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Davies: Computer Vision, 5th edition: Solutions to selected problems 3
3.3
This type of filter emulates the mode filter, and as such will enhance images by taking a
3.5
(a) See the main text in Chapter 3.
(b) Clear the whole histogram initially; then clear only the elements that have been filled,
after finding each median. Find the window minimum; then set sum equal to the
minimum before proceeding to find the median. Not having to clear the whole histogram
(c) See the main text in Chapter 3.
(d) original: 1 2 1 1 2 3 0 2 2 3 1 1 2 2 9 2 2 8 8 8 7 8 8 7 9 9 9
3 3 median: ? 1 1 1 2 2 2 2 2 2 1 1 2 2 2 2 2 8 8 8 8 8 8 8 9 9 ?
3.6
(a) On the background side of any edge, the background intensity dominates, so the
mode filter comes up with a central background intensity value. Similarly, it comes up
with a central foreground value on the foreground side of any edge. There is a critical
point on any edge where the dominant peak in the local intensity distribution changes over
(b) When a max filter is applied to an image, light background regions expand, and will
tend to reduce the size of objects, or even to eradicate small objects. It will also tend to
produce regions of constant (high) intensity in the image. In general, the mode filter will
Davies: Computer Vision, 5th edition: Solutions to selected problems 4
between foreground and background, so it will also cut into small regions of background
on high curvature boundaries.
(c) The purpose of the median filter is to suppress noise by eliminating local outliers in
intensity. In addition, it aims (successfully) to achieve this while not producing any local
(d) orig: 0 1 1 2 3 2 2 0 2 3 9 3 2 4 4 6 5 6 7 0 8 8 9 1 1 8 9
mean: ? ? 1 2 2 2 2 2 3 3 4 4 4 4 4 5 6 5 5 6 6 5 5 5 5 ? ?
max: ? ? 3 3 3 3 3 3 9 9 9 9 9 6 6 6 7 7 8 8 9 9 9 9 9 ? ?
(e) Further applications of a median filter in this case have no further effect. The max
3.7
(a) The 3 3 median filter removes the noise points (including the bump) and also
(b) To obtain a corner detector, first use the 3 3 median filter to remove the noise
points; then use the 5 5 median filter to find the corner points. This method has
3.8
(a) Mean filters average and return the mean intensity in the window; median filters
(b) The median algorithm is bookwork (see the main text in Chapter 3). It operates
Davies: Computer Vision, 5th edition: Solutions to selected problems 5
(c) By cascading max operations, the maximum value can be found in 8 operations. One
(d) However, finding the max in a 1 3 window takes 2 operations, and then finding it
again takes 1 operation, after setting the first max to zero. Thus the overall operation takes
3.9
(a) Results:
(i) 0 0 0 0 0 1 0 1 1 1 1 1 1 1 (input)
0 0 0 0 0 0 1 1 1 1 1 1 1 1 (output)
(b) General lessons:
(i) Median filters can eliminate impulse noise, but may introduce edge shifts. This
also applies to 2D images.
(c) Hint: Much of the computation for 2-D windows arises from loading the local
