The mathematics of digital photography

Mathematics in Digital Photography

The advances in both digital photography and computing have allowed more detailed and complex images to be shown on more realistic media than was ever previously possible. Through the use of more specialized equipment and digital imaging techniques the resulting photos of even the most novice user today can rival those of professionals from years before. This level of photographic precision could never have been achieved were it not for the tremendous advantages made in computing that allow both the hardware and software involved in digital photography to function at extremely high levels. Both hardware and software made today are capable of performing the necessary calculations in a fraction of a second, making focusing and editing pictures easier than ever before. The incorporation of mathematics directly into the digital photography process has been the primary impetus for the explosion of high-quality digital images available almost anywhere we look.

Mathematical Equations Used in Digital Photography

Digital cameras must receive light in much the same way the human does and, as a result of this, must find a way to model the typical responses of the cones in the eye. The mathematical model of this was first derived from John Dalton's description of his colorblindness. This response can be best represented by the following equation:

ci = max ?min si (?)f (?)d?, I = 1: 3, (Higham, 2007).

In this equation, f represents the distribution of light, si is the sensitivity of the ith cone, and ?max, ?min are the wavelengths of the visible spectrum. Another way of viewing this model of color can be shown by using three columns in a vector as the color primaries. They can be represented by:

P = [p1 p2 p3]}nx3, which can be further reduced to ST ? Pa (f). (Higham, 2007).

The color of any spectrum can then be matched by primaries. In these equations, it becomes clear that the cameras are quickly performing high-level calculations that allow them to view objects in the same way that a human eye would. However, the camera itself does not always perform the calculations. This part of the process is often left to digital editing software that renders the image in a way that is more pleasing and physically accurate. Often pictures can be altered so much that even very blurry images can be made clear. Blurred images can be caused by a number of different things, but the original image "g" can be seen as a small array "h" as shown in the following equation:

Blurred image = g*h-G-H (Hoggar, 2007, p.20).

Software can use this equation to remove the blurred area from the photo and make it appear more like it would normally to the human eye. Perhaps the area in which math is most easily recognized is in the compression of digital photographs. Compression technology is the primary driver behind the vast expansion of digital photography. Compression allows large amounts of data to be stored in a relatively small area so that more intricate details of the photo can be accurately reconstructed.

Compression is a complicated process that involves several steps. The first step is to convert red, green, and blue color channels to Y, Cb, and Cr channels while partitioning blocks into size 8x8 pixels. For a JPEG image, the compression relies on the Discrete Cosine Transformation, which is exemplified by the equation: C = UBU^T, where B. is one of the 8x8 blocks and U. is a special 8x8 matrix. Applying certain methods of encoding enables compression of about 70% in most instances, or even more in some circumstances. The process can then be inverted to expand the image and view it in full, by finding B. For the previous equation:

B^' = U^T C^'U for each block ("Image Compression," 2011).

One downside of this compression technique for JPEG images is that decoupling can occur causing some parts of the image to appear blocky. While this loss of resolution is often within the range of acceptability, sometimes it is not good enough for certain publications, such as magazines or other media ("Image Compression," 2011). It is possible that the compression algorithms will continue to improve as both the mathematical concepts and computing power rendering them continue to improve.