# MLTSP: New 3D Framework, Based on the Multilayer Tensor Spectrum Pyramid

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2×2×2}) [13]. Furthermore, one application of the presented pyramidal structure aimed at the accelerated search of 3D objects represented through MLTSP and 3D modified Mellin–Fourier Transform (3D MMFT), which is an upgrade from the previous research of the authors [15], is given in this work.

## 2. Basic Relations, Which Represent the Two-Layer Tensor Spectrum Pyramid with 3D OT and HTSVD

**X**—cubical tensor of size N = 2

^{n}for N = 8; r = 0, 1—the 2LTSP layer number;

**S**—cubical spectrum tensor of size N = 2

^{n}; F(.)—operator for direct 3D orthogonal transform (3D OT) (for example, one of the following: 3D FFT [16], 3D FO-AHKLT [17], 3D FO-FWHT [14], etc.); F

_{T}(.)—operator for direct truncated orthogonal transform (3D TOT); F

^{−1}(.)—operator for 3D inverse orthogonal transform (3D IOT); HTSVD

_{2×}

_{2×}

_{2}—hierarchical tensor singular value decomposition for a 2 × 2 × 2 elementary tensor [13].

#### 2.1. Description of the 2LTSP Coder Performance

**X**in the layer r = 0; ${\mathbf{E}}_{0}$—the difference tensor in the layer r = 0; ${\widehat{\mathbf{S}}}_{0,2\times 2\times 2}$—the spectrum sub-tensor of the size 2 × 2 × 2 in the layer r = 0, which comprises eight low-frequency spectrum coefficients of the selected 3D truncated orthogonal transform (3D TOT); $\widehat{\mathbf{X}}$—the approximation of tensor

**X**; ${\widehat{\mathbf{S}}}_{0,2\times 2\times 2}(\mathrm{t})$—the SVD component t for the sub-tensor ${\widehat{\mathbf{S}}}_{0,2\times 2\times 2}$.

^{th}difference sub-tensor of the size 4 × 4 × 4; ${\widehat{\mathbf{S}}}_{1}(\mathrm{u})$—the u

^{th}spectrum sub-tensor of the size 4 × 4 × 4, which is the approximation of the sub-tensor ${\mathbf{S}}_{1}(\mathrm{u})$ in the layer r = 1; ${\widehat{\mathbf{S}}}_{1,2\times 2\times 2}(\mathrm{u})$—the u

^{th}spectrum sub-tensor of the size 2 × 2 × 2 in the layer r = 1, which comprises eight low-frequency spectrum coefficients of the selected 3D TOT; ${\widehat{\mathbf{S}}}_{1,2\times 2\times 2}(\mathrm{u},\mathrm{t})$—the SVD component t of the sub-tensor ${\widehat{\mathbf{S}}}_{1,2\times 2\times 2}(\mathrm{u})$. At the coder output in the layer r = 0 and in correspondence with Equation (2), four HTSVD

_{2×}

_{2×}

_{2}components are obtained, arranged following the decreasing values of their dispersions. At the coder output in the layer r = 1, 32 components of HTSVD

_{2×}

_{2×}

_{2}are obtained, calculated in correspondence with Equation (5) for the sub-tensors u = 1, 2, …, 8, and arranged following the decreasing values of their dispersions.

#### 2.2. Description of the 2LTSP Decoder Performance

**X**is obtained, i.e., the transform of the input tensor

**X**through 2LTSP is reversible.

**X**:

- (1)
- The number of retained low-frequency spectrum coefficients, which compose the cubical spectrum tensors ${\widehat{\mathbf{S}}}_{0,2\times 2\times 2}$ and ${\widehat{\mathbf{S}}}_{1,2\times 2\times 2}(\mathrm{u})$ for u = 1, 2, …, 8 (in the last case, each spectrum tensor comprises eight coefficients);
- (2)
- The number of retained components of each HTSVD
_{2×2×2}, applied on the spectrum tensors ${\widehat{\mathbf{S}}}_{0,2\times 2\times 2}$ and ${\widehat{\mathbf{S}}}_{1,2\times 2\times 2}(\mathrm{u})$.

**X**.

## 3. HTSVD Algorithm for Spectrum Tensor Decomposition

**S**

_{2×}

_{2×}

_{2}, obtained at the outputs 0 and 1 of the 2LTSP coder shown in Figure 1a, HTSVD

_{2×}

_{2×}

_{2}is applied. As a result, a sum of four components is obtained, for which the energy of each tensor is concentrated mainly in the first and second components. This permits us to reduce (cut off) the number of low-energy components without lessening the quality of the restored tensor, in correspondence with Figure 1b.

