Image compression via least-squares 

We can use least-squares to represent an image in terms of a linear combination of ‘‘basic’’ images, at least approximately.

An image can be represented, via its pixel values or some other mechanism, as a (usually long) vector y \in \mathbb{R}^m. Now assume that we have a library of n ‘‘basic’’ images, also in pixel form, that are represented as vectors a_1, \ldots, a_n. Each vector a_j could contain the pixel representation of a unit vector on some basis, such as a two-dimensional Fourier basis; however, we do not necessarily assume here that the a_j ‘s form a basis.

Let us try to find the best coefficients x_j, j=1, \ldots, n, which allow approximating the given image (given by y \in \mathbb{R}^m) as a linear combination of the a_j ‘s with coefficients x_j. Such a combination can be expressed as the matrix-vector product Ax, where A=\left[a_1, \ldots, a_n\right] is the m \times n matrix that contains the basic images. The best fit can be found via the least-squares problem

    \[\min _x\|A x-y\|_2\]

Once the representation is found, and if the optimal value of the problem above is small, we can safely represent the given image via the vector x. If the vector x is sparse, in the sense that it has many zeros, such a representation can yield a substantial saving in memory, over the initial pixel representation y.

The sparsity of the solution of the problem above for a range of possible images y is highly dependent on the images’ characteristics, as well as on the collection of basic images contained in A. In practice, it is desirable to trade-off the accuracy measure above, against some measure of the sparsity of the optimal vector x.

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