JNTU B.Tech III DIGITAL IMAGE PROCESSING Previous Question Paper (November/December - 2020 - R16)

Given V={0,1}, the pixels with values in V are considered connected. The coordinates are p=(3,3) and q=(0,0) (assuming 0-indexed grid).

D4 distance (City-block distance): The distance is the sum of the absolute differences of the coordinates: D4(p,q) = |x - x_q| + |y - y_q| = |3 - 0| + |3 - 0| = 3 + 3 = 6. Since all intermediate pixels in the path have values in V={0,1}, the path is valid.

D8 distance (Chessboard distance): The distance is the maximum of the absolute differences of the coordinates: D8(p,q) = max(|x - x_q|, |y - y_q|) = max(|3 - 0|, |3 - 0|) = max(3, 3) = 3. This path is also valid as the pixels involved contain values in V.

The Slant transform is a non-sinusoidal orthogonal transform characterized by a sawtooth basis function. For an 8x8 matrix, it is defined recursively.

The 2x2 Slant matrix is:

The 2N x 2N Slant matrix S_2N is constructed from the N x N matrix S_N using the relation:

where I_N is the identity matrix and A_N is a specific sparse matrix containing the slant coefficients. For N=4, the coefficients are derived to maintain orthogonality and energy compaction. The 8x8 Slant matrix is obtained by performing this construction twice starting from S_2.

The order of sequence for the Slant transform refers to the arrangement of basis vectors based on the number of zero-crossings (sequency). The basis vectors are ordered such that the number of sign changes increases with the index, starting from the DC component (constant vector) to the highest frequency component, similar to the Walsh-Hadamard transform but with smoother transitions due to the slanted basis functions.