Use of chromium nanoparticles as a protector of digestive enzymes and biochemical parameters for various sources of fat in the diet of calves

Svyatoslav Lebedev; Elena Sheida; Irina Vershinina; Victoria Grechkina; Ilmira Gubaidullina; Sergey Miroshnikov; Oksana Shoshina; Svyatoslav Lebedev; Elena Sheida; Irina Vershinina; Victoria Grechkina; Ilmira Gubaidullina; Sergey Miroshnikov; Oksana Shoshina

doi:10.3934/agrfood.2021002

AIMS Agriculture and Food

2021, Volume 6, Issue 1: 14-31. doi: 10.3934/agrfood.2021002

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Research article

Use of chromium nanoparticles as a protector of digestive enzymes and biochemical parameters for various sources of fat in the diet of calves

Laboratory of Biological Testings and Expertises, Federal State Budget Scientific Institution «Federal Research Center for Biological Systems and Agro-technologies of the Russian Academy of Sciences», Orenburg, Russia

Received: 31 August 2020 Accepted: 05 November 2020 Published: 26 November 2020

The gastrointestinal tract acts as a digestive conveyor and carries out a specific stage of food processing throughout its length. A non-traditional approach to feeding cattle unprotected vegetable fats/oils may involve minerals (Ca and/or Mg). In our study, we studied the biological effects associated with the use of chromium nanoparticles at a dose of 200 μg/kg feed, together with soy and sunflower oils, in the diet of 10-month-old Kazakh white-headed breed calves. The calves were surgically fitted with duodenal fistula and pancreatic duct. Results demonstrated that inclusion of chromium nanoparticles in the diet increased the digestibility of crude protein, NDF, ADF and crude fat (p ≤ 0.05). At the same time, inclusion of chromium nanoparticles in the diet led to changes in the blood serum: a 2-fold increase in aspartate aminotransferase (p < 0.05), a 42.2% increase in lactate dehydrogenase and alanine levels similar to control values. We noted that the addition of lipids in the feed led to the mobilisation of triglycerides and cholesterol. At the same time, the inclusion of chromium nanoparticles contributed to a decrease in the level of total lipids in serum and inhibited peripheral metabolic pathways (a 2-fold increase in the de Ritis coefficient). These data may represent additional evidence of the activation of anabolic processes when chromium nanoparticles are introduced into the diet. The increase in the enzyme index was associated with an escalation of lipid peroxidation processes and a decrease in antioxidant defence activity. We showed that vegetable oils enhanced lipolytic and proteolytic activity of pancreatic juice enzymes in the gastrointestinal tract. Adding chromium nanoparticles to the diet reduced lipase activity, especially when used with soybean oil. These data indicate the inhibitory effect of chromium on the activity of pancreatic lipolytic enzymes.

Keywords:

Citation: Svyatoslav Lebedev, Elena Sheida, Irina Vershinina, Victoria Grechkina, Ilmira Gubaidullina, Sergey Miroshnikov, Oksana Shoshina. Use of chromium nanoparticles as a protector of digestive enzymes and biochemical parameters for various sources of fat in the diet of calves[J]. AIMS Agriculture and Food, 2021, 6(1): 14-31. doi: 10.3934/agrfood.2021002

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Abstract

1. Introduction

In the modern fields of scientific research and medical diagnostics ^[1,2,3], there is an increasing reliance on image restoration techniques ^[4,5,6,7], which are particularly prominent in the field of medical imaging. Ensuring image quality is paramount for the authenticity of data ^[8]. By processing X-ray projection data obtained from CT scans, we can reconstruct clear tomographic images with a resolution of $n\times n$ pixels. This technology encompasses two main schools: mathematical theoretical analysis ^[9] and iterative methods ^[10]. The former, like the filtered back projection method ^[11], is widely used in fields such as CT imaging, while the latter excels in dealing with noise interference and data loss. With technological advancements, image restoration is moving toward greater precision and efficiency.

