Matrix Calculations for Digital Photography

Different digital cameras have different spectral sensitivities and opto-electronic conversion functions (OECF), and therefore produce different data about the same scene. Correct interpretation of the data requires that it be presented in some sort of standard form. The obvious choice for a standard spectral space is a spectral space spanned by a set of color matching functions, a color space.
3x3 matrices are used for transformations from camera spectral spaces to a standard color space based on the ITU-R BT.709 red, green, and blue (RGB) primaries.

3x3 matrix transformations are used most because such transformations are most appropriate when the relationship between the scene radiance and the radiance incident on the sensor is variable, different illumination sources are used, and the colorants found in the scene are unknown or highly variable.


The first step in determining the desired transformations is to linearize the data obtained from the camera with respect to the source or scene radiance.

For this experiment, the camera data was linearized by measuring the camera and focal plane OECF’s of the
camera for a variety of scene dynamic ranges and mean reflectances, and constructing a flare model which predicts the camera flare based on focal plane image statistics.

Accurate OECF measurements for camera data linearization are extremely important, particularly with the LS methods, since the regression tries to transform the chart image data to aim linear values.


The numerical results of the experiments conducted are presented in the following table:

(WPPLS): white point preserving LS regressions.
(WTWPPLS):  weighted white point preserving LS regressions.

By Saria J. Beainy


Resource: "The Fifth Color Imaging Conference: Color Science, Systems, and Applications" by Paul M. Hubel and Jack Holm, Graham D. Finlayson, Mark S. Drew.

EIGENFACES AND DIGITAL IMAGE RECOGNITION

David Mumford, Alan Yuille, and Peter Hallinan at Harvard have taken a different tack in the computer matching of faces. One problem in analyzing a picture of a person's face is that the image depends largely on the presence or absence of shadows, which depend, in turn, on lighting conditions. Mumford and Hallinan address this by computing what they call "eigenfaces".
To obtain these, they photograph a subject under as many as a hundred different lighting conditions. Once digitized, these images can be interpreted as points in a very high-dimensional space (the dimension is equal to the number of pixels in the image). The resulting cloud of points is shaped something like an ellipse. The axes of the ellipse and their lengths can be determined by a computation in linear algebra (to be precise, by computing the eigenvectors and eigenvalues of the Matrix IIt where I is a column-by-column listing of the images and It is the transpose of I). The eigenfaces lie along these axes. The idea is to reconstruct a good approximation to a face under general lighting conditions by combining just a few eigenfaces and then "warping" the result.
The theory of linear agebra carries over into the high-dimensional world of digital images with the computation of "eigenfaces" in an application of control theory to an important problem in pattern recognition. (Figure courtesy of Harvard Robotics lab.)
(From: The Gentle Art of Control , in "What's Happening in the Mathematical Sciences", Vol. 3, by Barry Cipra, and published and available from the American Mathematical Society .)