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Bootstrapped learning of novel objects
Abstract
Recognition of familiar objects in cluttered backgrounds is a challenging computational problem. Camouflage provides a particularly striking case, where an object is difficult to detect, recognize, and segment even when in “plain view.” Current computational approaches combine low-level features with high-level models to recognize objects. But what if the object is unfamiliar? A novel camouflaged object poses a paradox: A visual system would seem to require a model of an object’s shape in order to detect, recognize, and segment it when camouflaged. But, how is the visual system to build such a model of the object without easily segmentable samples? One possibility is that learning to identify and segment is opportunistic in the sense that learning of novel objects takes place only when distinctive clues permit object segmentation from background, such as when target color or motion enables segmentation on single presentations. We tested this idea and discovered that, on the contrary, human observers can learn to identify and segment a novel target shape, even when for any given training image the target object is camouflaged. Further, perfect recognition can be achieved without accurate segmentation. We call the ability to build a shape model from high-ambiguity presentations bootstrapped learning.
History
Received December 20, 2002; published July 22, 2003
Citation
Brady, M. J. & Kersten, D. (2003). Bootstrapped learning of novel objects.
Journal of Vision, 3(6):2, 413-422,
Keywords
object recognition, learning, camouflage, segmentation, background, clutter, color, motion, mechanochemical, morphogenesis, novel objects, top down, bottom up
A fundamental function of
biological vision is to detect and recognize potential food items and predators
from naturally cluttered backgrounds. The task can be especially difficult when
the form and coloration of the target objects are similar to the background.
Camouflage provides a particularly striking example, where natural (or
artificial) mechanisms work to disguise an object even when in “plain
view.” It wasn’t until the advent of computer vision research in the
1960s and 1970s that it was realized that most objects, not just those that are
camouflaged, blend in with their backgrounds to a surprising extent. In fact, to
this day, the segmentation of static figures from cluttered backgrounds has no
robust machine vision solution. To the best computer vision system, almost all
objects are camouflaged, not just those intending to
hide. What we might call the
unintentional camouflage of everyday vision is the rule, rather than the
exception. Finding object contour boundaries is difficult because image edges
can be caused by illumination and material changes, and not just the depth
discontinuities defining object boundaries. Further, when an object is seen
against a cluttered background, its contour boundaries tend to merge with the
contours of background elements. There is an apparent scarcity of local image
features that remain invariant over viewpoint, lighting, and background changes.
The objective local ambiguity stands in contrast to the speed and accuracy with
which humans can identify objects in natural images (Thorpe, Fize, & Marlot,
1996).
It is generally believed that the extraordinary
competence of the primate visual system at detecting and recognizing objects
involves coupling local image measurements with global knowledge of the shapes
and properties of objects and object classes previously seen. Early
computational approaches used generic knowledge of surface properties
(piece-wise smoothness) to link together contour segments or texture
measurements likely to belong to the same surface (Poggio, Torre, & Koch, 1985). Systems
relying solely on generic grouping principles, however, tend not to be robust.
Greater robustness can be achieved, at the cost of generality, by relying on
specific knowledge of familiar objects for both segmentation and for
nonsegmented classification (Amit & Geman,
1999; Yuille, 1991). High-level object
knowledge is also important for dealing with occlusion and object articulations.
However, the importance of high-level object knowledge leaves us with a profound
paradox. When object learning begins, there is no model of the object. If
ambiguous local image measurements are not sufficient for object recognition,
then surely it is not sufficient for model building. We call this the
bootstrapped learning problem.
One way out of this paradox is to assume that learning
occurs under conditions of low ambiguity. For example, the movement or color of
an object may distinguish it from its background. Motion is a well known basis
for segmentation (Braddick,
1974; Lamme,
1995), and many animals display vivid
color patterns as warnings, or for social recognition (Brown, 1975). Therefore, it may be that
object learning is opportunistic, occurring when segmentation is possible, but
not under conditions of high ambiguity or camouflage, and it is only after
learning that an observer can recognize or segment a static camouflaged object.
We tested this hypothesis by training observers on images in which a camouflaged
novel object is presented against a cluttered background with (1) motion-defined
boundaries, (2) color-defined boundaries, or (3) ambiguous boundaries. The
hypothesis predicts that on testing, recognition performance should improve for
the low-ambiguity conditions (1 & 2), but not for the ambiguous
condition.
To generate unfamiliar camouflaged objects, we specify
object, camouflage, and scene models for image variation.