To reconstruct images, we must solve a large-scale system of linear equations with multiple righthand sides, which is presented as follows:

$\begin{array}{c}AX = B\end{array}$

(1)

In contemporary mathematics and engineering, solving systems of equations stands as a pivotal task, particularly in scientific research and industrial applications where efficiency and precision are of paramount importance. A variety of numerical methods have been widely adopted ^{[12,13,14,15]}, among which the Kaczmarz algorithm ^[16] is renowned for its iterative projection approach in approximating the true solution. The algorithm's simplicity has facilitated its application across various domains, including image reconstruction ^[12,17], medical imaging ^[11,18], and signal processing ^[19,20]. Advances in technology have given rise to multiple enhanced versions of the Kaczmarz algorithm ^{[21,22,23,24,25,26,27,28,29,30,31]}, improving performance in large-scale parallel computing and noisy data environments. Notably, with the development of free pseudo-inverse techniques, Du and Sun ^[24] further extended the randomized extended average block Kaczmar (REABK) method, proposing a class of pseudo-inverse-free stochastic block iterative methods for solving both consistent and inconsistent systems of linear equations, Free pseudo-inverse can accelerate convergence speed. Pseudo-inverse approximation is a specific case of generalized inverse techniques. For more theoretical analysis and applications, refer to literature ^[32]. Inspired by references ^[33,34], this paper introduces a faster lazy free greedy block Kaczmarz (LFGBK) method, exploring matrix sketching techniques as a key tool for accelerating matrix operations. This method employs sampling based on an approximate maximum distance criterion, excelling at selecting small, representative samples from large datasets for more efficient computation. The adaptive randomized block Kaczmarz (ARBK) ^[35] algorithm integrates adaptive and randomized block selection strategies. However, under certain specific matrix structures, ARBK may experience slow convergence or even fail to converge. Additionally, when dealing with large-scale sparse matrices, the ARBK algorithm incurs significant memory overhead. Our improved algorithm successfully overcomes these drawbacks, offering a more stable and efficient solution.

Additionally, it not only addresses single righthand side linear equations but also extends to multiple righthand side linear equations, solving the memory overflow issues that traditional algorithms face when dealing with image processing vectors. A comprehensive theoretical framework supports the convergence of these methods. Numerical experiments validate its effectiveness, demonstrating improved computational efficiency and laying a solid foundation for further optimization of matrix sketching techniques.

The structure of this paper is as follows: Section 2 introduces the necessary background knowledge. Section 3 presents the faster free pseudo-inverse GBK method for solving single righthand side linear equation systems. Section 4 proposes the leverage score sampling free pseudo-inverse GBK method for solving multiple righthand side linear equation systems. Section 5 details the numerical experiments, and Section 6 concludes the paper.

2. Knowledge preparation

In this article, we adopt the same notation as in reference ^[27]. For example, ${A}_{\left(i\right)}, {A}^{T}, {A}^{†}, ||A||, {||A||}_{F},$ and $\left[n\right]$ , respectively, represent the i-th row of the coefficient matrix A, transpose, generalized inverse, spectral norm, F-norm, and the set $\{\mathrm{1, 2}, \dots , n\}$ .

Recently, Niu and Zheng combined greedy strategy with the block Kaczmarz method, proposing the GBK ^[29] method for solving large-scale consistent linear equation systems. See Algorithm 1 for the specific process.

Algorithm1 GBK method

1: Input:

$A, b, l, {x}_{0}\in range\left({A}^{T}\right)\;and\;\eta \in \left(\mathrm{0, 1}\right)$ .
2: Output:

${x}_{l}$ .
3: for

$k = \mathrm{0, 1}, \dots l-1$ do
4: Compute

${\varepsilon }_{k} = \eta \genfrac{}{}{0pt}{}{max}{1\le i\le m}\left\{\frac{{\left|{b}_{i}-{A}_{\left(i\right){x}_{k}}\right|}^{2}}{{||{A}_{\left(i\right)}||}_{2}^{2}}\right\}$ .
5: Determine the sequence of indicator sets.

