A basis is a set of linearly independent vectors that can be linearly combined to represent any vector in a given vector space, such as the 8x8 matrix representing an image. A good basis for image compression has more low-frequency vectors that imply less variation in pixel values across the image, and fewer high-frequency vectors that imply more variation. There are several possible basis choices, including the Fourier basis where 'w' represents the nth root of unity, and the quality of the basis depends on how well it captures the general smoothness of images.

1. ImageCompression

3. Basis A set of linearly independent vectors whose linear combination can be used to express any vector in a given vector space (In our case, the vector space is the 8 x 8 matrix, X). There can be infinitely many bases for a given vector space. So, for representing our image, we are free to choose any basis that is convenient to us. The coefficient matrix will vary accordingly. ( Since X = BC and C = B-1X)

4. B = [ b0 | b1 | b2 | b3 | b4 | b5 | b6 | b7 ] where b0, b1,.., b7 are 8 x 1 linearly independent vectors.

5. A “Good” basis should have more of low frequency vectors or bis (ideal: all ones in the column; imply less variation of pixel values in space) and very few high frequency vectors (alternate +1s and -1s; imply maximum variation of pixel values in space) in order to account for the general smoothness of images.Bases To Choose From‘w’ in Fourier Basis is the nth root of unity for a basis of dimension n x n.

6. Choice of Basis

7. What makes a basis good?

10. ReferenceMIT OCW: Linear Algebra (Gilbert Strang) Lecture 31MIT OCW: Linear Algebra (Gilbert Strang) Lecture 26http://www.amara.com/IEEEwave/IW_wave_vs_four.htmlhttp://www.vidyasagar.ac.in/journal/maths/vol13/Art11.pdfhttp://users.rowan.edu/~polikar/WAVELETShttp://en.wikipedia.org