_{2×2×2}algorithm for decomposition of the elementary tensor

**S**

_{2×2×2}is shown in Figure 2. The decomposition is based on SVD for the matrix

**X**of size 2 × 2, denoted as SVD

_{2×2}, and described by the relation below [13]:

**X**, $\mathrm{A}=\sqrt{{\mathsf{\nu}}^{2}+4{\mathsf{\eta}}^{2}}\ne 0$, ${\mathsf{\sigma}}_{1}=\sqrt{\frac{\mathsf{\omega}+\mathrm{A}}{2}}$, ${\mathsf{\sigma}}_{2}=\sqrt{\frac{\mathsf{\omega}\hspace{0.33em}-\mathrm{A}}{2}}$, ${\mathrm{U}}_{1}=\frac{1}{\sqrt{2\mathrm{A}}}\left[\begin{array}{c}\sqrt{\mathrm{p}}\\ \sqrt{\mathrm{q}}\end{array}\right]$, ${\mathrm{U}}_{2}=\frac{1}{\sqrt{2\mathrm{A}}}\left[\begin{array}{c}-\sqrt{\mathrm{q}}\\ \hspace{0.17em}\sqrt{\mathrm{p}}\end{array}\right]$, ${V}_{1}=\frac{1}{\sqrt{2\mathrm{A}}}\left[\begin{array}{c}\sqrt{\mathrm{r}}\\ \sqrt{\mathrm{s}}\end{array}\right]$, ${V}_{2}=\frac{1}{\sqrt{2\mathrm{A}}}\left[\begin{array}{c}-\sqrt{s}\\ \hspace{0.17em}\sqrt{\mathrm{r}}\end{array}\right]$, $\mathrm{r}=\mathrm{A}+\mathsf{\nu},$$\mathrm{p}=\hspace{0.17em}\mathrm{A}+\mathsf{\mu},$$\mathrm{s}=\mathrm{A}-\mathsf{\nu},$$\mathrm{q}=\hspace{0.17em}\mathrm{A}-\mathsf{\mu},$$\mathsf{\nu}={\mathrm{a}}^{2}+{\mathrm{c}}^{2}-{\mathrm{b}}^{2}-{\mathrm{d}}^{2},$$\mathsf{\eta}=\mathrm{a}\mathrm{b}+\mathrm{c}\mathrm{d},$ $\mathsf{\mu}={\mathrm{a}}^{2}+{\mathrm{b}}^{2}-{\mathrm{c}}^{2}-{\mathrm{d}}^{2},$ $\mathsf{\omega}={\mathbf{a}}^{2}+{\mathbf{b}}^{2}+{\mathbf{c}}^{2}+{\mathbf{d}}^{2}$. Since ${\mathsf{\sigma}}_{1}>>{\mathsf{\sigma}}_{2}$, the energy of the first decomposition component in (12) is much larger than that of the second. The number of parameters which define SVD

_{2×2}is four ($\mathsf{\nu},$$\mathsf{\eta},$$\mathsf{\mu},$$\mathsf{\omega}$), i.e., the decomposition is not “over-complete”.

**S**

_{2×2×2}, the following is obtained:

**S**

_{2×2×2}(HTSVD

_{2×2×2}), for each matrix

**S**

_{1}and

**S**

_{2}, an SVD of the size 2 × 2 (SVD

_{2×2}) is executed, and as a result we obtain:

**C**

_{i,j}of the size 2 × 2 for i, j = 1, 2, calculated in accordance with Equation (12), are rearranged into new couples in correspondence with their singular values. After this, the first couple of matrices,

**C**

_{11}and

**C**

_{21}, which have high singular values, define the tensor

**S**

_{1(2×2×2)}by inverse matricization, and for the second couple,

**C**

_{12}and

**C**

_{22}, which have lower singular values, the corresponding tensor is

**S**

_{2(2×2×2)}. Then:

_{2×2×2}level, SVD

_{2×2}is applied for each matrix

**S**

_{i,j}of the size 2 × 2, and this result is obtained:

**C**

_{i,j,k}of the size 2 × 2 for i, j, k = 1, 2 are rearranged into four new couples, following the decrease in their singular values. After inverse matricization, each of these four couples of matrices defines a corresponding tensor of size 2 × 2 × 2.

