Sparse representations of astronomical of images, produced by standard greedy selection algorithms were investigated. The OMP2D was algorithm applied to approximate this image corpus, by selecting atoms from the combined RDC and RDBS dictionary. This resulted in hight quality sparse approximations. The suitability of this algorithm for quickly processing small blocks, was demonstrated by the low average processing time over this class of images. As the block size N was increased, so were both the average SR, and processing time. To reduce the approximation processing time for large N , the SPMP2D1 algorithm was proposed. SPMP2D1 was then shown experimentally to produce equivalent approximations to OMP2D, in a shorter period of time.
It has been shown that for a variety of dictionaries OMP2D results in significantly sparser approximations, of both astronomical and natural images. This is when it is compared to the DCT and CDF9/7 transforms, currently employed as part of the JPEG and JPEG2000 image compression standards.
The SR results produced by OMP2D with the RDC dictionary are encouraging. Fur-thermore its combination with supported B-splines significantly increase the resulting SR of astronomical image data.
The increase in the SR resulting from approximations made with OMP2D inspired the dictionary coding scheme. This was shown experimentally, in Section 2.8, to compress astronomical and natural images, significantly better than JPEG, for a variety approxi-mation qualities. More importantly, the proposed image coding scheme compressed higher quality approximations, of both astronomical and natural images, requiring significantly less bpp than JPEG2000.
The highest level of compression, for the proposed image coding scheme, resulted from
processing images with the largest block size, N = 32. Unfortunately processing im-ages with OMP2D in larger blocks, increases the processing time. An algorithm termed SPMP2D1 which produced equivalent results to OMP2D was then proposed. In the Ex-periment in Section 2.5, the SPMP2D1 algorithm required significantly less time than OMP2D, to approximate images partitioned into blocks with N = 32. The result in-dicates that SPMP2D1 is a valid alternative to OMP2D, for processing larger blocks to further increase the approximation sparsity.
The proposed approximation and coding scheme is competitive when compared to cur-rent image compression formats. This implies that sparse image approximations produced by greedy algorithms, could be an important first step in a new image compression scheme.
This result is important, and encouraging, because the proposed coding scheme is not yet at an advanced stage.
This Chapter describes and analyses a method for hiding information in the null space created by a sparse approximation of an image. The main idea stems from the fact that sparsity entails a projection onto a lower dimensional subspace, therefore creating a null space. Extra information can then be embedded and stably extracted from such a space.
The proposed idea can be applied to images as part of a partial encryption model taking advantage of image sparsity. The method termed image folding, takes a sparse approximation of an image and splits it into two sections, a host and embedded section.
The embedded section is added to the host section to produce a folded image, this can then be stored in any conventional lossless image format. Both sections can then be fully recovered from the folding process, by applying an orthogonal projection. The security comes from securing the embedded section before it is folded, thus partially encrypting the image.
Two methods are discussed for protecting the embedded image. The first approach, based on a previously outlined method, is successfully applied to this particular image processing application. The second, is the security scheme described in the paper Sparsity and “Something Else”: An Approach to Encrypted Image Folding [3].
The procedure can be applied to any sparse image representations including those realized by OMP2D in Chapter 2.
The contents of the Chapter are as follows: the first Section contains a general overview of the information embedding and recovery scheme. The next Section describes a specific application of this known as image folding. The following Section describes two methods for securing the folded information, including a number of simulations to determine the size of the keyspace for each method.
3.1 Information Embedding
Given an approximation IK ∈ VK of an image array I ∈ RNr×Nc and denoting V⊥ to be the orthogonal complement of VK in RNr×Nc, the embedding and retrieving principle is simple to describe: Any matrix E ∈ V⊥can be added and stably extracted from IK ∈ VK
with an orthogonal projector ˆPVK which acts by projecting onto VK and along V⊥ in the way that is shown below,
If1 = IK+ E,
PˆVKIf1 = ˆPVK(IK+ E) = IK, E = If1 − IK.
This suggests the possibility of using the sparse representation of an image IK ∈ VK ⊂ RNr×Nc as a host for embedding extra information. To achieve this a previously proposed scheme for embedding redundant representations [87], is applied to images as described below:
as the reconstruction of a sparse approximation of an image I ∈ RNr×Nc in the proper subspace VK = span{Sk}Kk=1. If the set of matrices {Sk}Kk=1 are linearly independent the dimension of V⊥, the orthogonal complement of VK in RNr×Nc, is Ne = N2 − K.
Therefore a vector of Ne numbers denoted as e(n), n = 1, . . . , Ne can be constructed to store coefficients for constructing a matrix E ∈ V⊥. The numbers e(n) can be hidden and extracted from this embedded vector as prescribed below:
• Take an orthonormal basis S⊥n, n = 1, . . . , Ne for V⊥ and form E as the linear
• Add E to IK to obtain If1 = IK+ E.
Information Retrieval: Given If1 retrieve the vector of numbers e(n), n = 1, . . . , Ne
as follows.
• Construct an orthogonal projection operator ˆPVK onto the subspace VK = span{Sk}Kk=1, and remove the components in V⊥ from If1 as IK = ˆPVKIf1.
• From If1 and the recovered image IK obtain E as E = If1 − IK.
• Retrieve the vector of numbers e(n), n = 1, . . . , Ne from the recovered E with the orthonormal basis S⊥n, n = 1, . . . , Ne, with the Frobenius inner product
e(n) = hS⊥n, EiF, n = 1, . . . , Ne. (3.3)
The procedure above can be applied to any sparse approximation in a known subspace VK. Next is an overview of its application to the blocked image approximation produced in Chapter 2, in a procedure called image folding. The term folding describes the way a sparse representation of some image blocks, provides space for the coefficients from other blocks, to be embedded or folded.