Due date: 25th May 2011
For the theoretical parts of the solutions either type them and send them as a pdf file, or write them down as a hard copy and deliver it in the box near the lecture room (48-462), the box is labelled "Exercises / AG. Prof. Breuel".
FIR Filter (30 points)
Implement a 2D Gaussian filter using a
shift-add as the computational kernel. That is, compute the output
image by adding shifted versions of the input image weighted by the
coefficients of the kernel. Take advantage of separability of the
Gaussian to speed things up.
Efficient Filtering Operations (50 points)
A common operation in image processing
is to convolve with the derivative of a Gaussian. This can be done
separably, convolving with the derivative of a Gaussian along one
axis and with a Gaussian along the other axis. Let us consider the
Gaussian derivative convolution. (a) Implement a FIR filter using a
kernel G' that is computed from the analytic form of the Gaussian
derivative. (b) Consider a simple derivative filter given by the
kernel D=[-1,1] and a regular Gaussian with kernel G. How do G' * S,
D * (G * S), (D * G) * S and (G * D) * S compare for a
1D signal S? How close are their impulse responses? (c) Assume that
D * G is a sufficiently close approximation for your purposes,
explain how to use combinations of Dx, Dy, Gx, and Gy to compute
directional derivatives of an image in both x and y directions most
efficiently (Dx and Dy are the discrete derivative operators in the x
and y direction, with kernels [-1,1], and Gx and Gy are the 1D
convolutions with a Gaussian in the x and y directions).
Laplacian and Unsharp Masking (50 points) This problem is due on 1st June 2011
the lecture, we discussed the use of the Laplacian operator for the
enhancement of image details. Unsharp masking works by smoothing an image with a Gaussian and then
subtracting the smoothed image from the original (look on the web for more detail).
(1) Derive the analytic form of the Laplacian-of-the-Gaussian in 2D. Show the kernel as an image.
(2) What kernel does unsharp masking correspond to? State an analytic form of the kernel.
(3) Discuss the relationship between unsharp masking and the Laplacian.