_{2×2×2}levels, the tensor ${\mathbf{S}}_{2\times 2\times 2}$ is represented as:

^{n}are decomposed [13].

## 4. Algorithm for Calculation of the 3D Frequency-Ordered Fast Walsh–Hadamard Transform

**X**of the size N × N × N, for N = 2

^{n}. To speed up the 1D FO-WHT calculations, the famous “fast” algorithm is used [16]. The number of basic operations “addition” (A

_{R}) needed for the execution of the “fast” 1D FO-WHT (1D FO-FWHT) in correspondence with the computational graph from Figure 3 is ${\mathrm{A}}_{\mathrm{R}}\left(\mathrm{n}\right)=\mathrm{N}\times {\mathrm{lg}}_{2}\mathrm{N}={2}^{\mathrm{n}}\mathrm{n}$. For the case when the “fast“ truncated 1D FO-TWHT (i.e., 1D FO-FTWHT) is used with a reduction (truncation) in some of the output coefficients from N down to 2, the number of “additions” ${\mathrm{A}}_{\mathrm{T}}(\mathrm{n})$ is defined by the relation:

**,**for the 3D WHT transform of the tensor

**X**, it is first divided into N frontal slices (2D matrices). Then, for each column of the sequence of 2D matrices, 1D FWHT is executed, followed by a similar operation for each row of the corresponding transformed matrices. The calculated tensor

**X**comprises all matrices transformed this way. Then, the tensor is divided again, but into N horizontal slices (2D matrices), which are transformed column-by-column through 1D FWHT. After that, on each row of the obtained 2D matrices, 1D TFWHT is applied again (for unfolding mode3 only), which, in correspondence with Equation (20), needs 2(N-1) additions only. Hence, the total number of additions needed to transform the tensor

**X**of the size N × N × N (N = 2

^{n}) through 3D FO-FWHT is correspondingly:

**X**of the size N × N × N through 3D WHT is:

**X**of the size 8 × 8 × 8, is 2.6 times.

## 5. Comparative Evaluation of 3D-FWHT Computational Complexity

_{3H}) for the hierarchical 3D-FWHT [14] when N = 2

^{n}(n denotes the number of decomposition levels) is:

^{n}, size N = 2

^{n}, and order d = 3 requires ${\mathrm{O}}_{\mathrm{H}\mathrm{T}}(\mathrm{n})=(3\hspace{0.17em}\times {2}^{3\mathrm{n}}+2\times {2}^{4\mathrm{n}})$ operations. For the TT decomposition [7], the needed operations for the same tensor are ${\mathrm{O}}_{\mathrm{T}\mathrm{T}}(\mathrm{n})=(3\hspace{0.17em}\times {2}^{4\mathrm{n}})$, i.e., the CC is approximately 1.5 times higher than that of the H-Tucker. This is why the H-Tucker transform was selected for the CC comparison with the analyzed 3D deterministic orthogonal transforms.

## 6. Global Algorithm for 3D Object Search in a Database of 3D Objects Represented through MLTSP

^{n}), calculated through the n-layer MLTSP. For the query, the 3D object used is a deep neural network [18]. The segmented object image is scaled so as to match the size of the objects in the DB, and its tensor is represented through the n-layer MLTSP. The basic principle which we used to search for the unknown 3D object is to detect the closest similar objects in the DB through sequential multilayer selection. In the first MLTSP layer (r = 0), the search is full, and, as a result, a group of closest similar objects is selected. The selection is based on the similarity degree between homonymous HTSVD components of the spectrum tensors of the size 2 × 2 × 2 in layers r = 0 of the corresponding MLTSPs. The small size of the compared THSVD components does not demand the use of significant computational resources for similarity evaluations. The search in the next MLTSP layer (r = 1) is also based on the similarity evaluation of the HTSVD components, but for the already selected group in the layer r = 0 only. As a result, the total number of computational operations needed for the similarity evaluation of the query tensor is reduced compared to the group from the previous layer. The search continues in the next MLTSP layers and stops when a 3D object is detected in the DB, whose similarity is higher than a predefined threshold. The thresholds used to select the groups of similar tensors in each layer of MLTSP are defined through training the corresponding neural network (NN).

