Cognitive Science 126/ Psych 126
Section quiz – week of 4/17/00
1. What is shape equivalence?
Shape equivalence is the ability to perceive distinct objects as having the same shape
despite differences due to various spatial transformations.
2. Differentiate shape equivalence from the concept of shape constancy (studied in
chapter 7).
Shape constancy involves seeing the SAME object as having the same shape under
different viewing conditions. Shape equivalence always involves different objects.
Shape equivalence might involve perceiving the similar shape of toy cars and real
cars despite the large differences in their size.
3. Define objective shape.
Objective shape is defined as the spatial structure of an object that does not change
over the following spatial transformations: translations, rotations, dilations, and
reflections.
4. Draw the appropriate transformations of each of the objects:
Rotation:
Translation:
Dilation:
Reflection:
See figure 8.1.1 in the text
Draw some transformations that are not members of the similarity group:
5. What are the 3 approaches to determining shape equivalence. Briefly describe each
approach.
Invariant Features Approach - This approach assumes that shape perception depends
on encoding properties that are invariant across transformations, in this case the
similarity transformations. Look at all of the different properties and see which ones
don't vary across the transformations. If all of these are the same, then the shape is the
same.
Transformational Alignment - try to bring each object into alignment with
comparison object via performing some combination of the similarity
transformations.
Object-centered Reference Frame - each object has a self-defined reference frame.
Objects are compared within their own reference frames, essentially cancelling out all
of the similarity transformations and thus allowing direct comparison.
6. Why does Mach's square-diamond example pose a problem for both invariant features
and transformational approaches?
A diamond has the same set of invariant features as a square, thus the invariant
features approach would predict that these two different shapes should be the same.
A diamond is simply a rotational transformation of a square. There is no basis for
distinction between the two types of shapes.
7. List the 4 features of a coordinate system and relate them to the 4 types of
transformations discussed above.
Origin = translational transformation
Reference Orientation = rotational transformation
Reference Distance (unit size) = dilation transformation
Reference Sense (defines direction along axis) = reflection transformation
8. How can the object-centered hypothesis account for Mach's Square/Diamond
problem?
Each shape has its own reference frame. Thus, diamonds have a frame defined by the
vertical symmetry while squares have a separate frame defined by their symmetries.
Because there are frames defined by the intrinsic properties, the problem is solved.
Based on the frame, diamonds are fundamentally different from squares. The
determination of whether it is a diamond or a square will depend on the frame of
reference assigned.
9. What are some of the problems of template models of shape representation?
Multiple sensory channels - not everything about shape can be found in the forms of
luminance edges. This creates a problem of mapping the templates across different
sensory channels that correspond to different types of information like illusory
contours, and stereoscopic depth edges.
Spatial Transformations - if one has to have templates for every orientation and other
spatial transformation, one finds a combinatorial explosion of templates needed.
Part Structure -
Three-dimensions - templates are generally 2D things. Thus, how does one map them
onto 3D objects that can show up in various different views?