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Section 6.6 Notes 194

Applications for shape from texture have been largely absent, explaining its
status as a minority interest. However, we believe that image-based rendering of
clothing is an application with substantial promise. Cloth is difficult to model
for a variety of reasons. It is much more resistant to stretch than to bend: this
means that dynamical models result in stiff differential equations (for example,
see (Terzopolous et al. 1987)) and that it buckles in fine scale, complex folds (for
example, see (Bridson et al. 2002)). However, rendering cloth is an important
technical problem, because people are interesting to look at and most people wear
clothing. A natural strategy for rendering objects that are intrinsically difficult to
model satisfactorily is to rearrange existing pictures of the objects to yield a render-
ing. In particular, one would wish to be able to retexture and reshade such images.
Earlier work on motion capturing cloth used stereopsis, but faced difficulties with
motion blur and calibration (Pritchard 2003, Pritchard and Heidrich 2003). More
recent work prints a fine pattern on the cloth (White et al. 2007), or uses volume
intersections (Bradley et al. 2008b). We believe that, in future, shape from texture
methods might make it possible to avoid some of these problems.

PROBLEMS
6.1. Show that a circle appears as an ellipse in an orthographic view, that the minor
axis of this ellipse is the tilt direction, and that the aspect ratio is the cosine
of the slant angle.
6.2. We will study measuring the orientation of a plane in a perspective view, given
that the texture consists of points laid down by a homogeneous Poisson point
process. Recall that one way to generate points according to such a process is
to sample the x and y coordinate of the point uniformly and at random. We
assume that the points from our process lie within a unit square.
(a) Show that the probability that a point will land in a particular set is
proportional to the area of that set.
(b) Assume we partition the area into disjoint sets. Show that the number of
points in each set has a multinomial probability distribution.
We will now use these observations to recover the orientation of the plane. We
partition the image texture into a collection of disjoint sets.
(c) Show that the area of each set, backprojected onto the textured plane,is a
function of the orientation of the plane.
(d) Use this function to suggest a method for obtaining the plane’s orientation.

PROGRAMMING EXERCISES
6.3. Texture synthesis: Implement the non-parametric texture synthesis algo-
rithm of Algorithm 6.4. Use your implementation to study:
(a) the effect of window size on the synthesized texture;
(b) the effect of window shape on the synthesized texture; and
(c) the effect of the matching criterion on the synthesized texture (i.e., using
a weighted sum of squares instead of a sum of squares, etc.).