${\mathcal{T}}_{k} = \left\{{i}_{k}:{\left|{b}_{ik}-{A}_{\left({i}_{k}\right)}{x}_{k}\right|}^{2}\ge {\varepsilon }_{k}{||{A}_{\left({i}_{k}\right)}||}_{2}^{2}\right\}$ .
6: Compute

${x}_{k+1} = {x}_{k}+{A}_{{\mathcal{T}}_{k}}^{†}\left({b}_{{\mathcal{T}}_{k}}-{A}_{{\mathcal{T}}_{k}}{x}_{k}\right)$ .
7: end for

Convergence analysis of the GBK method is described as follows:

Theorem 1 (^[29]) If the linear system of Eq (1) is consistent, then the iterative sequence ${\left\{{x}_{k}\right\}}_{k = 0}^{\infty }$ generated by algorithm 1 converges to the minimum norm solution ${x}_{*} = {A}^{†}b$ of the system of equations, and satisfies for any $k\ge 0$ ,

${||{x}_{k+1}-{x}_{\mathrm{*}}||}_{2}^{2}\le {\left(1-{\gamma }_{k}\left(\eta \right)\frac{{\sigma }_{min}^{2}\left(A\right)}{{||A||}_{F}^{2}}\right)}^{k+1}{||{x}_{0}-{x}_{\mathrm{*}}||}_{2}^{2} .$

The formula for ${\gamma }_{k}\left(\eta \right)$ is defined as follows: ${\gamma }_{k}\left(\eta \right) = \eta \frac{{||A||}_{F}^{2}}{{||A||}_{F}^{2}-{||{A}_{{\mathcal{T}}_{k-1}}||}_{F}^{2}}\frac{{||{A}_{{\mathcal{T}}_{k}}||}_{F}^{2}}{{\sigma }_{max}^{2}\left({A}_{{\mathcal{T}}_{k}}\right)}$ . Here, ${\gamma }_{0\left(\eta \right)}$ is defined as $\eta \frac{{||{A}_{{\mathcal{T}}_{0}}||}_{F}^{2}}{{\sigma }_{max}^{2}\left({A}_{{\mathcal{T}}_{0}}\right)}$ , where $\eta$ is in the range $\left(\mathrm{0, 1}\right]$ , and ${\sigma }_{min}\left(A\right)$ and ${\sigma }_{max}\left(A\right)$ represent the nonzero minimum singular value and maximum singular value of matrix $A$ .

In the matrix sketching technique, as described in leverage score sampling ^[36,37], we select samples based on the leverage score of each row. Specifically, we will choose each row with a probability proportional to its leverage score. Therefore, rows with higher scores (i.e., rows with greater influence in the dataset) will have a greater chance of being selected.

Algorithm2 Leverage score sampling method based on manifold

1: Input:

$A\in {\mathbb{R}}^{m\times n}$ .
2: Initialize

$C$ as a

$d\times n$ zero matrix.
3: Initialize the variable

$sum$ to

$0$ .
4: Calculate the singular value decomposition of

$A$ as

$U$ ,

$S$ , and

$V$ .
5: for

$k = 1, \dots , m$
6: Calculate the sum of the squares of each row in

$U$ and add it to the

$sum$ .
7: Calculate the probability of each data point being sampled: Calculate the sum of squares for each row of

$U$ , then divide by the total

$sum$ .
8: Based on the calculated probabilities, sampling is conducted to obtain the sampling

$indices$ .

9: Utilize

$C = [indices, :]$ for

$indexing$ .
10: end for
11: Return

$C\in {\mathbb{R}}^{d\times n}$ .

The advantage of this sampling method lies in its ability to select a small, representative subset of samples from a large dataset, enabling more efficient computation. The resulting sample set $S\in {R}^{d\times m}$ , where $d$ is the chosen number of samples, can be utilized to estimate various properties of the original matrix $A$ , such as singular value decomposition, principal component analysis, and so on. The LFGBK algorithm is similar to the derivation process in reference ^[38].

