Viola-Jones' face detection claims 180k features - algorithm

Question

I've been implementing an adaptation of Viola-Jones' face detection algorithm. The technique relies upon placing a subframe of 24x24 pixels within an image, and subsequently placing rectangular features inside it in every position with every size possible.
These features can consist of two, three or four rectangles. The following example is presented.
They claim the exhaustive set is more than 180k (section 2):
Given that the base resolution of the detector is 24x24, the exhaustive set of rectangle features is quite large, over 180,000 . Note that unlike the Haar basis, the set of rectangle
features is overcomplete.
The following statements are not explicitly stated in the paper, so they are assumptions on my part:
There are only 2 two-rectangle features, 2 three-rectangle features and 1 four-rectangle feature. The logic behind this is that we are observing the difference between the highlighted rectangles, not explicitly the color or luminance or anything of that sort.
We cannot define feature type A as a 1x1 pixel block; it must at least be at least 1x2 pixels. Also, type D must be at least 2x2 pixels, and this rule holds accordingly to the other features.
We cannot define feature type A as a 1x3 pixel block as the middle pixel cannot be partitioned, and subtracting it from itself is identical to a 1x2 pixel block; this feature type is only defined for even widths. Also, the width of feature type C must be divisible by 3, and this rule holds accordingly to the other features.
We cannot define a feature with a width and/or height of 0. Therefore, we iterate x and y to 24 minus the size of the feature.
Based upon these assumptions, I've counted the exhaustive set:
const int frameSize = 24;
const int features = 5;
// All five feature types:
const int feature[features][2] = {{2,1}, {1,2}, {3,1}, {1,3}, {2,2}};
int count = 0;
// Each feature:
for (int i = 0; i < features; i++) {
int sizeX = feature[i][0];
int sizeY = feature[i][1];
// Each position:
for (int x = 0; x <= frameSize-sizeX; x++) {
for (int y = 0; y <= frameSize-sizeY; y++) {
// Each size fitting within the frameSize:
for (int width = sizeX; width <= frameSize-x; width+=sizeX) {
for (int height = sizeY; height <= frameSize-y; height+=sizeY) {
count++;
}
}
}
}
}
The result is 162,336.
The only way I found to approximate the "over 180,000" Viola & Jones speak of, is dropping assumption #4 and by introducing bugs in the code. This involves changing four lines respectively to:
for (int width = 0; width < frameSize-x; width+=sizeX)
for (int height = 0; height < frameSize-y; height+=sizeY)
The result is then 180,625. (Note that this will effectively prevent the features from ever touching the right and/or bottom of the subframe.)
Now of course the question: have they made a mistake in their implementation? Does it make any sense to consider features with a surface of zero? Or am I seeing it the wrong way?

Answer 1

Upon closer look, your code looks correct to me; which makes one wonder whether the original authors had an off-by-one bug. I guess someone ought to look at how OpenCV implements it!
Nonetheless, one suggestion to make it easier to understand is to flip the order of the for loops by going over all sizes first, then looping over the possible locations given the size:
#include <stdio.h>
int main()
{
int i, x, y, sizeX, sizeY, width, height, count, c;
/* All five shape types */
const int features = 5;
const int feature[][2] = {{2,1}, {1,2}, {3,1}, {1,3}, {2,2}};
const int frameSize = 24;
count = 0;
/* Each shape */
for (i = 0; i < features; i++) {
sizeX = feature[i][0];
sizeY = feature[i][1];
printf("%dx%d shapes:\n", sizeX, sizeY);
/* each size (multiples of basic shapes) */
for (width = sizeX; width <= frameSize; width+=sizeX) {
for (height = sizeY; height <= frameSize; height+=sizeY) {
printf("\tsize: %dx%d => ", width, height);
c=count;
/* each possible position given size */
for (x = 0; x <= frameSize-width; x++) {
for (y = 0; y <= frameSize-height; y++) {
count++;
}
}
printf("count: %d\n", count-c);
}
}
}
printf("%d\n", count);
return 0;
}
with the same results as the previous 162336
To verify it, I tested the case of a 4x4 window and manually checked all cases (easy to count since 1x2/2x1 and 1x3/3x1 shapes are the same only 90 degrees rotated):
2x1 shapes:
size: 2x1 => count: 12
size: 2x2 => count: 9
size: 2x3 => count: 6
size: 2x4 => count: 3
size: 4x1 => count: 4
size: 4x2 => count: 3
size: 4x3 => count: 2
size: 4x4 => count: 1
1x2 shapes:
size: 1x2 => count: 12 +-----------------------+
size: 1x4 => count: 4 | | | | |
size: 2x2 => count: 9 | | | | |
size: 2x4 => count: 3 +-----+-----+-----+-----+
size: 3x2 => count: 6 | | | | |
size: 3x4 => count: 2 | | | | |
size: 4x2 => count: 3 +-----+-----+-----+-----+
size: 4x4 => count: 1 | | | | |
3x1 shapes: | | | | |
size: 3x1 => count: 8 +-----+-----+-----+-----+
size: 3x2 => count: 6 | | | | |
size: 3x3 => count: 4 | | | | |
size: 3x4 => count: 2 +-----------------------+
1x3 shapes:
size: 1x3 => count: 8 Total Count = 136
size: 2x3 => count: 6
size: 3x3 => count: 4
size: 4x3 => count: 2
2x2 shapes:
size: 2x2 => count: 9
size: 2x4 => count: 3
size: 4x2 => count: 3
size: 4x4 => count: 1

