇ +  ε2Sε0   L    {x  (t),x  (t)⎇ +  . . .
where L is an n-linear mapping, can be read from an appropriate sequence
of such elementary graphs. Fig. 8b shows the graphs that implement the
fly's orientation behavior, studied by Reichardt and Poggio. Several
findings suggest that they may correspond to separate physiological modules.
Characteristic functional and computational properties can be associated to each
interaction type. (From Poggio and Reichardt, 1976).

9. A pair of random dot stereograms (left and right), the initial state of
a network that implements the algorithm illustrated in figure 4, and various
iterations of the network operating on this stereo pair. To understand how the figures
represent states of the network, imagine looking down on it from above. The different
disparity layers in the network are in parallel planes spread out horizontally,
and the viewer is looking down through them.
In each plane, some nodes are on and some are off. Each layer in the network
has been assigned a different gray level, so that a node that is switched
on in the lowest layer contributes a light point to the image, and one that is switched
on in the top layer contributes a darker point. Initially (iteration 0) the
the network is disorganised, but in the final state order has been achieved
(iteration 14). The central squares have a divergent disparity relative
to the background, and they therefore appears lighter. The density of the original
random dot stereogram was 50%, but the algorithm succeeds in extracting
disparity values at densities down to less than 5%.  Let    C↓xyd↔ denote the
state of a cell (either 0 or 1) in the 3-D array of fig. 4b in position ε1(x, y)ε0
and disparity ε1dε0 at the n-th iteration. Then the algorithm used here reads
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C↓xyd↔   =  u   { ε2Sε0     C↓x'y'd'↔     -  ε2e  Sε0     C↓x'y'd'↔     +       C↓xyd↔      ⎇,
.sp
where u  (z) = 0  if  z < ε2qε0,  and  u  (z)  = 1  otherwise; S(xyd)  is  a
neighborhood of cell (xyd) on the same disparity layer; O(xyd)
represents the neighborhood of cell (xyd) defined by the "orthogonal"
directions shown in Fig. 4b. Excitation between parallel disparity
layers may also be present.
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