Perfect sequences over the real quaternions

Kuznetsov, O. P.

doi:10.1109/iwsda.2009.5346443

Cited by 19 publications

(18 citation statements)

References 2 publications

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“…One could consider more general scalings of the real canonical Huffman arrays, like the quaternions, however issues such as non-commutation render this extension non-trivial and outside the scope of this work. Indeed there exist interesting constructions, including exhaustive searches [21], of uni-modular perfect sequences over the quaternions [22], [23], [24], which similarly call for generalized notions of right-and left-auto-correlation which respect the absence of commutation [25]. Recently Bright et al [26] derived new constructions for arbitrarily long perfect sequences over the quaternions, contradicting a prior existence conjecture concerning the allowable lengths given constrained alphabets of sequences in this context [14].…”

Section: Dual Of Delta-correlation To Harness Complex-scaled Canonica...mentioning

confidence: 99%

Extending the known families of scalable Huffman sequences

Petersen,

Paganin,

Svalbe

2021

Preprint

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A canonical Huffman sequence is characterized by a zero inner-product between itself and each of its shifted copies, except at their largest relative shifts: their aperiodic auto-correlation then becomes delta-like, a single central peak surrounded by zeros, with one non-zero entry at each end. Prior work showed that the few known families of Huffman sequences (of length N = 4n − 1, for integers n > 1, with continuously scalable elements) are based upon Fibonacci polynomials. Related multi-dimensional (nD) Huffman arrays were designed, as well as non-canonical quasi-Huffman arrays that also possess delta-like auto-correlations. We examined links between these discrete sequences and delta-correlated functions defined on the continuum, and provided simple non-iterative approaches to successfully deconvolve nD data blurred by diffuse Huffman arrays. Here we describe new constructions for canonical Huffman sequences. Examples of length N = 4n + 1, N = 2n and families of arbitrary length are given, including scaled forms, as well as for Fibonacci-based arrays with perfect periodic auto-correlations, that are zero for all nonzero cyclic shifts. A generalization to include canonical sequences with complex scale factors invokes an equally useful dual form of delta-correlation. We also present 1D arrays with a much smaller dynamic range than those where the elements are built using Fibonacci recursion. When Huffman arrays (that are comprised of inherently signed values) are employed as diffuse probe beams for image acquisition, a new two-mask de-correlating step is described here that significantly reduces the total incident radiation dose compared to a prior method that added a positive pedestal-offset.

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Section: Dual Of Delta-correlation To Harness Complex-scaled Canonica...mentioning

confidence: 99%

Extending the known families of scalable Huffman sequences

Petersen,

Paganin,

Svalbe

2021

Preprint

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show abstract

“…There is considerable interest in the generation of 2D sequences that have perfector near-perfect correlation properties [8,9,13]. A perfect sequence on an m × n array has a normalised peak auto-correlation value of m × n at one location and is zero everywhere else.…”

Section: D Correlations Between Minimal Ghost Imagesmentioning

confidence: 99%

“…Near-perfect sequences have off-centre correlation values of ±1 or ±2, rather than zero. The theoretical interest here extends to practical applications in encryption, message-coding and watermarking applications [13].…”

Section: D Correlations Between Minimal Ghost Imagesmentioning

confidence: 99%

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Properties of Minimal Ghosts

Svalbe

Normand

2011

Discrete Geometry for Computer Imagery

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International audienceA ghost image is an array of signed pixel values so positioned as to create zero-sums in all discrete projections taken across that image for a pre-defined set of angles. The discrete projection scheme used here is the finite Radon transform. Minimal ghosts employ just 2N pixels to generate zero-sum projections at N projection angles. We describe efficient methods to construct Nth order minimal ghost images on prime-sized 2D arrays. Ghost images or switching components are important in discrete image reconstruction. Ghosts usually grow larger as they are constrained by more projection angles. When ghosts become too large to be added to an image, image reconstruction from projections becomes unique and exact. Ghosts can be used to synthesize image/anti-image data that will also exhibit zero-sum projections at N pre-defined angles. We examine the remarkable symmetry, cross- and auto-correlation properties of minimal ghosts. The geometric properties of minimal ghost images may make them suitable to embed in data as watermarks

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“…In reality, a peak of +2N and 1 values at 2N locations gives a balanced (zero-mean) result. Nearperfect correlations [1][2][3] give a +2N peak value and low offpeak values (typically values from 2 to +2).…”

Section: Introductionmentioning

confidence: 99%

Near Perfect Correlation Functions Based on Zero-Sum Projections

Svalbe

2011

2011 International Conference on Digital Image Computing: Techniques and Applications

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Minimal projective ghost functions make good watermark labels for embedding into images. Although fragile to hacking attacks, they are near-to-invisible because of their distributed, random appearance and their binary, zero-mean statistics. They have the strong correlation properties needed to extract a low-intensity watermark from bright image data. We present a method to embed, concurrently, up to (p 1)/2 independent minimal ghosts within the same pxp image space, where p is prime. Compounding ghosts inside the same pxp space increases, by (p 1)/2, the robustness with which these watermarks can be recovered. This result then approaches the optimal peak correlation result obtainable using 2D perfect or near-perfect sequences. However, unlike perfect sequences, minimal ghosts are simple to construct and compound. A large number of independent compounded minimal ghosts can be generated for each prime p, thus each watermark is sufficiently individual to prevent confusion when multiple labels are present.

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Perfect sequences over the real quaternions

Cited by 19 publications

References 2 publications

Extending the known families of scalable Huffman sequences

Extending the known families of scalable Huffman sequences

Properties of Minimal Ghosts

Near Perfect Correlation Functions Based on Zero-Sum Projections

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