Erasure Coding with the Finite Radon Transform Nicolas Normand∗† , Imants Svalbe∗ , Benoît Parrein† and Andrew Kingston‡ ∗ School
of Physics, Monash University, Wellington Road, Victoria 3800, Australia Email:
[email protected] † IRCCyN/IVC CNRS UMR 6597, Polytech’Nantes, La Chantrerie, Nantes, France ‡ Research School of Physical Sciences and Engineering, Australian National University, Canberra, Australia
Abstract—The Mojette transform and the finite Radon transform (FRT) are discrete data projection methods that are exactly invertible and are computed using simple addition operations. The FRT is also Maximum Distance Separable (MDS). Incorporation of a known level of redundancy into data and projection spaces enables the use of forward error correction to recover the exact, original data when network packets are lost or corrupted during data transmission. By writing the above transforms in Vandermonde form, explicit expressions for discrete projection and inversion as matrix operations have been obtained. A cyclic, prime-sized Vandermonde form for the FRT approach is shown here to yield explicit polynomial expressions for the recovery of image rows from projected data and vice-versa. These polynomial solutions are consistent with the heuristic algorithms for “rowsolving” that have been published previously. This formalism also opens the way to link “ghost” projections in FRT space and “anti-images” in data space that may provide a key to an efficient method of encoding and decoding general data sets in a systematic form.
I. I NTRODUCTION Al-Shaikhi and Ilow [1] proposed an efficient codec that employs shift operators within a Vandermonde matrix as a means to detect and correct for packets of data that are lost or damaged during data transmission over a network. This technique has several properties that are favourable relative to the traditional Forward Error Correcting schemes (FEC) built on Reed-Solomon codes. Translation within a Vandermonde formalism is shown here to be isomorphic to the exact, discrete data projection and inversion method called the Mojette Transform (MT) [2]. The MT has been applied to data transmission, encryption and file storage, to reconstruct images from arbitrary sets of projected views of discrete data, and for tomography using real x-ray data [3]. The MT, in turn, is closely related to the Finite Radon Transform (FRT) [4], an exact, discrete projection method restricted to data embedded in prime-sized (p) arrays under periodic boundary conditions. Applications of the FRT parallel those of the MT [5]. For each array size p, the FRT has a total of p + 1 basis projection vectors. The FRT has tight links, through the slice theorem, to the 1:1 discrete Fourier and number theoretic transforms [6]. Under quite weak constraints (for example, the use of a column of parity pixels to ensure zero data row-sums) the FRT can be shown to be maximum distance separable (MDS) and, written in a normalised form, the projection operator is the same as its inverse. The periodic, prime length structure
enables all mappings between data and FRT space to be computed modulo p. We propose here a variation of the Al-Shaikhi-Ilow method to create an approach where FRT projection and inversion are represented by cyclic shifts of the prime length rows of a Vandermonde matrix. Applied to codec design, this approach should offer improved compactness and lower computational overheads, great flexibility in the degree of redundancy one can encode and in how this information is both encoded and decoded. In image data that contains some redundancy, (for example blank or constant areas), Kingston et al. [5] demonstrated that the FRT can be used to recover lost projections or missing data using what are called row (or, equivalently column) solving algorithms [7]. These algorithms involve back-projecting the incomplete sets of projection data and then shifting, subtracting and integrating the rows of data that lie inside the redundant image area that contain projection artefacts (or image “ghosts”) to untangle and hence recover the missing projections. The cyclic prime Vandermonde structure is used here to mathematically formalise the FRT projection and inversion operators and then to provide explicit expressions for the above de-ghosting algorithms. An example is given where, for i ≤ r, i arbitrary rows of image data are recovered from any i of r received rows of redundant encoded data. Similarly, any i FRT projections can be recovered from any i of their r redundant rows. We also presage use of this method to find new and more general techniques for data recovery. These methods are based on data structures called anti-images (or ghosts) that form the primitive basis functions needed to construct any data element, given some maximum number of missing projections. Ghost images contain signed pixel values that, when projected, sum to zero for one or more projection directions. An example of a ghost image is shown in figure 1. The coupling of image structure and projected data is the key to exploit redundancy in discrete projections. Section II describes briefly the FRT formalism while section III maps the projection operator into its polynomial form as a Vandermonde matrix and derives an explicit form for the inverse operator for image reconstruction. Section IV details the method used to pack data into a systematic form. Section V outlines the existing algorithm used to untangle mixed FRT rows while section VI shows how the Vandermonde approach
Fig. 1. The grayscale image on the left is a ghost image represented by signed image data (where black is more negative and white is more positive). The image on the right is its FRT where the second quarter of all projections are zero.
where t I stands for the transpose of the image I. Multiplying a polynomial by x is equivalent to performing a one step to the right cyclic permutation of the corresponding row. Generally speaking, multiplying two polynomials is equivalent to a spatial cyclic convolution. In particular, multiplying a polynomial P (x) by a difference of monomials, xma − xmb , produces the cyclic derivative of the vector p represented by P . Conversely, dividing by xma −xmb performs the cyclic integration of p, as described in section V and depicted in figure 4. Except for m = p (which has projected sums equal to zero), the mth row of the FRT, Rm , is the vector sum of each image row l ∈ [0, p] after a circular shift of ml. Rm (x) = A(x) + x−m B(x) + x−2m C(x) + · · · .
replicates the row solving algorithm. II. F INITE R ADON T RANSFORM The FRT transforms a p×p image into a (p+1)×p representation. For the FRT, the image size p is always prime. Periodic boundary conditions are applied so that (p, p) = (0, 0). The FRT is defined as: [FRT p f ](t, m) = % f (k, l) · ∆(k − ml ≡ t k,l
(mod p)) %
0
