Lab 2
DUE: Nov 10 23:55

Problems are from Gonzales et al. 4th Edition. Read this document carefully. Ask if you are unsure what to do, otherwise use common sense to solve the problems. Upload your solutions to Moodle; both typed and photocopied hand-ins are fine.
Warning! Plagiarism will not be tolerated.

1. 10.10 (10 points)

   Consider a horizontal intensity profile though the middle of a binary image that contains a vertical step edge through the center of the image. Draw what the profile would look like after the image has been blurred by an averaging kernel of size \(n \times n\) with coefficients equal to \(1/n^2\). For simplicity assume that the image was scaled so that its intensity levels are \(0\) and the left of the edge and \(1\) on the its right. Also, assume that the size of the kernel is much smaller than the image, so that image border effects are not a concern near the center of the image.

2. 10.20 LoG (10 points)

   One often finds in the literature a derivation of the Laplacian of Gaussians (LoG) that start with the expression \begin{equation} G(r) = e^{-r^2/2\sigma^2} \end{equation} where \(r^2=x^2+y^2\). The LoG is then derived by taking the second partial derivative with respect to \(r\): \begin{equation} \nabla G(r) = \partial^2 G(r)/\partial r^2 \end{equation} Finally, \(x^2+y^2\) is substituted for \(r^2\) to get to the final (incorrect) result: \begin{equation} \nabla^2 G(x,y) = \left[(x^2+y^2-\sigma^2)/\sigma^4\right]\exp\left[-(x^2+y^2)/2\sigma^2\right] \end{equation} Derive this result and explain the reason for the difference between this expression and Eq. (10-29) shown also below \begin{equation} \nabla^2 G(x,y) = \left[(x^2+y^2-2\sigma^2)/\sigma^4\right]\exp\left[-(x^2+y^2)/2\sigma^2\right] \end{equation}

3. 12.19 (10 points)

   For the figure shown,

   (a) What is the order of the shape number?

   (b) Obtain the shape number.

   

4. 12.25 (10 points)

   Consider a binary image of size \(200 \times 200\) pixels, with a vertical black band extending from columns 1 to 99 and a vertical white band extending from column 100 to 200.

   (a) Obtain the co-occurrence matrix of this image using the position operator “one pixel to the right.”

   (b) Normalize this matrix so that its elements become probability estimates, as explained in Section 12.4.

   (c) Use your matrix from (b) to compute the six descriptors in Table 12.3.