Visualizing Factors and Prime Factorization.

Update 8/2014: The calculator discussed below is also available in the Chrome store here (for free of course).

I’ve previously discussed Brent Yorgey’s factor diagrams. As the father of a 6 year old, I’ve found they are a great way to introduce the concepts of primes and factorization.

Since then, I dabbled with the javascript animations by Sean Seefried to create 2 related products:
1. a calculator, and
2. a factorization game.

Factor Diagram Calculator
The calculator does multiplication and division and allows the young ‘uns to explore the diagrams. I also recently added the ability to do exponentiation after watching Mike Lawler’s video on powers of 3 and Sierpinski’s triangle.

Multiplication:
Multiplication

Division:
Division

Calculator is here.

Factorization Game
Kids learn by playing, that is well known. So how to make a game out of all of this? I scripted up something simple whereby you’d be presented with a large number and have to factor it while the clock is ticking. Do this a few times, get a score. Then compare with friends collect badges, etc. That last bit (prizes, badges) is not written and is a whole separate app of course.

game

Game is here.

So as it now stands it is simple, a germ of an idea really. Any thoughts on how to improve the learning experience?

 

Factor Dominoes

Many have discovered Brent Yorgey’s very cool factorization diagrams. They seem like a great way to teach multiplication and factorization to children.

We also were excited by the domino game suggested by Malke Rosenfeld on her blog and decided to try to take her ideas and see if we could create viable game that also would be a fun way to think about factors and primes. To this end we brought together a math geek, a visual artist and a 6 year old ninja and started playing. Through trial and error we arrived at a set of rules.

Simple version of the rules:

Start by printing out a deck of cards. (We have attached a pdf to this post with numbers up to 24 that can be printed on card stock and cut up.)

Each player gets 6 cards. One card is turned face up in the middle. The first player tries to match the turned up card, following the match rules below. After that, players can choose to match the card at either end of the growing chain. If you can’t match then you draw another card. First person to run out of cards wins.

The whole game comes down to the matching rules:
– The number 1 matches anything
– The number 2 matches any even number
– Primes match primes (or as you can explain to a child: circles match circles)
– Other numbers have a major and minor group. For instance 9 has a major group of 3 and a minor group of 3. 15 has a major group of 5 and a minor group of 3. 18 has a major group of 3 and a minor group of 6.

expl

You can match based on major or minor group. If the card you want to play has the same major group or same minor group as the card to be matched, then you can play it. So 10 can match 15 (major group of 5 matches), and 9 can match 15 (minor group of 3 matches). A prime number can match either the minor or the major group, thus 5 can match 10 (major group), but 3 can match 15 (minor group).

Beyond the simple rules:

We started to think about a rule set that might work if you had numbers higher than 24.

Types of number visualizations: 

Primes are represented as circles.

Here are the types of Minor Groups:

Simple Minor groups are low primes:
1, 2, 3, 5, 7
low primes

Compounded minor groups:
4 (2×2), 6 (3×2)
compound

Doubly compounded minor groups:
8 (2x2x2), 9 (3x3x3), 16 (4x4x4), 18 (2x3x3)

Triple compounded minor groups:
24 (2x2x2x3), etc.

Major groups

A major group is a combination of N copies of one of these minor or compounded groups.

So here are the generalized matching rules:
– 1 matches anything
– 2 matches any even number
– Minor group types can be matched with each other within types, but not across.
– Major groups can be matched if they have an equal number of elements. For that, minor group types do not have to match.

Attachment:

Here is the pdf you can print out on card stock to make your own set of cards:

cards1-24

 

Constrain Face Detection for Better Face Recognition

Getting good results with PCA-based facial recognition algorithms depends on correcting for differences in lighting and alignment between the faces. Widely used techniques for correcting for lighting include using histogram equalization or discarding the first eigen vector. Techniques for correcting for alignment differences often involve locating facial features such as the eyes and then rotating the face.

I found these techniques problematic. The corrections for lighting can indeed reduce the impact of overall illumination effects but don’t work for side-lighting scenarios. Face alignment methods are complicated and error prone. In the case of eye alignment one would presumably use additional haar cascades to locate the right eye and the left eye which in turn are error-prone and repeat many of correction problems we have with faces.

It seems that face recognition problems stem from an overly permissive face detection algorithm. The haar cascade that comes with Open CV (called ‘frontalface_alt2’) is indeed very good at detection faces, including rotated faces. It seems it was trained with a sample set that included rotated faces and faces in all different illumination conditions.

Thus constraining face detection so that it only detects horizontal well lit faces would make face recognition much more accurate. That was my hypothesis at any rate and I decided to give it a try.

To this end I used the Open CV tools to build a training set that included only horizontal, full frontal and well lit faces. In the negative sample set I included faces rotated along the Z axis. A better negative set would have included faces rotated along the X and Y axes as well, but I didn’t do that. (By Z axis I mean an axis running vertically through the face).

The resultant cascade is available on https://github.com/rwoodley/iOS-projects/blob/master/FaceDetectionAlgoCompare/OpenCVTest/haarcascade_constrainedFrontalFace.xml. There is a sample iOS application there as well that allows you to compare this cascade with frontalface_alt2 as well as one other cascade.

I was quite pleased with the results. The detection only works on well lit, well aligned faces. This puts the onus on the user to submit a good face. I realize this scenario won’t work with everyone’s system requirements, but in my case it was quite useful.

See the sample output below from my test iPhone application. I aim the camera at a photo of Joe Biden. Above his face are 3 smaller faces in grey-scale. These represent the faces detected by an LBP cascade, the Open CV alt2 cascade and my constrained cascade respectively.

Now see the sample below where Joe Biden’s photo is tilted. The first 2 cascades still find his face, but the 3rd one (the constrained one) does not.

 

Face Recognition System Design Considerations

Building a face recognition algorithm involves a whole string of design decisions, each of which will impact how well the system can identify faces.

Here is a partial list of these decisions (*):

– Choice of faces used to train the detection algo. Note that both a positive and negative set is needed. In other words you need to show it faces it should recognize, and faces (or other images) it should reject.
– Image preprocessing decisions (size, greyscale, normalization).
– Face alignment corrections.
– Number of eigen vectors to use. In other words, the number of dimensions of your face space.
– Number of low order eigen vectors to exclude. The idea being that low order vectors encode irrelevant (for face recognition) information like lighting.
– Similarity measures. Aka nearest neighbor measures. (Euclidean vs Mahalanobis etc)
– Choice of PCA methodology (Eigenfaces vs Fisherfaces).

After laboriously doing numerous comparisons across multiple training sets on my own, I discovered this paper published in 2001 where some researchers investigated these issues in a very systematic fashion.
Computational and Performance Aspects of PCA-based Face Recognition Algorithms, by H. Moon and P. J. Phillips, Perception, 2001, Vol 30, pg 303-321 and (NISTIR 6486)
http://www.nist.gov/customcf/get_pdf.cfm?pub_id=151458

I was reassured to see that their conclusions were close to my own (which I’ll write up later). It is worth reading the entire paper, but some of main conclusions I took away are:

– Image Normalization helps, but the particular implementation is not critical.
– Performance increases until approximately 200 eigenvectors, then decreases slightly. However much of the performance gain can be captured with 100 eigenvectors.
– Removing the first low-order eigen vectors is best, removing the 2nd helps slightly.
– The similarity measure has a huge effect. Using an enhanced Mahalanobis classifier gives the best result.

(*) note that I’m following the framework implemented in the OpenCV software, namely using Haar classifiers for detection and some form of PCA for recognition.