Hierarchical Adaptive KL-Based Transform: Algorithms and Applications

Kountchev, Roumen; Kountcheva, Roumiana

doi:10.1007/978-3-319-10653-3_4

Cited by 6 publications

(10 citation statements)

References 27 publications

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“…[12][13][14]. This approach is distiny its flexibility regarding the choice of the transform based on the processed data is work, we present alternative new hierarchical 3D tensor decompositions the famous statistical orthogonal Karhunen-Loeve Transform (KLT) [15,16]. close to optimal (which ensures full decorrelation of the decomposition compot do not need iterations and have lower computational complexity.…”

Section: Methods For 3d Adaptive Frequency-ordered Hierarchical Klt O...mentioning

confidence: 99%

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Third-Order Tensor Decorrelation Based on 3D FO-HKLT with Adaptive Directional Vectorization

Kountchev

Mironov

Kountcheva³

2022

Symmetry

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In this work, we present a new hierarchical decomposition aimed at the decorrelation of a cubical tensor of size 2n, based on the 3D Frequency-Ordered Hierarchical KLT (3D FO-HKLT). The decomposition is executed in three consecutive stages. In the first stage, after adaptive directional vectorization (ADV) of the input tensor, the vectors are processed through one-dimensional FO-Adaptive HKLT (FO-AHKLT), and after folding, the first intermediate tensor is calculated. In the second stage, on the vectors obtained after ADV of the first intermediate tensor, FO-AHKLT is applied, and after folding, the second intermediate tensor is calculated. In the third stage, on the vectors obtained from the second intermediate tensor, ADV is applied, followed by FO-AHKLT, and the output tensor is obtained. The orientation of the vectors, calculated from each tensor, could be horizontal, vertical or lateral. The best orientation is chosen through analysis of their covariance matrix, based on its symmetry properties. The kernel of FO-AHKLT is the optimal decorrelating KLT with a matrix of size 2 × 2. To achieve higher decorrelation of the decomposition components, the direction of the vectors obtained after unfolding of the input tensor in each of the three consecutive stages, is chosen adaptively. The achieved lower computational complexity of FO-AHKLT is compared with that of the Hierarchical Tucker and Tensor Train decompositions.

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Section: Methods For 3d Adaptive Frequency-ordered Hierarchical Klt O...mentioning

confidence: 99%

“…In this work, we present alternative new hierarchical 3D tensor decompositions based on the famous statistical orthogonal Karhunen-Loeve Transform (KLT) [15,16]. They are close to optimal (which ensures full decorrelation of the decomposition components), but do not need iterations and have lower computational complexity.…”

Section: Introductionmentioning

confidence: 99%

Third-Order Tensor Decorrelation Based on 3D FO-HKLT with Adaptive Directional Vectorization

Kountchev

Mironov

Kountcheva³

2022

Symmetry

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“…Nevertheless, it is possible to represent these sets in terms of their numerical characteristics, such as the expectation and covariance matrices. Typical examples are stochastic signal processing [15][16][17][18][19], statistics [20][21][22][23][24], engineering [25][26][27][28] and image processing [29,30]; in the latter case, a digitized image, presented by a matrix, is often interpreted as the sample of a stochastic signal.…”

Section: Motivationmentioning

confidence: 99%

“…While the theory of operator approximation with any given accuracy is well elaborated (see, e.g., [11], [5], [6], [14]), [2], [10], [3], [1], [4], [13], [8], [7], [9], [12]), the theory of best constrained constructive operator approximation is still not so well developed, although this is an area of intensive recent research (see, e.g., [31][32][33][34][35][36]). Despite increasing demands from applications [17][18][19][21][22][23][25][26][27][28][30][31][32][33][34][36][37][38][39][40][41][42][43][44][45][46] this subject is hardly tractable because of intrinsic difficulties in best approximation techniques, especially when the approximating operator should have a specific structure implied by the underlying problem.…”

Section: Motivationmentioning

confidence: 99%

Best approximations of non-linear mappings: Method of optimal injections

Torokhti¹,

Soto-Quiros²

2018

Preprint

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While the theory of operator approximation with any given accuracy is well elaborated, the theory of best constrained constructive operator approximation is still not so well developed. Despite increasing demands from applications this subject is hardly tractable because of intrinsic difficulties in associated approximation techniques. This paper concerns the best constrained approximation of a non-linear operator in probability spaces. We propose and justify a new approach and technique based on the following observation. Methods for best approximation are aimed at obtaining the best solution within a certain class; the accuracy of the solution is limited by the extent to which the class is suitable. By contrast, iterative methods are normally convergent but the convergence can be quite slow. Moreover, in practice only a finite number of iteration loops can be carried out and therefore, the final approximate solution is often unsatisfactorily inaccurate. A natural idea is to combine the methods for best approximation and iterative techniques to exploit their advantageous features. Here, we present an approach which realizes this.The proposed approximating operator has several degrees of freedom to minimize the associated error. In particular, one of the specific features of the approximating technique we develop is special random vectors called injections. They are determined in the way that allows us to further minimize the associated error.

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“…The SVD calculation for blocks of size 22 is based on the adaptive KLT [28]. The НSVD algorithm [29,30] is aimed at the achievement of decomposition with high computational efficiency, which is also suitable for parallel recursive processing with relatively simple algebraic operations, and permits calculation speedup through cutting-off the branches with very small eigenvalues.…”

Section: Related Workmentioning

confidence: 99%

Hierarchical Singular Value Decomposition for Halftone Images

Kountchev

Kountcheva²

2015

The 7th International Conference on Information Technology

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This work is devoted to one new approach for decomposition of images represented by matrices of size 2 n 2 n , based on the multiple application of the Singular Value Decomposition (SVD) over image blocks of relatively small size (22), obtained after division of the original image matrix. The new decomposition, called Hierarchical SVD, has tree structure of the kind binary tree of n hierarchical levels. Its basic advantages over the famous SVD are: the reduced computational complexity, the opportunity for parallel and recursive processing of the image blocks, based on relatively simple algebraic relations, the high concentration of the image energy in the first decomposition components, and the ability to accelerate the calculations through cutting-off the tree branches in the decomposition levels, where the corresponding eigen values are very small. The HSVD algorithm is generalized for images of unspecified size. The new decomposition opens numerous opportunities for fast image processing in various application areas: image compression, filtration, segmentation, merging, digital watermarking, extraction of minimum number of features sufficient for the objects recognition, etc.

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Hierarchical Adaptive KL-Based Transform: Algorithms and Applications

Cited by 6 publications

References 27 publications

Third-Order Tensor Decorrelation Based on 3D FO-HKLT with Adaptive Directional Vectorization

Third-Order Tensor Decorrelation Based on 3D FO-HKLT with Adaptive Directional Vectorization

Best approximations of non-linear mappings: Method of optimal injections

Hierarchical Singular Value Decomposition for Halftone Images

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