In order to study how objects are learned in
camouflage, it is necessary to have stimuli that retain the generic properties
of naturally important surfaces and yet are unfamiliar. Plants and animals are
fundamental to our survival; unfortunately, there are no guarantees that any set
of plants or animals will be novel to all observers in an experiment. We solved
this problem by simulating some aspects of embryological development to grow 3D
shapes, rendered using computer graphics (M.
J. Brady, 1999). For a demonstration
of the growth process, see M. Brady, 1999. Our morphogenic algorithm is
mechanochemical (i.e., intracellular forces as well as diffusion of a chemical
morphogen are both simulated, such that chemical pattern formation and shape
forming movements of cells occur simultaneously). For other examples of
mechanochemical morphogenesis, see Ball (1999) as well as Murray (1993). We call our resulting
objects digital embryos. (See right panel of Figure
1). Digital embryos appear to be organic forms but do not resemble a
familiar class of organism.
Figure 1. A. A photograph of a flat-tailed
gecko, with and without background. B.
An artificial morphogenic object, or “digital embryo,” also with and
without background. Despite the fact that both objects are unoccluded, they
cannot be segmented without prior knowledge of the object. Digital embryo scenes
mimic aspects of nature’s more severe forms of camouflage. Gecko
photograph by M. Kramer (http://home.wxs.nl/~mkramer/).
Camouflage occurs when the surface texture and/or shape
of an animal or object appears similar to the background (left panel of Figure 1).
Background
model. We adopted the extreme form of camouflage in which the target
object is set against a background of similar objects, all drawn from the same
class, in our case, other digital
embryos that also had albedo variations that we describe next (right panel of Figure 1).
Surface texture
model. In nature, intra-species albedo variation may seem minor (as in
deer) or it may be major (as in zebra and giraffe). Upon close inspection, many
of the seemingly minor variations prove to be quite extensive. Within
individuals, albedo variation may occur due to mud wallowing, precipitation,
sweat, molting, shedding, changing clothes, or wearing makeup. Shadows cast from
forest canopies may also be confounded with these albedo variations. In this
experiment, we mimic the more rigorous conditions found in the learning of both
object classes and individuals by the use of major changes in albedo patterns.
The appearance of a given object of fixed shape was varied by painting, or
texture mapping, different gray-level albedo patterns on the surface. The
camouflage was made particularly challenging by using albedo patterns that were
themselves images of other digital embryos. One consequence of this manipulation
is to introduce albedo edges, internal to the object, that mimic shading seen at
self-occluding boundaries or folds (Ben-Shahar, Huggins, & Zucker, 2002).
Using albedo patterns, which mimic 3D-surface shading and discontinuities, might
seem to be an unrealistically difficult form of camouflage. However, such
mimicry appears frequently in nature (Thayer,
1909), a dramatic example of which is found in the moths of the genus
Callionima (http://dlp.cs.berkeley.edu/photos/). Also, albedo patterns, which
mimic the 3D surfaces of vegetation, are the basis of many popular patterns on
modern hunting
clothing.
Our camouflage model ensured that no geometric forms
other than embryos need to be introduced or their effects explained.
We fixed object orientation, distance, and
illumination, but varied the background objects, position of the object of
interest, and camouflage patterns from trial to trial. In the color training
scenes, objects of interest were colored green. In motion training scenes,
objects of interest were animated over a parabolic path with quasi-random
coefficients. (See “Appendix A” for rendering details). The complete
set of stimuli can be viewed and downloaded (Brady & Kersten, 2000).
We wanted to characterize algorithmically the extent of
albedo and contrast variation, in order to measure the extent to which portions
of an object appear repeatedly in a sequence of training images. To do this, we
applied a translation invariant consistency algorithm, which detects objects or
object parts that appear repeatedly in a series of scenes. The results are shown
in Figure 2. Object fragments reappear to
varying degrees. However, there is considerable uncertainty whether repeating
fragments are from the object or from the background.