**X**and

**Y**, must be executed. In cases that the total number of vectors obtained after the unfolding is Q, for the similarity evaluation, the mean value of SCSim could be used for each couple of vectors

**x**

_{q}and

**y**

_{q}of the same sequential number q = 1, 2, …, Q, which belongs to

**X**and

**Y**, correspondingly:

## 7. Invariant 3D Object Representation, Based on MLTSP with 3D Modified Mellin–Fourier Transform

_{r}, β

_{r}, γ

_{r}, 3D translation shift (T) in directions x

_{s}, y

_{s}, z

_{s}, 3D scaling (S), and contrast (C) changes. To obtain the invariant representation of a 3D object through MLTSP, the spectrum coefficients at the outputs of the decomposition layers r = 0, 1, …, n−1 must be calculated by using the 3D modified Mellin–Fourier transform (3D MMFT), which is an upgrade from the previous work of the authors, introduced in [15].

- (1)
- Bi-polar transform of the voxels x(i, l, k) of the input tensor image
**X**:$$\mathrm{L}(\mathrm{i},\mathrm{l},\mathrm{k})=\mathrm{x}(\mathrm{i},\mathrm{l},\mathrm{k})-({x}_{\mathrm{max}}+1)/2for\mathrm{i},\mathrm{l},\mathrm{k}=0,1,..,\mathrm{N}-1,$$_{max}is the maximum value in the voxel quantization scale. - (2)
- First direct 3D discrete Fourier transform (3D DFT):$$\mathrm{F}(\mathrm{a},\mathrm{b},\mathrm{c})={\displaystyle \sum _{\mathrm{i}=0}^{\mathrm{N}-1}{\displaystyle \sum _{\mathrm{l}=0}^{\mathrm{N}-1}{\displaystyle \sum _{\mathrm{k}}^{\mathrm{N}-1}\mathrm{L}(\mathrm{i},\mathrm{l},\mathrm{k})\hspace{0.17em}exp\{-\mathrm{j}\hspace{0.17em}(2\pi /\mathrm{N}(ia+\mathrm{l}\mathrm{b}+\mathrm{k}\mathrm{c})]\}}}}for\mathrm{a},\mathrm{b},\mathrm{c}=0,1,..,\mathrm{N}-1$$$${\mathrm{F}}_{0}(\mathrm{a},\mathrm{b},\mathrm{c})=\mathrm{F}(\mathrm{a}-{\scriptscriptstyle \frac{\mathrm{N}}{2}},\mathrm{b}-{\scriptscriptstyle \frac{\mathrm{N}}{2}},\mathrm{c}-{\scriptscriptstyle \frac{\mathrm{N}}{2}})for\mathrm{a},\mathrm{b},\mathrm{c}=0,1,..,\mathrm{N}-1$$
- (3)
- Retaining the centered low-frequency spectrum Fourier coefficients:$${\mathrm{F}}_{0\mathrm{R}}(\mathrm{a},\mathrm{b},\mathrm{c})=\left\{\begin{array}{c}{\mathrm{F}}_{0}(\mathrm{a},\mathrm{b},\mathrm{c}),\mathrm{i}\mathrm{f}(\mathrm{a},\mathrm{b},\mathrm{c})\in \mathrm{r}\mathrm{e}\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{d}\hspace{0.33em}\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{i}\mathrm{o}\mathrm{n};\\ 0\hspace{1em}\hspace{1em}\hspace{1em}-\hspace{1em}\hspace{1em}\mathrm{i}\mathrm{n}\hspace{0.33em}\mathrm{a}\mathrm{l}\mathrm{l}\hspace{0.17em}\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\hspace{0.33em}\mathrm{c}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{s}.\end{array}\right.$$