Remark: In addition to the method proposed in this paper, there are other sampling techniques such as random Gaussian matrices, subsampled randomized Hadamard transform (SRHT), and uniform sampling. In previous experiments, we also tried other methods like random sparse sampling. Through comparison, we found that the method proposed in this paper excels in extracting the main features of matrices. Other methods not only have higher computational complexity but may also extract all-zero vectors during actual image processing, rendering calculations impossible. Our method effectively avoids these issues, which is why we chose to adopt it.

3. Fast free pseudo-inverse GBK method for solving single righthand side linear equation systems

This chapter introduces an algorithm, namely, the leverage score sampling free pseudo-inverse GBK method for single righthand side, and provides a proof of the corresponding convergence theory for the algorithm.

First, each step follows the approximate maximum illustration principle.

$\frac{{\left|{b}_{{i}_{k}}-{A}_{\left({i}_{k}\right)}{x}_{k}\right|}^{2}}{{||{A}_{\left({i}_{k}\right)}||}_{2}^{2}}\ge \eta \underset{1\le i\le m}{\mathrm{max}}\left\{\frac{{\left|{b}_{i}-{A}_{\left(i\right)}{x}_{k}\right|}^{2}}{{||{A}_{\left(i\right)}||}_{2}^{2}}\right\} ,$

Second, select the index set ${\mathcal{T}}_{k}$ for the block matrix ${A}_{{\mathcal{T}}_{k}}$ ; then, project the current estimate onto each row forming the block matrix ${A}_{{\mathcal{T}}_{k}}$ ; finally, calculate the average of the projections to determine the next iteration.

${x}_{k+1} = {x}_{k}+\left(\sum _{i\in {\mathcal{T}}_{k}}{w}_{i}\frac{{b}_{i}-{A}_{\left(i\right)}{x}_{k}}{{||{A}_{\left(i\right)}||}_{2}^{2}}{A}_{\left(i\right)}^{T}\right) .$

Before presenting the convergence theory for Algorithm 3, let us first introduce a lemma.

Algorithm 3 SLFGBK method

1: Let

$A, b, l, {x}_{0}$ be within the range of

$\left({A}^{T}\right)$ , parameter

$\eta$ within

$\left(\mathrm{0, 1}\right)$ , and consider the sequences of step sizes

${\left({\alpha }_{k}\right)}_{k}\ge 0$ and weights

${\left({w}_{k}\right)}_{k}\ge 0$ .
2: Output:

${x}_{l}$ .
3: Initialization:
Leverage score sampling enables the selection of a small, representative subset from large datasets, facilitating more efficient computation.
Algorithm 2 generates a leverage score sampling transformation matrix

$\widetilde {A} = SA\in {\mathbb{R}}^{d\times n}$ and vector

$\widetilde {b} = Sb$ , where

$d\ll m$ .
4: for

$k = \mathrm{0, 1}, \dots l-1$ do
5: Calculation

${\widetilde {\varepsilon }}_{k} = \eta \genfrac{}{}{0pt}{}{max}{1\le i\le m}\left\{\frac{{\left|{\widetilde {b}}_{i}-{\widetilde {A}}_{\left(i\right){x}_{k}}\right|}^{2}}{{||{\widetilde {A}}_{\left(i\right)}||}_{2}^{2}}\right\}$ .
6: Define the index set sequence

${\mathcal{T}}_{k} = \left\{{i}_{k}:{\left|{\widetilde {b}}_{{i}_{K}}-{\widetilde {A}}_{\left({i}_{k}\right)}{x}_{k}\right|}^{2}\ge {\varepsilon }_{k}{||{\widetilde {A}}_{\left({i}_{k}\right)}||}_{2}^{2}\right\}$ .
7: Computation

