JNTU B.Tech IV COMPUTER VISION Previous Question Paper (September - 2022 - R16)

To respond to the second derivative in the horizontal direction while maintaining invariance to the gradient (first derivative) and absolute intensity in the horizontal direction, and ensuring total invariance in the vertical direction, we design a 1D kernel that acts on the horizontal axis and is convolved with a smoothing or identity kernel in the vertical direction. The standard discrete second derivative operator is [1, -2, 1].

To achieve the required properties, the 3x3 filter kernel H is defined as:

Justification:

• Horizontal Second Derivative: The row [1, -2, 1] approximates the Laplacian or second-order difference, which detects edges by identifying zero-crossings.
• Invariance to Gradient/Intensity: Because the sum of the coefficients is zero (1 - 2 + 1 = 0), the filter output is zero for any constant intensity region or linear gradient (first derivative), satisfying the invariance requirement.
• Vertical Invariance: By setting the top and bottom rows to zero, the filter ignores all vertical variations, effectively acting only on the horizontal scan line.

Given the image A and the 3x3 structuring element B (all ones), the morphological operations are performed by sliding B over A. Since B is a 3x3 matrix of ones, erosion takes the minimum value in the 3x3 neighborhood, and dilation takes the maximum.

Image A:

1. Erosion (Min operation): For a 3x3 window, the output at position (i,j) is the minimum of the overlapping pixels. Assuming zero-padding or boundary handling where only valid full overlaps are considered:

Calculation: The top-left 3x3 block [11, 18, 13; 12, 2, 22; 22, 22, 22] has a minimum of 2. The other windows similarly yield 2.

2. Dilation (Max operation): For a 3x3 window, the output at position (i,j) is the maximum of the overlapping pixels:

Calculation: The top-left 3x3 block [11, 18, 13; 12, 2, 22; 22, 22, 22] has a maximum of 22. The bottom-right 3x3 block [2, 22, 22; 22, 22, 2; 68, 70, 6] has a maximum of 70.