The Numbers Behind NUMB3RS by Keith Devlin Page A

(The tattoo identified him as a member of the notorious Los Angeles gang Eight Tray Gangster Crips.) Unfortunately, although some frames of the newsreel video did show the assailant’s left arm, the image was not sharp enough to discern the tattoo.
    At that point, the frustrated prosecutors got a major break. A Santa Monica reporter supplied them with some still photographs shot from a helicopter with a 400-millimeter long-distance lens. Thanks to the much higher resolution of still photographs, close scrutiny of one of the photographs, both with the naked eye and a magnifying glass, did reveal a vague gray region on the assailant’s left arm as he stood over the prone body of Williams. (See figure 5.) The gray region—a mere one six-thousandth of the overall area of the photograph—might indeed have been a tattoo; unfortunately, it could just as easily have been a smudge of dirt or even a blemish on the photo. Enter mathematics.
    Using highly sophisticated mathematical techniques, developed initially to enhance surveillance photographs taken by military satellites, the crucial portion of the photograph was processed on a high-performance computer to generate a much clearer image. The resulting image revealed that the apparent mark on the suspect’s left arm had a shape and color that, above the usual legal threshold of “beyond a reasonable doubt,” was indeed a rose tattoo like the one on Damian Williams’ arm.

    Figure 5. Mathematically enhanced aerial photograph of the Reginald Denny beating, with feature enlargement showing a blurred mark on the assailant’s left arm.
    The techniques used to process the photographic images in the Reginald Denny case fall in the general area known as image enhancement . This is not a technique for adjusting brightness, color, or contrast, or otherwise tweaking photographs familiar to computer users in the form of programs such as Photoshop, nor is it the proprietary photograph-handling software that often comes with new digital cameras. In image enhancement, mathematical techniques are used to reconstruct image details that were degraded by optical blurring in the original photograph.
    The term “reconstruct” as used here can be misleading to laypersons unfamiliar with the technique. One of the key steps in the trial of Damian Williams was for the experts to convince the judge, and then the jury, that the process was reliable, and that the resulting image did not show “what might have been,” but did in fact reveal “what was. ” The judge’s ruling in the case, that images produced by enhancement techniques were indeed allowable evidence, was a landmark in legal history.
    The general idea behind image enhancement is to use mathematics to supply features of the image that were not captured in the original photograph. No photograph will represent everything in a visual scene. Most photographs capture enough information that the human eye is often unable to discern any differences between the photograph and the original scene, and certainly enough for us to identify an individual. But as cognitive scientists have demonstrated, much of what we see when we look at either a real-life scene or a photograph is supplied by our brains, which fill in—generally reliably and accurately—anything that (for one reason or another) is missing from the visual signal that actually enters our eyes. When it comes to certain particular features in an image, mathematics is far more powerful, and can furnish—also reliably and accurately—details that the photograph never fully captured in the first place.
    In the Damian Williams trial, the key prosecution witness who identified the defendant was Dr. Leonid Rudin, the cofounder in 1988 of Cognitech, Inc., a Santa Monica–based company specializing in image processing. As a doctoral student at Caltech in the mid-1980s, Rudin developed a novel method for deblurring photographic