Feature Extraction & Image Processing, 2nd Edition

have, and then the shadow will become much fainter. We can also include spatial averaging as

in Section 3.3.2, to reduce further the effect of the walking subject, as shown in Figure 3.22(h).

This gives spatiotemporal averaging. For this, we have not required any more images, but the

penalty paid for the better improvement in the estimate of the background is lack of detail. We

cannot see the numbers in the clock, because of the nature of spatial averaging. However, if we

form the background image by taking the median of the six images, a temporal median,weget

a much better estimate of the background, as shown in Figure 3.22(i). A lot of the image detail

is retained, while the walking subject disappears. In this case, for a sequence of images where

the target walks in front of a static background, the median is the most appropriate operator. If

we did not have a sequence, we could just average the single image with a large operator and

that could provide some estimate of the background.

(a) Image 1 (b) Image 2 (c) Image 3 (d) Image 4 (e) Image 5 (f) Image 6

(g) Temporal averaging (h) Spatiotemporal averaging (i) Temporal median

Figure 3.22 Background estimation by mean and median filtering

3.5.3 Mode filter

The mode is the final statistic of interest, although there are more advanced filtering operators

to come. The mode is very difficult to determine for small populations and theoretically does

not even exist for a continuous distribution. Consider, for example, determining the mode of

the pixels within a square 5×5 template. It is possible for all 25 pixels to be different, so each

could be considered to be the mode. As such, we are forced to estimate the mode: the truncated

median filter, as introduced by Davies (1988), aims to achieve this. The truncated median filter

is based on the premise that for many non-Gaussian distributions, the order of the mean, the

median and the mode is the same for many images, as illustrated in Figure 3.23. Accordingly,

if we truncate the distribution (i.e. remove part of it, where the part selected to be removed in

Figure 3.23 is from the region beyond the mean) then the median of the truncated distribution

will approach the mode of the original distribution.

The implementation of the truncated median, trun_med, operator is given in Code 3.14.

The operator first finds the mean and the median of the current window. The distribution of

intensity of points within the current window is truncated on the side of the mean so that the

median now bisects the distribution of the remaining points (as such, not affecting symmetrical

94 Feature Extraction and Image Processing

No. of points

Mode

Median

Mean

Brightness

Figure 3.23 Arrangement of mode, median and mean

ha←floor

wsze

2

⎛

⎟

⎝

⎞

⎟

⎠

win←submatrix(p,y−ha,y+ha,x−ha,x+ha)

med←median(win)

upper←

2

⋅med−min(win)

lower←

2

⋅med−max(win)

cc←

0

cc←cc+

1

trun

cc

←win

j,i

trun

cc

←win

j,i

cc←cc+

1

if

(

win

j,i

>lower

)

⋅(med>ave)

if

(

win

j,i

<upper

)

⋅(med<ave)

for i∈

0

..wsze−

1

for j∈

0

..wsze−

1

newpic

y,x

←median(trun) if cc>

0

newpic

y,x

←med otherwise

for y∈ha..rows(p)−ha−

1

for x∈ha..cols(p)−ha−

1

newpic

trun_med(p,wsze):= newpic←zero(p)

ave←mean(win)

Code 3.14 The truncated median operator

distributions). So that the median bisects the remaining distribution, if the median is less than

the mean, the point at which the distribution is truncated, upper,is

upper = median +median −mindistribution

=2 ·median −mindistribution (3.26)

Basic image processing operations 95