${x}_{k+1} = {x}_{k}+\left(\sum _{i\in {\mathcal{T}}_{k}}{w}_{i}\frac{{\widetilde {b}}_{i}-{\widetilde {A}}_{\left(i\right)}{x}_{k}}{{||{\widetilde {A}}_{\left(i\right)}||}_{2}^{2}}{\widetilde {A}}_{\left(i\right)}^{T}\right)$ .
8: end for

Lemma 1 (^[27]): If any vector $u$ belongs to the range of ${A}^{T}$ , then

${||Au||}_{2}^{2}\ge {\lambda }_{min}\left({A}^{T}A\right){||u||}_{2}^{2}$

Theorem 2: The leverage score transformation S satisfies $d = O({n}^{2}/\mathrm{\delta }{\theta }^{2})$ , and ${x}_{*} = {A}^{†}b$ is the minimum norm solution of the single righthand side linear equation systems, for any k ≥ 0.

$\begin{array}{c}{||{x}_{k+1}-{x}_{\mathrm{*}}||}_{2}^{2}\le \left(1-{\widetilde {\psi }}_{k}\left(\eta \right)\frac{{\sigma }_{min}^{2}\left(\widetilde {A}\right)}{{||\widetilde {A}||}_{F}^{2}}\right){||{x}_{k}-{x}_{\mathrm{*}}||}_{2}^{2}\end{array}$

(2)

The function ${\widetilde {\psi }}_{k}\left(\eta \right) = \eta t\left(\frac{2}{t}-\frac{1}{{t}^{2}}{\sigma }_{max}^{2}\left({\widehat{A}}_{{\mathcal{T}}_{k}}^{T}\right)\right){\sigma }_{min}^{2}, \eta \in \left(\mathrm{0, 1}\right],$ where $range\left(A\right), {\sigma }_{min}\left(A\right),$ and ${\sigma }_{max}\left(A\right)$ represent the range, nonzero minimum singular value, and maximum singular value of matrix $A$ , respectively.

Proof. With algorithm 3 and ${\widetilde {r}}_{k} = \widetilde {b}-\widetilde {A}{x}_{k}$ , we can obtain

$\begin{array}{c}{x}_{k+1} = {x}_{k}+\left(\sum _{i\in {\mathcal{T}}_{k}}\frac{1}{\left|{\mathcal{T}}_{k}\right|}\frac{{\widetilde {r}}_{k}^{i}{\widetilde {A}}_{\left(i\right)}^{T}}{{||{\widetilde {A}}_{\left(i\right)}||}_{2}^{2}}\right)\end{array} .$

(3)

Expand Eq (3) with the set $\left|{\mathcal{T}}_{k}\right| = t$ and ${\mathcal{T}}_{k} = \left\{{j}_{{k}_{1}}, \dots , {j}_{{k}_{t}}\right\}$ .

$\begin{array}{l} {x}_{k+1} = {x}_{k}+\sum _{i\in {\mathcal{T}}_{k}}\frac{1}{t}\frac{{\widetilde {A}}_{\left(i\right)}^{T}{e}_{i}^{T}{\widetilde {r}}_{k}}{{||{\widetilde {A}}_{\left(i\right)}||}_{2}^{2}} \\ \;\;\;\;\;\;\;\;{ = x}_{k}+\frac{1}{t}\left(\frac{{\widetilde {A}}_{\left({j}_{{k}_{1}}\right)}^{T}{e}_{{j}_{{k}_{1}}}^{T}{\widetilde {r}}_{k}}{{||{\widetilde {A}}_{\left({j}_{{k}_{1}}\right)}||}_{2}^{2}}+\dots +\frac{{\widetilde {A}}_{\left({j}_{{k}_{t}}\right)}^{T}{e}_{{j}_{{k}_{t}}}^{T}{\widetilde {r}}_{k}}{{||{\widetilde {A}}_{\left({j}_{{k}_{t}}\right)}||}_{2}^{2}}\right) \\ \;\;\;\;\;\;\;\; = {x}_{k}+\frac{1}{t}{\widehat{A}}_{{\mathcal{T}}_{k}}^{T}{\widehat{I}}_{{\mathcal{T}}_{k}}{\widetilde {r}}_{k} . \end{array}$