Figure 2. Output of the consistency
algorithm. The consistency algorithm takes a set of training images and attempts
to find object fragments, which appear repeatedly in the training sequence (see
“Appendix B”). The algorithm is
challenged because the object images vary due to changing camouflage,
translation in 3D space, and minor changes of relative viewpoint induced by the
translation. The algorithm attempts to translate a series of training images to
align the object of interest in each. In the image resulting from the averaging
of the translated training images, background pixels will tend toward a mean
value (minimum consistency score), whereas the darker and lighter portions of
the object of interest will remain near the extreme pixel values (maximum
consistency score). Pseudo-coloration A allows the reader to consider various
thresholds for object versus background. B-D. Distance-from-the-mean images from
a training set of 3, 7, and 20 images, respectively. The object of interest is
shown in E. Some hypothetical fragments may be selected from these images but
the uncertainty is high. F-H. Algorithm output for 3, 7, and 20 training images
with the same object shown in E, except that the object was not camouflaged
during training. The algorithm performs much better, showing the effect of
camouflage on image consistency. J-L. Output from 3, 7, and 20 training images
containing the object shown in I. Uncertainty is high after 3 training images
but decreases with further training. A suitably chosen threshold could segment
out a diagnostic fragment for use in object recognition. Other thresholds would
produce a mix of false positives and false negatives. Performance was similar on
4 out of 9 object training sets. N-P. Results from the training set for the
object shown in M. Uncertainty remains very high. Four out of 9 training sets
produced similar results. In all training sets, uncertainty is very high at the
onset of training and remains high even after 3 images. One would expect that an
observer would not be able to segment the first image of a training set.
We trained six adult observers on scenes of novel
objects and then tested their ability to recognize those objects. The training
and testing scenes were generated by placing a digital embryo at random in a
scene, applying camouflage, then placing other camouflaged embryos to fill in
the background (Figure 1, upper right). The
camouflage patterns consisted of scenes of novel objects. Every scene had a new
set of background objects and new camouflage patterns on the objects.
The goal of training was to imitate a natural object
learning scenario, where an animal views an object, such as another animal, the
object may disappear for some seconds, then reappear. It may not reappear for
some hours or days, when learning may resume. Each time it appears it has a new
background and if it is a different instance of the same object class, it may
have a new surface pattern. The object may emit a sound or an odor, which helps
to identify it. In this experiment, we played a sound with each
appearance.
Each observer was trained on three sets of three
objects. One set was shown as moving against its background, one set was shown
in color against its background, and one set was shown with no supplemental
segmentation clues. We call this last case the ambiguous
case.
Movie 1. The first six scenes of a
training session, in the ambiguous segmentation case.
Training took place over 4 days. Each training session
consisted of showing object A for 10 s, B for 10 s, C for 10 s, and then A, B,
C, repeating until each object had appeared 5 times. In each appearance of an
object, the camouflage pattern, background, and object location was changed. In
order to compare any two scenes of the same object, observers had to hold one
scene in memory during a 20-s delay and two intervening learning
tasks. Observers were tested on their ability to recognize
objects. Test stimuli consisted of a camouflaged object with a background of
other camouflaged objects (Figure 1, right).
The object of interest was one of the three current training objects or a
completely novel object. Each trial was a four alternative forced choice where
the choices were “A, B, C, or other.” No sounds or segmentation
clues were provided and the camouflage patterns and backgrounds changed with
each scene. Testing took place on four days. Tests were given 24 hr after each
training session and before any new training. Thus, testing was of long term
memory.
Recall our initial hypothesis that recognition learning
cannot occur under conditions of high ambiguity or camouflage, but only when
segmentation is possible, and it is only after such opportunistic learning that
an observer can recognize or segment a static camouflaged object. Our scenes
were constructed so that when segmentation clues were not present, observers
could not be certain which elements belong to the object of interest and which
belong to background. How do observers perform in learning objects from images
with and without segmentation clues?
Figure 3 shows
observers’ ability to utilize segmentation clues during object learning.
However, our initial hypothesis is proven incorrect in general (Figure 3B), because three out of six observers did
not depend upon segmentation clues in order to learn new objects. Apparently,
they are able to combine information from a number of ambiguous sources to
produce a reliable object model. We call this phenomenon bootstrapped learning.
Figure 3B shows just how reliable bootstrapped
object models can be for some
observers. Figure 3. Novel object learning over 4
days of testing. Training occurred on days 1-4. Observers were divided into two
groups according to performance on the ambiguous case. A. Weak learners SE, PR,
and IB show little or no learning in the ambiguous case, but they were able to
exploit situations where segmentation clues were present. Chance performance is
.25 for an observer who guesses at all choices with equal probability. However,
chance performance is .5 for an observer who always guesses
”other” and this is the
upper bound of any guessing strategy. Statistics are by analysis of variance
(ANOVA) with clue and day as factors. Clue and day effects are significant, F
ratio has a p value < .0001.
Interaction between clue and day is not significant,
p value = .324. B.
