Highly Nonlinear Approximations for Sparse Signal Representation
35], we discuss an application involving the null space yielded by the sparse representation of an image, for storing part of the image itself. We term this application `Image Folding'.
Consider that by an appropriate technique one finds a sparse representation of an image. Let be the -dictionary's atoms rendering such a representation and the space they span. The sparsity property of a representation implies that is a subspace considerably smaller than the image space . We can then construct a complementary subspace , such that , and compute the dual vectors yielding the oblique projection onto along . Thus, the coefficients of the sparse representation can be calculated as:
Now, if we take a vector in and add it to the image forming the vector to replace in (37), since is orthogonal to the duals , we still have
This suggests the possibility of using the sparse representation of an image to embed the image with additional information stored in the vector . In order to do this, we apply the earlier proposed scheme to embed redundant representations , which in this case operates as described below.
We can embed numbers into a vectors as follows.
- Take an orthonormal basis
form vector as the linear combination
- Add to to obtain
Given retrieve the numbers as follows.
- Use to compute the coefficients of the sparse representation of as in (38). Use this coefficients to reconstruct the image
- Obtain from the given and the reconstructed
. Use and the orthonormal
the embedded numbers