Operators for Locating an Iris

Iris analysis begins with reliable means for establishing whether an iris is visible in the video image, and then precisely locating its inner and outer boundaries (pupil and limbus). Because of the felicitous circular geometry of the iris, these tasks can be accomplished for a raw input image I(x,y) by integrodifferential operators that search over the image domain (x,y) for the maximum in the blurred partial derivative, with respect to increasing radius r, of the normalized contour integral of I(x,y) along a circular arc ds of radius r and center coordinates (x_o,y_o):

where * denotes convolution and is a smoothing function such as a Gaussian of scale . The complete operator behaves in effect as a circular edge detector, blurred at a scale set by , that searches iteratively for a maximum contour integral derivative with increasing radius at successively finer scales of analysis through the three parameter space of center coordinates and radius (x_o,y_o,r) defining the path of contour integration.

At first the blurring factor is set for a coarse scale of analysis so that only the very pronounced circular transition from iris to (white) sclera is detected. Then after this strong circular boundary is more precisely estimated, a second search begins within the confined central interior of the located iris for the fainter pupillary boundary, using a finer convolution scale and a smaller search range defining the paths (x_o,y_o,r) of contour integration. In the initial search for the outer bounds of the iris, the angular arc of contour integration ds is restricted in range to two opposing 90° cones centered on the horizontal meridian, since eyelids generally obscure the upper and lower limbus of the iris. Then in the subsequent interior search for the pupillary boundary, the arc of contour integration ds in operator (1) is restricted to the upper 270° in order to avoid the corneal specular reflection that is usually superimposed in the lower 90° cone of the iris from the illuminator located below the videocamera. Taking the absolute value in (1) is not required when the operator is used first to locate the outer boundary of the iris, since the sclera is always lighter than the iris and so the smoothed partial derivative with increasing radius near the limbus is always positive. However, the pupil is not always darker than the iris, as in persons with normal early cataract or significant back-scattered light from the lens and vitreous humour; applying the absolute value in (1) makes the operator a good circular edge-finder regardless of such polarity-reversing conditions. With automatically tailored to the stage of search for both the pupil and limbus, and by making it correspondingly finer in successive iterations, the operator defined in (1) has proven to be virtually infallable in locating the visible inner and outer annular boundaries of irises.

For rapid discrete implementation of the integrodifferential operator in (1), it is more efficient to interchange the order of convolution and differentiation and to concatenate them, before computing the discrete convolution of the resulting operator with the discrete series of undersampled sums of pixels along circular contours of increasing radius. Using the finite difference approximation to the derivative for a discrete series in n,

where r is a small increment in radius, and replacing the convolution and contour integrals with sums, we can derive through these manipulations an efficient discrete operator for finding the inner and outer boundaries of an iris:

where is the angular sampling interval along the circular arcs, over which the summed I(x,y) pixel intensities represent the contour integrals expressed in (1).

A nonlinear enhancement of this operator makes it more robust for detecting the inner boundary of the iris. Because the circular edge that defines the pupillary boundary is often very faint, especially in dark-eyed persons, it is advantageous to divide each term in the convolution summation over k in (3) by a further contour integral around a smaller radius (k-2) r. This divisor becomes very small and stable as the parameters (nr,x_o,y_o) of contour integration become well-matched to the true location and size of the pupil, and this helps the resulting sum of ratio terms to achieve a distinctive maximum that reliably locates the pupillary boundary:

In essence, dividing by the second contour integral exploits the fact that the interior of the pupil is generally both homogeneous and dark. This creates a suddenly very small divisor when the parameters (nr,x_o,y_o) are optimal for the true pupil, thus producing a sharp maximum in the overall search operator (4).

Using multigrid search with gradient ascent over the image domain (x,y) for the center coordinates and initial radius of each series of contour integrals, and decimating both the incremental radius r interval and the angular sampling interval in successively finer scales of search spanning four octaves, these iris locating operations become very efficient without loss of reliability. The total processing time on a RISC-based CPU for iris detection and localization to single-pixel precision using such operators, starting from a 640 x 480 image, is about one-quarter of a second (250 msec) with optimized integer code.