A short quiz on convolution and friends

Adrian F. Clark

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

   • the image is unchanged

   • the image is shifted down one pixel

   • the image is blurred
   • the image is shifted up one pixel

   At each pixel, the new value becomes the value in the line above, which has the effect of moving the image down by one pixel.

2. What is the effect of convolution with the following mask? $$\frac19 \pmatrix{1 & 1 & 1\cr 1 & 1 & 1\cr 1 & 1 & 1\cr}$$

   • it will blur edges

   • it will enhance edges
   • it has no effect
   • it will detect isolated points

   Averaging over a region will tend to blur an image.

3. What is a morphological grey-level shrink?

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

   • taking the minimum when performing a convolution

   • taking the mode 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.

4. 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?

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

   • dark and light spots 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.

5. What is matched filtering?

   • designing a mask that matches a desired image feature

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

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

6. 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 not affected by 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

   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.

7. What is a morphological grey-level expand?

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

   • taking the maximum when performing a convolution

   • taking the minimum 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.

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

   • light spots will be enhanced

   • dark 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.

9. What does a morphological opening tend to do?

   • delete isolated dark pixels within light regions
   • find the boundary between dark and light regions

   • delete isolated light pixels within dark regions

   • find the skeleton of 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.

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

    • X-shaped features will be enhanced

    • +-shaped features will be enhanced

    • light spots will be enhanced
    • dark spots 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.