- (4)
- Calculation of modules and phases of the retained coefficients ${\mathrm{F}}_{0\mathrm{R}}\hspace{0.17em}(\mathrm{a},\mathrm{b},\mathrm{c})={\mathrm{D}}_{{\mathrm{F}}_{0\mathrm{R}}}(\mathrm{a},\mathrm{b},\mathrm{c}{)\mathrm{e}}^{\mathrm{j}{\mathsf{\phi}}_{{\mathrm{F}}_{0\mathrm{R}}}(\mathrm{a}\hspace{0.17em},\mathrm{b},\mathrm{c})}$:$${\mathrm{D}}_{{\mathrm{F}}_{0\mathrm{R}}}(\mathrm{a},\mathrm{b},\mathrm{c})=\left|\sqrt{{[{\mathrm{A}}_{{\mathrm{F}}_{0\mathrm{R}}}(\mathrm{a},\mathrm{b},\mathrm{c})]}^{2}+{[{\mathrm{B}}_{{\mathrm{F}}_{0\mathrm{R}}}(\mathrm{a},\mathrm{b},\mathrm{c})]}^{2}}\right|,\phantom{\rule{0ex}{0ex}}{\mathsf{\phi}}_{{\mathrm{F}}_{0\mathrm{R}}}(\mathrm{a},\mathrm{b},\mathrm{c})=arctan\hspace{0.17em}[{\mathrm{B}}_{{\mathrm{F}}_{0\mathrm{R}}}(\mathrm{a},\mathrm{b},\mathrm{c})/{\mathrm{A}}_{{\mathrm{F}}_{0\mathrm{R}}}(\mathrm{a},\mathrm{b},\mathrm{c})]$$
- (5)
- Calculation of the retained coefficients’ normalized modules:$$\mathrm{D}(\mathrm{a},\mathrm{b},\mathrm{c})=\mathrm{p}\mathrm{ln}{\mathrm{D}}_{{\mathrm{F}}_{0\mathrm{R}}}(\mathrm{a},\mathrm{b},\mathrm{c})for\mathrm{p}-\mathrm{the}\mathrm{normalization}\mathrm{coefficient};$$
- (6)
- Replacement of the orthogonal 3D discretization grid, which contains the voxels $\mathrm{D}(\mathrm{a},\mathrm{b},\mathrm{c})$, by a new grid, defined through 3D logarithmic spherical polar transform (3D LSPT), using the relations below:$$\mathsf{\rho}=\mathrm{log}\sqrt{{\mathrm{a}}^{2}+{\mathrm{b}}^{2}+{\mathrm{c}}^{2}},\mathsf{\theta}=\mathrm{arccos}\hspace{0.17em}[\mathrm{c}/\sqrt{{\mathrm{a}}^{2}+{\mathrm{b}}^{2}+{\mathrm{c}}^{2}}],\mathsf{\phi}=\mathrm{arctan}\hspace{0.17em}[\mathrm{b}/\mathrm{a}]$$

- (7)
- Replacement of the discretization grid for voxels $\mathrm{D}({\mathsf{\rho}}_{\mathrm{i}},{\mathsf{\theta}}_{\mathrm{j}},{\mathsf{\varphi}}_{\mathrm{j}})$ defined through 3D EPT, by the orthogonal discretization 3D grid for voxels D(x,y,z), calculated through trilinear interpolation (Figure 6). In this case, each interpolated voxel D(x
_{1},y_{1},z_{1}) is calculated taking into consideration the closest eight neighbor voxels on the grid:$$\mathrm{D}({\mathrm{x}}_{1},{\mathrm{y}}_{1},{\mathrm{z}}_{1})=(1/{\mathrm{H}}^{3})\{{\mathrm{z}}_{1}[{\mathrm{x}}_{1}({\mathrm{y}}_{1}\mathrm{B}+{\mathrm{y}}_{2}\mathrm{F})+{\mathrm{x}}_{2}({\mathrm{y}}_{1}\mathrm{E}+{\mathrm{y}}_{2}\mathrm{A})]+{\mathrm{z}}_{2}[{\mathrm{y}}_{1}({\mathrm{x}}_{1}\mathrm{G}+{\mathrm{x}}_{2}\mathrm{L})+{\mathrm{y}}_{2}({\mathrm{x}}_{1}\mathrm{C}+{\mathrm{x}}_{2}\mathrm{K})]\}$$

_{1},y

_{1},z

_{1})—linearly interpolated voxel for ${\mathrm{x}}_{1}=\mathsf{\alpha}-{\mathrm{x}}_{2}$, ${\mathrm{y}}_{1}=\mathsf{\beta}-{\mathrm{y}}_{2}$, ${\mathrm{z}}_{1}=\mathsf{\gamma}-{\mathrm{z}}_{1}$;

