Midterm CMSC828J 2006
Assigned, 10/19/06
Due, 10/26/06
1. Convolution: Consider a 1D signal that is a Gaussian, centered at 0, with a
standard deviation of σ1.
2
1
2
2
2
1
σ
πσ
x
e−
What is the result of convolving this with a Gaussian with a standard deviation of σ2?
Give an analytic expression.
2. Diffusion: In this problem we will consider the distribution of particles in 1D
among a discrete number of positions, just as when we first learned about
diffusion. Suppose we have a very large number of particles at -100 and an
equally large number of particles at 100. Every second, 1% of the particles jump
1 to the left, and 1% jump 1 to the right.
a. Write an expression for the distribution of particles after T seconds. You
may approximate this with a continuous distribution, in a way that is only
valid for large T.
b. How much time will pass before the distribution is unimodal?
3. Edge Detection: These questions concern the Canny edge detector.
a. Suppose we have an image that consists of a black region and a white
region, separated by a vertical line. That is, imagine the image is black for
negative values of x, and white for zero or positive values of x. Is there
any choice of parameter settings that will result in edges consisting of two,
parallel vertical lines? Give an example, or explain why this is
impossible.
b. Suppose we have a 2D image that consists of a sharp corner. On the left,
the image is black. On the right, it is white, with a region that has a corner