“Super
observers” AN, WA, and SM provide
an existence proof for a powerful bootstrapped learning algorithm. They achieve
near perfect performance, even in the ambiguous case. All observers have normal
or better acuity and contrast sensitivity. Statistics are by ANOVA. Factors were
clue and day. Clue and day effects were significant, with
p values <.0001. Interaction is also
significant, with p value < .0001.
In an ANOVA comparing groups, supers are significantly better than weak
learners, p value <.0001.
The surprising result of Experiment 1 was that there
exist observers who could learn to recognize and segment objects in the
ambiguous training condition. In order to further explore the generality of this
finding, we did a second experiment with just the ambiguous training. Further,
we want to empirically establish that, prior to training, human observers could
not segment the objects with any degree of certainty. Experiment 2 quantified
the observers’ ability to segment scenes before and after training.
The same stimuli as in Experiment 1 were used, except
that the color and motion cases were omitted.
Six observers were trained as in Experiment 1, except
that there was no training on colored or motion scenes.
Observers were tested in their ability to recognize and
segment objects. Recognition testing was the same as in Experiment 1.
For segmentation testing observers were asked to trace
object contours in test scenes. Prior to any training, observers were asked to
trace three novel objects in test scenes. These objects were not part of an
individual subject’s subsequent training or testing set, but were used by
the other subjects as part of the balanced experimental design to control for
object-specific effects. After 4 days of training, observers were asked to trace
the three training objects of interest in novel scenes and were asked to trace
three still novel objects in test scenes. Tracing errors were of two types,
missed contour segments and extraneous tracing. Tracings were scored by
combining tracing path lengths as
follows:
At Trace Score = .5, the amount of correct tracing
equals the amount of tracing error.
Example tracings are shown in Figure 4. Prior to training, observers are
uncertain of what is the object and what is the background. They may miss the
object entirely (Figure 4A) or, because the object of interest is in the
foreground, they may trace part of the true boundary. However, part of the
background (Figure 4B and 4C) is included as
well. After training, both segmentation and recognition performance improve
significantly (Figure 5).
Figure 4. Tracing examples. Green and red
segments were originally drawn by the observer and color coded by the
experimenter. Green segments are correct and red are extraneous. Yellow segments
were missing in the observer’s trace and added by the experimenter. A.
Observer YH’s trace of object D on day 1, prior to any training. Score is
0.0. B. Observer YH’s tracing of object A on day 1. Score is .21. C.
FF’s tracing of object 1 on day 1. Score is .25. D. BK’s tracing of
D on day 5 after 4 days training on the object. Score is .41. E. KH’s
tracing of A on day 10 after 4 days training on the object. Score is .60. F.
BK’s tracing of 1 on day 10 after 4 days training on the object. Score is
1.0.
Figure 5. Recognition (top) and tracing
(bottom) of camouflaged objects with background and without segmentations clues.
Observers start with a new set of objects on day 6. Observers are significantly
better at recognizing and tracing familiar objects. A two factor ANOVA of the
tracing data, with novelty and subject factors, shows significant effects for
novelty (p =.00041), subject
(p =.00042), and insignificant
interaction (p =.64). An ANOVA was
performed on data averaging the two 1-week blocks. The difference between day 1
novel tracing and day 5 novel tracing was not significant in a
t test
(p =.29). The difference between day 5
trained tracing and day 10 trained tracing was significant in a
t test
(p =.02) but may be due to either a
generalized task training effect or an object effect, because object effects are
not controlled for within a single week. For examples of these tracing score
values, see Figure 4. Observers learn to recognize objects in spite of an
initial inability to segment them. In 5 of 12 day-1 tracings, super observers
had a 0.0 tracing score. Segmentation ability evolves in parallel with
recognition ability, as expected in the case of bootstrapped learning. There
were 4 super learners in Experiment 2, who learned to perfect or near perfect
performance, and 1 weak learner, whose performance did not improve by a
measurable amount. A sixth observer went from being a weak learner in week 1 to
being a super learner in week 2. Data is
averaged over super and weak groups. Novel scene tracing at day 5 is a control
for general task learning.
Over several days, and with relatively few image
presentations (20 scenes per object in Experiments 1 and 2), observers are able
to learn to recognize and segment camouflaged objects. Further, they were able
to do this with images that were highly ambiguous as shown in observers’
inability to initially segment any given training image. Evidence of
segmentation uncertainty comes not only from the objective consistency score and
the subjective tracing tasks, but also from the recognition task. Recognition of
objects after ambiguous training is near chance, even after a day of training.