(4)

Among them

$\begin{array}{c}{\widehat{A}}_{{\mathcal{T}}_{k}}^{T} = \left[\frac{{\widetilde {A}}_{\left({j}_{{k}_{1}}\right)}^{T}}{{||{\widetilde {A}}_{\left({j}_{{k}_{1}}\right)}||}_{2}}, \frac{{\widetilde {A}}_{\left({j}_{{k}_{2}}\right)}^{T}}{{||{\widetilde {A}}_{\left({j}_{{k}_{2}}\right)}||}_{2}}, \dots , \frac{{\widetilde {A}}_{\left({j}_{{k}_{t}}\right)}^{T}}{{||{\widetilde {A}}_{\left({j}_{{k}_{t}}\right)}||}_{2}}\right]\in {\mathbb{R}}^{n\times t}\end{array} ,$

(5)

and

$\begin{array}{c}{\widehat{I}}_{{\mathcal{T}}_{k}} = {\left[\frac{{e}_{\left({j}_{{k}_{1}}\right)}}{{||{\widetilde {A}}_{\left({j}_{{k}_{1}}\right)}||}_{2}}, \frac{{e}_{\left({j}_{{k}_{2}}\right)}}{{||{\widetilde {A}}_{\left({j}_{{k}_{2}}\right)}||}_{2}}, \dots , \frac{{e}_{\left({j}_{{k}_{t}}\right)}}{{||{\widetilde {A}}_{\left({j}_{{k}_{t}}\right)}||}_{2}}\right]}^{T}\in {\mathbb{R}}^{t\times d}\end{array} .$

(6)

Subtracting ${x}_{*}$ from Eq (4) simultaneously yields

$\begin{array}{c} {x}_{k+1}-{x}_{\mathrm{*}} = {x}_{k}-{x}_{\mathrm{*}}-\frac{1}{t}{\widehat{A}}_{{\mathcal{T}}_{k}}^{T}{\widehat{I}}_{{\mathcal{T}}_{k}}\widetilde {A}\left({x}_{k}-{x}_{\mathrm{*}}\right) \\ = \left(I-\frac{1}{t}{\widehat{A}}_{{\mathcal{T}}_{k}}^{T}{\widehat{A}}_{{\mathcal{T}}_{k}}\right)\left({x}_{k}-{x}_{\mathrm{*}}\right) \end{array}$

(7)

Taking the spectral norm and squaring both sides of Eq (7), and for any positive semi-definite matrix Q, satisfying ${Q}^{2}\preccurlyeq {\lambda }_{max}\left(Q\right)Q$ , we can obtain

$\begin{array}{l} {||{x}_{k+1}-{x}_{\mathrm{*}}||}_{2}^{2} = {||{(x}_{k}-{x}_{\mathrm{*}})-\frac{1}{t}{\widehat{A}}_{{\mathcal{T}}_{k}}^{T}{\widehat{A}}_{{\mathcal{T}}_{k}}{(x}_{k}-{x}_{\mathrm{*}})||}_{2}^{2} \\ \;\;\;\;\;\;\;\;\;\; = {||{(x}_{k}-{x}_{\mathrm{*}})||}_{2}^{2}-\frac{2}{t}{||{\widehat{A}}_{{\mathcal{T}}_{k}}{(x}_{k}-{x}_{\mathrm{*}})||}_{2}^{2}+\frac{1}{{t}^{2}}{||{\widehat{A}}_{{\mathcal{T}}_{k}}^{T}{\widehat{A}}_{{\mathcal{T}}_{k}}\left({(x}_{k}-{x}_{\mathrm{*}}\right)||}_{2}^{2} \\ \;\;\;\;\;\;\;\;\;\; \le {||{(x}_{k}-{x}_{\mathrm{*}})||}_{2}^{2}-\left(\frac{2}{t}-\frac{1}{{t}^{2}}{\sigma }_{max}^{2}\left({\widehat{A}}_{{\mathcal{T}}_{k}}^{T}\right)\right){||{\widehat{A}}_{{\mathcal{T}}_{k}}{(x}_{k}-{x}_{\mathrm{*}})||}_{2}^{2} \end{array}$

