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## 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

In 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.