If observers had a high confidence in object feature selection or segmentation,
they are unable to exploit this during recognition. Given that Experiment 1
shows observers’ ability to utilize segmentation information when
available, an inability to utilize
segmentation information would be even more surprising than bootstrapped
learning.
Previous psychophysical studies of unsupervised novel
shape learning have tended to focus on scenes without background clutter, with
components that are segmentable without learning. Fiser and Aslin (2001) showed that human observers can learn
shape-composites based on probabilistic co-occurences of potential parts.
Bootstrapped learning is distinguished from other types of object learning in
that observers learn from scenes where the classification of edge segments into
boundary segments and background segments is uncertain.
Computer models of novel object learning with
background clutter are rare. However, Weber, Welling, and Perona
(2002) have developed an algorithm that can
learn uncamouflaged objects in cluttered scenes. This algorithm does not
necessarily compute edges, whereas our human observers do in the segmentation
tracing task. Shams, Brady, and Schaal
(2001) have developed a somewhat similar algorithm, which has shown some
ability to learn uncamouflaged digital embryos in background. However, it is
unlikely to be able to cope with the degree of camouflage found in the present
experiment. Both of these algorithms make use of a constellation of features
approach, in which a set of features is collected, along with their relative
positions.
We believe that bootstrapped learning for our observers
may have been accomplished by a process such as the following: 1.The first
image containing object A (image A1) is presented. 2.Features or object parts
are extracted from image A1, by a method such as that described by Malik et al., (2001) and Tu and Zhu (2002a, 2002b), and stored in a
working memory buffer. Such features must be more subtle than templates of
object images or templates of object image parts. (See algorithmic
characterization of scenes above, especially Figure 2M-2P.) 3.Working memory
content persists while two or more unrelated images (B1, C1, etc.) are processed
over a period of 20 s or more. 4.Image A2 is presented. 5.Features are
extracted from A2. 6.The intersection of the two feature sets is tested for
preservation of relations between features within each image. 7.The resulting
subset of features is bound together and stored in long-term working model
memory as an evolving model. This long-term working model memory can persist for
at least 24 hr. 8.Steps 1-7 are repeated for images A2, A3, etc., except that
an evolving high-level model is now available to help segment relevant features
via a top down mechanism.Any algorithm used by the
bootstrapping observers must depend upon two memories: the working memory buffer
and the long-term working model memory.
A major challenge for machine vision research is to
create a system that can learn to recognize objects from example images. But,
which way should one proceed? The target system might need to be opportunistic
and subsist on a diet of well-segmented scenes. These scenes would have to be
prepared manually and therefore require a great deal of human labor.
Alternatively, the opportunistic system might be placed in a rich environment
and simply wait for well-segmented images to occur. This could take a
considerable amount of time and require its own form of automation. Fortunately,
we have been able to demonstrate the existence of a learning algorithm, which
does not depend on special opportunities. Instead, it proceeds directly to learn
objects given only the most ambiguous, yet commonly occurring, images of those
objects. By defining what information is required by an object learner, machine
vision researchers are able to pursue one avenue to machine object learning
rather than several.
Natural scenes tend to be highly cluttered, which
presents a challenge to observers learning to see new objects. Yet there are
opportunities when segmentation may be easier, such as when color or motion
segmentation clues are present. Our study began with the hypothesis that
learners of novel objects would necessarily rely on such opportunities to
overcome segmentation problems, especially when dealing with severely
camouflaged objects. We found, on the contrary, that there exist two routes to
object recognition. One is opportunistic whereas the other relies on
bootstrapped learning.
Six observers participated in each experiment. All were
20/20 or better on the standard Lighthouse test at 4 m and in a modified
Lighthouse at the experimental viewing distance of 61 cm.
Images were 18-cm square and viewed on an iMac computer
at 72-dpi resolution.
Images were generated using the Inventor library on a
Silicon Graphics computer. The position of objects of interest were randomized
at an
(x,y)
location with mean
(x,y,z)
=(0,0,0). The
(x,y)
position of the object of interest varied
±.375. In each scene, 21
background objects had
(x,y)
coordinates ±1.5 and z coordinates from -1.8 to -9. The projection was
perspective with the virtual camera at (0,0,5). Units are arbitrary Inventor
units.
Lighting was directional with fixed direction vector
(1, -1, -1). Objects were rendered with Phong shading using Inventor parameters
diffuse color = (.8, .8, .8), specular color = (1, 1, 1), ambient color = (.5,
.5, .5), and shininess = 1.
