Convex cuboid hulls in RGB space

Question

Greetings,
imagine an RGB (Red, Green, Blue) cube with quite a bit of resolution, i.e. all R, G, and B can have
values between [0, (1<<20)-1] (twenty bits resolution). These values are read from three ADC's
(Analog to Digital Converters), part of a 'colour sensor'.
These sensors are fast: they can 'sense' a colour value (RGB) within less than a millisecond and
above all: they can read those values very accurately. The hardware stuff is beyond me, I'm just a
silly math guy, but in short, every colour RGB is represented by three twenty bits values.
A sensor has an Atmel micro processor on board. I've written some software for that Atmel,
testing those sample RGB colour values against a set of RGB cuboids, i.e. if a colour value
RGB (a three dimensional point) lies within some cuboid (a finite sub space of the entire RGB cube)
something is wrong/wright.
Allow me to elaborate on this: one application for those sensors is controlling the quality of broken eggs
(don't laugh ;-) Eggs are broken and egg-white is separated from the rest. Sometimes this breaking
process (a mechanical process) fails, i.e. there's no pure egg-white. We're talking a few million eggs
per machine here. Pure egg-white is used for a zillion purposes; don't ask me to elaborate here ...
Egg-white has a certain (RGB) colour range with a certain illumination intensity. In this RGB cube (see
above), the colour of egg-white can be represented by a set of cuboids within this RGB cube, i.e. if
a sensored values set lies within a set of cuboids, the sensor just sampled egg-white, otherwise
something else must've passed the sensor's 'eye' (think of shell particles, blood in case of a 'naughty
egg', etc. etc.)
Suppose there's a set of n samples rgb_1, rgb_2, ... rgb_n, given a number m, the maximum number
of cuboids that contain all samples rgb_i, find the cuboids c_1, c_2 ... c_m such that the sum of the
'volumes' of those cuboids is minimal. (n > m).
I have a nice and fast heuristic up and running right now, but I'm very interested in more ideas, formal
problem definitions, applicability of an LP (simplex) problem definition, oct-trees, etc. etc.
If you need more information, don't hesitate to ask (my email is available somewhere in my profile also).
Thanks in advance for any sensible reply and,
kind regards,
Jos
ps. this hardware stuff also does its dirty deeds in plastic-granulate quality testing, BMW/Mercedes
dashboard colour quality control and a few other applications.

Answer 1

given your space constraints the cuboids probably make better sense than a regression tree, none the less, you have still given no justification for why you want to minimize the volume of any cuboid.
Presumably the thing you are trying to optimize is the classification value of the resultant set of boxes and it seems unlikely to me that minimum cuboid size in any way does that. In fact, if you are not careful, choosing a strict tolerance based on an existing data set, will make your system too fragile.
I assume that in your model you will have cuboids that are indicators of good points and other cuboids that are indicators of bad points, and I further assume that you have a set of training data of properly labled points.
You could take any collection of cuboids whatsoever, overlapping or otherwise and you get a bit vector that is the result of the simultaneous tests against all the cuboids. (if you were imagining a disjoint set of cuboids you were imagining a bit vector with at most a single bit set. In the more general model where more than one bit could be set, you introduce some means of conflict resolution, like choose the first on bit, or you vote, or you take some weighted average)
In any case, a single bit vector represents a region in space (a complicated voting model will affect the shapes of the regions that you are modeling), and for any such region you can get a correct baysian probability for the classification of objects within that region.
You can then talk about the classification probabilities for that particular set of cubes. You can measure the number correctly classified and the number incorrectly classified. Presumably you want a high ratio on that.
I have perhaps confused the issue by introducing this bit vector representation. My real point is this. It makes little sense to me to write a routine to optimize one thing (like minimization of cuboid volume) and then see if the results conform to your intuition and just happen to optimize the thing you are actually trying to optimize for which is a good color classifier, rather than just starting right out and attempting to construct the optimal classifier.
If I were faced with this problem, even if I were to stick with the cuboid representation that you have choosen, I would focus on defining the classification function that I want to optimize and then attempt to maximize that. Any grubby old optimization technique like a GA should be able to give you a decent set of cuboids.
You also wanted some other names. Kohonen networks, Neural Gas and of course good old Nearest Neighbor classifications are perfectly fine classification algorithms and could get you there.
The Neural Gas model, if you are not familar with it already, is perhaps the easiest one to quickly grasp. Is it essentially a continuous K-means process. You create m random cuboids. You choose a sample point at random from you training set. Find the box that is nearest the sample point. Slide the center of the winning box a little closer (like maybe a tenth of the way) toward the sample point and expand the box slightly (or shrink it if it was already inside). Of course if the point was a "bad sample" you do the reverse, you push the box away a bit a bit. You can also keep an age on each box, (length of time since last hit) and retire a box if it "gets too old" and thus is not capturing and representing the samples. When you retire a box the tradition is to introduce a new one near the boxes that are getting the most activity.
Of course you can simultaneously grow both a set of cubes for positive results as well as for negtative results.
But trust me, having built classifiers before, you would do well to spend some time up front in defining what you mean by a well performing classifier. With a quantative measure for the quality of your final classifier, you will be in a much better position to search for and to evaluate design solutions that fit your space and speed constraints.
Enjoy!