(8)

Using Eq (5) and the inequality ${\left|{\widetilde {b}}_{{j}_{k}}-{\widetilde {A}}_{\left({j}_{k}\right)}{x}_{k}\right|}^{2}\ge {\widetilde {\varepsilon }}_{k}{||{\widetilde {A}}_{\left({j}_{k}\right)}||}_{2}^{2}$ , a straightforward calculation yields.

$\begin{array}{c}{||{\widehat{A}}_{{\mathcal{T}}_{k}}{(x}_{k}-{x}_{\mathrm{*}})||}_{2}^{2} = \sum _{{j}_{k}\in {\mathcal{T}}_{k}}\frac{1}{{||{\widetilde {A}}_{\left({j}_{k}\right)}||}_{2}^{2}}{\left|{\widetilde {r}}_{k}^{{j}_{k}}\right|}^{2}\end{array}$

(9)

$\begin{array}{c}\ge \sum _{{j}_{k}\in {\mathcal{T}}_{k}}\frac{1}{{||{\widetilde {A}}_{\left({j}_{k}\right)}||}_{2}^{2}}{\widetilde {\varepsilon }}_{k}{||{\widetilde {A}}_{\left({j}_{k}\right)}||}^{2}\end{array}$

(10)

Substituting ${\widetilde {\varepsilon }}_{k} = \eta \genfrac{}{}{0pt}{}{max}{1\le i\le d}\left\{\frac{{\left|{\widetilde {b}}_{i}-{\widetilde {A}}_{\left(i\right){x}_{k}}\right|}^{2}}{{||{\widetilde {A}}_{\left(i\right)}||}_{2}^{2}}\right\}$ into the above equation yields the following result:

$\begin{array}{l} {||{\widehat{A}}_{{\mathcal{T}}_{k}}{(x}_{k}-{x}_{\mathrm{*}})||}_{2}^{2}\ge \sum \limits_{{j}_{k}\in {\mathcal{T}}_{k}}\eta \underset{1\le i\le d}{\mathrm{max}}\left\{\frac{{\left|{\widetilde {r}}_{k}^{i}\right|}^{2}}{{||{\widetilde {A}}_{\left(i\right)}||}_{2}^{2}}\right\}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\; \ge \eta t\sum \limits_{i = 1}^{d}\frac{{\left|{\widetilde {r}}_{k}^{i}\right|}^{2}}{{||{A}_{\left(i\right)}||}_{2}^{2}}\frac{{||{\widetilde {A}}_{\left(i\right)}||}_{2}^{2}}{{||A||}_{F}^{2}}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\; = \eta t\frac{{||{\widetilde {r}}_{k}||}^{2}}{{||\widetilde {A}||}_{F}^{2}} \end{array}$

(11)

Given ${x}_{0}\in range\left({A}^{T}\right)$ and ${x}_{*} = {A}^{†}b$ , we have ${{x}_{k}-x}_{*}\in range\left({A}^{T}\right)$ . Therefore, by Lemma 1, we can conclude that

${||{\widetilde {r}}_{k}||}^{2} = {||\widetilde {A}\left({{x}_{k}-x}_{\mathrm{*}}\right)||}_{2}^{2}$

$\ge {\lambda }_{min}^{2}\left({\widetilde {A}}^{T}\widetilde {A}\right){||{{x}_{k}-x}_{\mathrm{*}}||}_{2}^{2}$ Combining the above equation, we can obtain the following

$\begin{array}{c}{||{\widehat{A}}_{{\mathcal{T}}_{k}}{(x}_{k}-{x}_{\mathrm{*}})||}_{2}^{2}\ge \eta t\frac{{\sigma }_{min}^{2}\left(\widetilde {A}\right)}{{||\widetilde {A}||}_{F}^{2}}{||{{x}_{k}-x}_{\mathrm{*}}||}_{2}^{2}\end{array}$

(12)

From Eqs (8) and (12), we can obtain Eq (2), and the iterative sequence ${\left\{{x}_{k}\right\}}_{k = 0}^{\infty }$ converges to the minimum norm solution ${x}_{*} = {A}^{†}b$ of the system of equations. Hence, Theorem 2 is proved.

