A short quiz on convolution and friends

Adrian F. Clark

1. What is matched filtering?

   • finding matches by convolution
   • finding the morphological skeleton
   • using correlation as the 'statistic' at each pixel

   • designing a mask that matches a desired image feature

   Matched filtering is designing a convolution mask that will produce a strong response at specific image features.

2. What does a morphological opening tend to do?

   • delete isolated dark pixels within light regions

   • delete isolated light pixels within dark regions

   • find the skeleton of light regions
   • find the boundary between dark and light regions

   An opening is a shrink followed by an expand. The shrink reduces the size of light regions, so an isolated pixel (for a $3 \times 3$ mask) within a dark region will disappear. The expand then grows any light regions that remain.

3. Given a white feature on a black background, how could you find its boundary?

   • subtract the result of shrink from the original image

   • subtract the result of opening from the original image
   • subtract the result of closing from the original image
   • subtract the result of expand from the original image
   • subtract the result of opening from the original image

   A shrink using a $3 \times 3$ mask reduces the size of light regions by a single pixel around its boundary. Subtracting this from the original image leaves just the boundary.

4. What is the effect of convolving an image with the mask $$\pmatrix{0 & 1 & 0\cr 1 & 1 & 1\cr 0 & 1 & 0\cr}$$

   • light spots will be enhanced

   • +-shaped features will be enhanced

   • dark spots will be enhanced
   • X-shaped features will be enhanced

   This produces a large output when the value below the middle of the mask lies at the centre of a light +-shaped feature.

5. What is a morphological grey-level shrink?

   • taking the minimum when performing a convolution

   • taking the mode when performing a convolution
   • taking the standard deviation when performing a convolution
   • taking the maximum when performing a convolution

   A morophological shrink involves taking the minimum when performing a convolution. It has the effect of reducing the size of light-coloured regions on a dark background.

6. What is the effect of convolving an image with a +-shaped mask?

   • +-shaped features will be enhanced

   • dark spots will be enhanced
   • light spots will be enhanced
   • X-shaped features will be enhanced

   This produces a large output when the value below the middle of the mask lines at the centre of a light +-shaped feature.

7. What is the effect of convolving an image with the mask $$\pmatrix{0 & 1 & 0\cr 1 & -4 & 1\cr 0 & 1 & 0\cr}$$

   • dark spots will be enhanced

   • light spots will be enhanced
   • +-shaped features will be enhanced
   • X-shaped features will be enhanced

   This produces a large output when the value below the middle of the mask is much lower than the pixels on its four sides --- and this will tend to enhance dark spots in an image.

8. How does the time taken to perform a convolution vary with the dimension of a (square) mask?

   • it is proportional to the dimension of the mask

   • it is proportional to the square of the dimension of the mask

   • it is proportional to the logarithm of the dimension of the mask
   • it is not affected by the dimension of the mask

   The execution time is proportional to the number of pixels in a mask, so for a mask of dimension 3 there will be $3 \times 3 = 9$ operations per pixel, and so on. Hence, the time taken is proportional to the square of the dimension of the mask.

9. What is a morphological grey-level expand?

   • taking the minimum when performing a convolution
   • taking the mode when performing a convolution

   • taking the maximum when performing a convolution

   • taking the standard deviation when performing a convolution

   A morophological shrink involves taking the maximum when performing a convolution. It has the effect of increasing the size of light-coloured regions on a dark background.

10. What is the effect of convolving an image with a +-shaped mask where the central pixel is -4 and its neighbours to north, south, east and west are 1?

    • dark and light spots will be enhanced

    • X-shaped features will be enhanced
    • the image will be lightened
    • +-shaped features will be enhanced

    This produces a large output when the value below the middle of the mask is much lower than the pixels on its four sides -- and this will tend to enhance dark spots in an image.