**D**for x, y, z = 0, 1, 2, .., H − 1.

- (8)
- Second direct 3D DFT for a tensor
**D**, with voxels D(x,y,z):$$\mathrm{S}(\mathrm{a},\mathrm{b},\mathrm{c})=(1/{\mathrm{H}}^{3}){\displaystyle \sum _{\mathrm{x}=0}^{\mathrm{H}-1}{\displaystyle \sum _{\mathrm{y}=0}^{\mathrm{H}-1}{\displaystyle \sum _{\mathrm{z}=0}^{\mathrm{H}-1}\mathrm{D}(\mathrm{x},\mathrm{y},\mathrm{z})\hspace{0.17em}exp\{-\mathrm{j}\hspace{0.17em}(2\pi /\mathrm{N}(xa+\mathrm{y}\mathrm{b}+\mathrm{z}\mathrm{c})]\}}}}for\mathrm{a},\mathrm{b},\mathrm{c}=0,1,..,\mathrm{H}-1.$$ - (9)
- Calculation of the complex coefficient $\mathrm{S}(\mathrm{a},\mathrm{b},\mathrm{c})$ modules:$${\mathrm{D}}_{\mathrm{S}}(\mathrm{a},\mathrm{b},\mathrm{c})=\left|\sqrt{{[{\mathrm{A}}_{\mathrm{S}}(\mathrm{a},\mathrm{b},\mathrm{c})]}^{2}+{[{\mathrm{B}}_{\mathrm{S}}(\mathrm{a},\mathrm{b},\mathrm{c})]}^{2}+{[{\mathrm{C}}_{\mathrm{S}}(\mathrm{a},\mathrm{b},\mathrm{c})]}^{2}}\right|,$$
- (10)
- Calculation of the normalized modules ${\mathrm{D}}_{{\mathrm{S}}_{0}}(\mathrm{a},\mathrm{b},\mathrm{c})$ of the Fourier coefficients $\mathrm{S}(\mathrm{a},\mathrm{b},\mathrm{c})$:$${\mathrm{D}}_{{\mathrm{S}}_{0}}(\mathrm{a},\mathrm{b},\mathrm{c})={\mathrm{x}}_{max}\times \hspace{0.17em}[({\mathrm{D}}_{\mathrm{S}}(\mathrm{a},\mathrm{b},\mathrm{c})/{\mathrm{D}}_{\mathrm{S}max}(\mathrm{a},\mathrm{b},\mathrm{c}))],$$
- (11)
- Calculation of the vector for RSTC-invariant 3D object representation based on the coefficients ${\mathrm{D}}_{{\mathrm{S}}_{0}}(\mathrm{a},\mathrm{b},\mathrm{c})$ of highest energy in the amplitude spectrum 3D MMFT, for $\mathrm{a},\mathrm{b},\mathrm{c}=0,1,..,\mathrm{H}-1$. The v
_{m}components for m = 1, 2, .., R of the corresponding RSTC-invariant vector $V=\hspace{0.17em}{[{\mathrm{v}}_{1},{\mathrm{v}}_{2},..,\hspace{0.17em}{\mathrm{v}}_{\mathrm{R}}]}^{\mathrm{T}}$ are defined by coefficients ${\mathrm{D}}_{{\mathrm{S}}_{0\mathrm{R}}}(\mathrm{a},\mathrm{b},\mathrm{c})$, arranged as a 1D massif after scanning the 3D-MMFT spectrum in the frame of the 3D mask area with R < H^{3}voxels ${\mathrm{D}}_{{\mathrm{S}}_{0}}(\mathrm{a},\mathrm{b},\mathrm{c})$.

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 5.**Block diagram of the algorithm for calculation of 3D MMFT coefficients for a cubical tensor of size N × N × N.

**Figure 6.**Calculation of the voxel D(x

_{1}, y

_{1}, z

_{1}) through trilinear interpolation on the basis of the closest eight neighbor voxels (colored in green) of the 3D discretization grid $({\mathsf{\rho}}_{\mathrm{i}},{\mathsf{\theta}}_{\mathrm{j}},{\mathsf{\varphi}}_{\mathrm{j}})$.

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Kountcheva, R.A.; Mironov, R.P.; Kountchev, R.K. MLTSP: New 3D Framework, Based on the Multilayer Tensor Spectrum Pyramid. *Symmetry* **2022**, *14*, 1909.
https://doi.org/10.3390/sym14091909

**AMA Style**

Kountcheva RA, Mironov RP, Kountchev RK. MLTSP: New 3D Framework, Based on the Multilayer Tensor Spectrum Pyramid. *Symmetry*. 2022; 14(9):1909.
https://doi.org/10.3390/sym14091909

**Chicago/Turabian Style**

Kountcheva, Roumiana A., Rumen P. Mironov, and Roumen K. Kountchev. 2022. "MLTSP: New 3D Framework, Based on the Multilayer Tensor Spectrum Pyramid" *Symmetry* 14, no. 9: 1909.
https://doi.org/10.3390/sym14091909