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Hierarchical third‐order tensor decomposition through inverse difference pyramid based on the three‐dimensional Walsh–Hadamard transform with applications in data mining

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A new approach is presented for hierarchical decomposition of third‐order tensors through their transformation into the generalized three‐dimensional (3D) spectrum space based on the inverse difference pyramid (IDP). For this, we choose the 3D Walsh–Hadamard transform (3D‐WHT). As result, each tensor is represented as a spectral tensor of m hierarchical levels which contains selected low‐frequency 3D‐WHT coefficients. Calculating sequentially the inverse 3D‐WHT for the coefficients from each pyramid level starting from its top, the tensor is approximated with increasing accuracy until its full restoration is achieved. To illustrate the new approach, given is the algorithm for hierarchical three‐level tensor decomposition based on the reduced IDP. The proposed approach permits simultaneous decorrelation of tensor elements in three mutually orthogonal directions. The energy of the tensor elements is concentrated in a small number of spectral coefficients which build the top of the inverse pyramid. The use of the 3D‐WHT permits to achieve minimum computational complexity, compared to deterministic 3D orthogonal transforms. The main applications of the new method for data mining in the contemporary intelligent systems are in the processing and analysis of large sets of different kinds of data/images/videos in the following areas: Compression of correlated image sequences, computer tomography, thermo vision, ultrasound and multichannel medical signals; search of 3D objects in image databases; extraction of features for recognition of 3D objects; multidimensional data denoising; multilayer watermarking of video sequences; and so on. This article is categorized under: Fundamental Concepts of Data and Knowledge > Big Data Mining Algorithmic Development > Spatial and Temporal Data Mining
Decomposition of the third‐order tensor X in the space of the three dimensional Walsh–Hadamard transform (3D‐WHT) basic functions Wu,v,l which correspond to the low‐frequency spectral coefficients s(u,v,l) for u,v,l = 0, 1. The weight of the bright cubes is (+1), and of the dark (−1)
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Block diagram of the multilevel search illustrated for two‐level three‐dimensional reduced inverse difference pyramid (3D‐RIDP)/Walsh–Hadamard transform (WHT)
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Speedup ratio of the “fast” one‐dimensional Walsh–Hadamard transform (1D‐WHT) computation towards the “fast” 1D‐ТWHT
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Scanning the coefficients of the spectrum tensor S of size 4 × 4 × 4
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Octotree representation of the tensor Xr of size 4 × 4 × 4 in the spectrum space of the three‐dimensional reduced inverse difference pyramid (3D‐RIDP)/Walsh–Hadamard transform (WHT)
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Of restoration the subtensor Xr of size 8 × 8 × 8, represented through the three‐level local 3D‐IDP(r). 3D‐IDP: three‐dimensional inverse difference pyramid
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Decomposition of three hierarchical levels for the subtensor Xr of size 8 × 8 × 8, through local 3D‐IDP(r). 3D‐IDP: three‐dimensional inverse difference pyramid
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(a) Tensor X and (b) division of the tensor X into eight subtensors
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Fundamental Concepts of Data and Knowledge > Big Data Mining
Algorithmic Development > Spatial and Temporal Data Mining

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