Texture wrapping uses Inventor’s default method.
First, the bounding box for each object is computed. Next, the texture image is
projected onto each side of the box and then onto the object’s
polygons.
Each work week (5 days) observers were trained on day
1, tested for recognition and then trained on days 2-4, and they were tested for
recognition on day 5. This was repeated for each segmentation clue type in
Experiment 1.
In Experiment 2, observers performed (a) a novel
tracing task on day 1, week 1; (b) a novel and a familiar object tracing task on
day 5, week 1; and (c) a familiar tracing task on day 5, week 2. During each
week, the observers used a different set of three training objects. Object sets
were permuted evenly among observers to control for object difficulty and
ordering effects.
This program measures the consistency with which object
regions in a series of n images appear,
relative to random background fragments. In the interest of modeling the
iterative nature of human learning, the algorithm collects information one image
at a time and consolidates it with a current assessment of image consistency.
The algorithm first takes the pixel-wise log of each
image. It then computes a sequence of composite images. Composite pixel
intensity value
c(k,i,j)
at
(i,j)
in the kth composite is
defined recursively
as
where
p(k,i,j)
is a pixel intensity value in the
kth
training image and
(tx(k),ty(k))
is a translation of the
kth
training image.
c(1,i,j)
is simply
p(1,i,j).
(tx(k),ty(k))
is chosen so as to
minimize
The final translated mean image
M is simply the
nth
composite, where n
is the number of training images.
Let the mean within
M be
_(M)
Background pixels will tend toward this mean. Therefore, any pixels, which tend
toward the extremes are likely to be on object fragments. To visualize this,
compute a new image
A threshold may be applied to select candidate
object fragments, or as we have done in this paper, the image may be
pseudo-colored to portray numerous possible thresholds at once.
Digital embryos are generated using simulated hormonal
diffusion, simulated physical forces, and polygon fission. These operations are
applied repeatedly to an evolving polyhedron. Any polyhedron can be used as a
starting shape. In the current application, a regular icosahedron was
used.
Two loops operate concurrently. One loop controls the
lifecycles of morphogen secreting cells and morhogen diffusion among cells. A
second loop controls cell division and simulates the physical dynamics of the
cells. Cells are represented by vertices in the polyhedron geometry.
The morphogen secretor lifecycle and loop is simple. A
fixed number of vertices (3 in the current experiment) are maintained as
morphogen secretors. Each generator is assigned a finite lifespan at random. At
the end of a particular generator’s lifespan, it is replaced by another
generator somewhere else on the surface of the embryo. The location is
determined at random.
Active morphogen secretors retain a fixed high
morphogen concentration, which diffuses to adjacent calls or vertices. Morphogen
flows between vertices, which are connected by edges. The flow rate is
proportional to the difference in the concentrations of the adjacent vertices.
There is also a constant leakage of morphogen, out of each cell, into the
surrounding fluid. The result of these effects is that the morphogen
concentration in any nonsecreting cell
i, with neighbors
j is
at time
t+1.
R is a diffusion
rate and n is the
number of vertices connected to vertex
i. The
polygon fission operation proceeds as follows: All polygons in the present
implementation are triangles. A triangle is marked for fission if the average
morphogen concentration of its constituent vertices is above some threshold. The
triangle is split into four new triangles as shown in Figure 6. After fission, vertex I is a
full-fledged vertex but vertices K and J are not. They cannot be allowed to move
as a normal vertex would because it might cause triangles AED and DFC to become
quadrangles, and non-planar ones at that. Therefore, vertices K and J remain
dependent vertices. What this means, in the case of K, for example, is that K
must remain on a line between D and E regardless of what forces act upon it. K
will be promoted to a nondependent vertex when AED is split.
Vertices move about in space according to the sum of
forces that act upon them. The amount of motion per time increment is
proportional to the magnitude of the force, whereas the direction of motion is
determined by the total force vector. All vertices in an embryo repel all other
vertices according to an inverse square law. At the same time, vertices, which
are attached by an edge, are attracted according to Hooke’s law.
Figure 6. Triangle DEF before and after
fission. DEF will eventually be replaced by KEI, IFJ, JDK, and KIJ. However, DEF
may persist for awhile as the neighbor of AED and DFC.
We thank an anonymous reviewer for suggesting the log
transform in the consistency algorithm. This research was supported by National
Institutes of Health Grant RO1 EY02857. Commercial relationships: none.
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