# Highly Nonlinear Approximations for Sparse Signal Representation

# Sparsity and `something else'

We present here a `bonus' of sparse representations by alerting that they can be used for embedding information. Certainly, since a sparse representation entails a projection onto a subspace of lower dimensionality, it generates a null space. This feature suggests the possibility of using the created space for storing data. In particular, in a work with James Bowley [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 [34], which in this case operates as described below.

**Embedding Scheme**

We can embed numbers into a vectors as follows.

- Take an orthonormal basis
for
and
form vector as the linear combination
- Add to to obtain

**Information Retrieval**

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
as
. Use and the orthonormal
basis
to retrieve
the embedded numbers
as

*key*for generating it. An example is given in the next section.

**Subsections**