0.4 Iris recognition: gabor filtering

Gabor wavelets

To understand the concept of Gabor filtering, we must first start with Gabor wavelets. Gabor wavelets are formed from two components, a complex sinusoidal carrier and a Gaussian envelope. $$g(x,y)=s(x,y)w_r(x,y)$$ The complex carrier takes the form: $$s(x,y)=e^{j(2(u_{0}x+v_{0}y)+P)}$$ We can visualize the real and imaginary parts of this function seperately as shown in this figure.

The second component of a gabor wavelet is its envelope. The resulting wavelet is the product of the sinusoidal carrier and this envelope. The envelope has a gaussian profile and is described by the following equation:

To put it all together, we multiply $s(x,y)$ by $w_r(x,y)$ . This produces a wavelet like this one:

Generating an iris code

Now that we have Gabor wavelets, lets do something interesting with them. Lets start with an image of an eye and then unroll it (map it to cartesian coordinates) so we have something like the following:

Any given iris has a unique texture that is generated through a random process before birth. Filters based on Gabor wavelets turn out to be very goodat detecting patterns in images. We'll use a fixed frequency 1D Gabor filter to look for patterns in our unrolled image. First, we'll take a one pixel widecolumn from our unrolled image and convolve it with a 1D Gabor wavelet. Because the Gabor filter is complex, the result will have a real and imaginary partwhich are treated seperately. We only want to store a small number of bits for each iris code, so the real and imaginary parts are each quantized. If a givenvalue in the result vector is greater than zero, a one is stored; otherwise zero is stored. Once all the columns of the image have been filtered and quantized,we can form a new black and white image by putting all of the columns side by side. The real and imaginary parts of this image (a matrix), the iris code, areshown here:

Comparing iris codes

The problem of comparing iris codes arises when we want to authenticate a new user. The user's eye is photographed and the iris code produced from the image. It would be nice to be able to compare the new codeto a database stored codes to see if this user is allowed or to see who they are. To perform this task, we'll attempt to measure the Hamming distance between two iris codes. The Hamming distance between any twoequal length binary vectors is simply the number of bit positions in which they differ divided by the length of the vectors. This way, two identical vectors have distance 0 while two completely different vectors havedistance 1. Its worth noting that on average two random vectors will differ in half their bits giving a Hamming distance of 0.5. The Hamming distance is mathematically defined in this equation: $$D=\frac{AB}{\text{length(A)}}$$

In theory, two iris codes independently generated from the same iris will be exactly the same. In reality though, this doesn't happen vary often for reasons such as imperfect cameras, lighting or small rotationalerrors. To account for these slight inconsistencies, two iris codes are compared and if the distance between them is below a certain threshold we'll call them a match. This is based on the idea of statisticalindependance. The iris is random enough such that iris codes from different eyes will be statistically independent (ie: have a distance larger than the threshold) and therefore only iris codes of the same eyewill fail the test of statisical independance. Empirical studies with millions of images have supported this assertion. In fact, when these studies used the threshold used in our method (.3) false positive ratesfell below 1 in 10 million.