Mathematics Behind Image Compression

Abstract

This research paper is written in the summer of 2013 while conducting a summer research funded by NYCRI, NASA, NSF, NOAA and Department of Education. The team consists of four members, a faculty, an undergraduate student, a high school teacher and a high school student. The research topic is “image compression” and more precisely the mathematics behind image compression. Image compression is fundamental to NASA and the world’s daily operations. Images are transmitted to NASA from satellites and even Mars, making it very important to send data as efficiently as possible through the low-bandwidth links to these locations. This project focuses its studies in three areas. First, a hands-on mathematical analysis of the singular value decomposition (SVD) compression.  Second, on the area of two field experiments that explore the effect of light conditions, shot composition and content, as well as the time of day and other variables on the file sizes of images generated in a digital camera that implements JPEG compression.  Third, is about an in-depth study of the JPEG algorithm.

In the SVD study, the team analyzed mathematically how matrices are manipulated to return to its equivalent original matrix and the theory about SVD is reinforced by using the software Wolfram Mathematica to compress images from NASA satellites and Mars rover.  Mathematica analyzed the file size and timing data for the compression process.

In the field experiment, a camera with fixed focus, aperture, and other shooting parameters was used to take pictures at various times of day of the same scene to see how the amount and quality of daylight influenced JPEG’s ability to compress images. The same camera with the parameters still fixed was used to shoot various locations, indoors and outdoors, at the same time of day to see how the content of the photo influenced JPEG file sizes.

Finally, the team looked at JPEG’s compression algorithm using Wolfram Mathematica to better understand its efficiency and power, since NASA’s radiation-hardened computer processors are generally not powerful enough to compress images with JPEG. Loosely, the team found that JPEG is best able to compress images with little variation pixel to pixel in color or brightness, and that it provides better looking images at the same file size than SVD compression.