Answer 2

all. There is still some confusion in Viola and Jones' papers.
In their CVPR'01 paper it is clearly stated that
"More specifically, we use three
kinds of features. The value of a
two-rectangle feature is the difference between the sum of the
pixels within two rectangular regions.
The regions have the same size and
shape and are horizontally or
vertically adjacent (see Figure 1).
A three-rectangle feature computes the sum within two outside
rectangles subtracted from the sum in
a center rectangle. Finally a
four-rectangle feature".
In the IJCV'04 paper, exactly the same thing is said. So altogether, 4 features. But strangely enough, they stated this time that the the exhaustive feature set is 45396! That does not seem to be the final version.Here I guess that some additional constraints were introduced there, such as min_width, min_height, width/height ratio, and even position.
Note that both papers are downloadable on his webpage.

Answer 3

Having not read the whole paper, the wording of your quote sticks out at me
Given that the base resolution of the
detector is 24x24, the exhaustive set
of rectangle features is quite large,
over 180,000 . Note that unlike the
Haar basis, the set of rectangle
features is overcomplete.
"The set of rectangle features is overcomplete"
"Exhaustive set"
it sounds to me like a set up, where I expect the paper writer to follow up with an explaination for how they cull the search space down to a more effective set, by, for example, getting rid of trivial cases such as rectangles with zero surface area.
edit: or using some kind of machine learning algorithm, as the abstract hints at. Exhaustive set implies all possibilities, not just "reasonable" ones.

Answer 4

There is no guarantee that any author of any paper is correct in all their assumptions and findings. If you think that assumption #4 is valid, then keep that assumption, and try out your theory. You may be more successful than the original authors.

Answer 5

Quite good observation, but they might implicitly zero-pad the 24x24 frame, or "overflow" and start using first pixels when it gets out of bounds, as in rotational shifts, or as Breton said they might consider some features as "trivial features" and then discard them with the AdaBoost.
In addition, I wrote Python and Matlab versions of your code so I can test the code myself (easier to debug and follow for me) and so I post them here if anyone find them useful sometime.
Python:
frameSize = 24;
features = 5;
# All five feature types:
feature = [[2,1], [1,2], [3,1], [1,3], [2,2]]
count = 0;
# Each feature:
for i in range(features):
sizeX = feature[i][0]
sizeY = feature[i][1]
# Each position:
for x in range(frameSize-sizeX+1):
for y in range(frameSize-sizeY+1):
# Each size fitting within the frameSize:
for width in range(sizeX,frameSize-x+1,sizeX):
for height in range(sizeY,frameSize-y+1,sizeY):
count=count+1
print (count)
Matlab:
frameSize = 24;
features = 5;
% All five feature types:
feature = [[2,1]; [1,2]; [3,1]; [1,3]; [2,2]];
count = 0;
% Each feature:
for ii = 1:features
sizeX = feature(ii,1);
sizeY = feature(ii,2);
% Each position:
for x = 0:frameSize-sizeX
for y = 0:frameSize-sizeY
% Each size fitting within the frameSize:
for width = sizeX:sizeX:frameSize-x
for height = sizeY:sizeY:frameSize-y
count=count+1;
end
end
end
end
end
display(count)

Answer 6

In their original 2001 paper they only state that they used three kinds of features:
we use three kinds of features
with two, three and four rectangles respectively.
Since each kind has two orientations (that differ by 90 degrees), perhaps for the computation of the total number of features they used 2*3 types of features: 2 two-rectangle features, 2 three-rectangle features and 2 four-rectangle features. With this assumption there are indeed over 180,000 features:
feature_types = [(1,2), (2,1), (1,3), (3,1), (2,2), (2,2)]
window_size = (24,24)
total_features = 0
for f_type in feature_types:
for f_height in range(f_type[0], window_size[0] + 1, f_type[0]):
for f_width in range(f_type[1], window_size[1] + 1, f_type[1]):
total_features += (window_size[0] - f_height + 1) * (window_size[1] - f_width + 1)
print(total_features)
# 183072
The second four-rectangle feature differs from the first only by a sign, so there is no need to keep it and if we drop it then the total number of features reduces to 162,336.