4. Solving multiple righthand side linear systems with leverage score sampling free pseudo-inverse GBK method

In this chapter, we introduce an algorithm known as the multi-righthand-side leverage score sampling free pseudo-inverse GBK method, and we provide a proof for the corresponding convergence theory of the algorithm.

For most image reconstruction, the system of equations is formulated as follows:

$\begin{array}{c}AX = B\end{array}$

(13)

In the context, $A\in {R}^{m\times n}, X = \left[{x}_{1}, {x}_{2}, \dots , {x}_{d}\right]\in {R}^{n\times d}, B = \left[{b}_{1}, {b}_{2}, \dots , {b}_{d}\right]\in {R}^{m\times d}, d > 1$ . This paper addresses the solution of multiple righthand side linear equation systems and initially presents the index set for each iteration selection in the multiple righthand side linear equation systems.

$\begin{array}{c}\underset{1\le i\le m}{\mathrm{max}}\left\{\frac{\left|{b}_{\left(i\right)}-{A}_{\left(i\right)}{x}_{k}\right|}{{||{A}_{\left(i\right)}||}_{2}^{2}}\right\}\end{array}$

(14)

Iterate through the following steps:

$\begin{array}{c}{x}_{k+1} = {x}_{k}+\frac{{b}_{\left(i\right)}-{A}_{\left(i\right)}{x}_{k}}{{||{A}_{\left(i\right)}||}_{2}^{2}}{A}_{\left(i\right)}^{T}\end{array}$

To solve systems of linear equations with multiple righthand sides, our approach involves working with the $j$ righthand side vector ${b}_{j}$ by selecting the working row ${\mathcal{T}}_{{k}_{j}}$ according to the criterion established in Eq (15).

$\begin{array}{c}\underset{1\le i\le m}{\mathrm{max}}\left\{\frac{\left|{B}_{\left(i, j\right)}-{A}_{\left(i\right)}{x}_{j}^{\left(k\right)}\right|}{{||{A}_{\left(i\right)}||}_{2}^{2}}\right\}, j = \mathrm{1, 2}, \dots , {k}_{b}.\end{array}$

(15)

The iterative formula corresponding to the solution at the $j$ right endpoint is as follows:

$\begin{array}{c}{x}_{j}^{\left(k+1\right)} = {x}_{j}^{k}+\frac{{\widetilde {B}}_{\left({\mathcal{T}}_{{k}_{j}}, j\right)}-{\widetilde {A}}_{\left({\mathcal{T}}_{{k}_{j}}\right)}{x}_{j}^{\left(k\right)}}{{||{\widetilde {A}}_{\left({\mathcal{T}}_{{k}_{j}}\right)}||}_{2}^{2}}{\widetilde {A}}_{{\mathcal{T}}_{{k}_{j}}}^{T}\end{array}$

(16)

According to the criterion for selecting the index set in Eq (15), each term on the righthand side corresponds to the selection of a working row. For the system of linear equations with multiple righthand terms as in Eq (13), the Kaczmarz method can be extended to an iterative format that simultaneously solves for the righthand side system of linear equations.

In this context, ${X}^{(k+1)} = ({x}_{1}^{\left(k+1\right)}, {x}_{2}^{\left(k+1\right)}, \dots , {x}_{d}^{\left(k+1\right)})$ represents the approximate solution at the $k+1$ iteration.

${X}^{k+1} = {X}^{k}+{\left({\widetilde {A}}_{{j}_{{k}_{1}}}^{T}, \dots , {\widetilde {A}}_{{j}_{{k}_{d}}}^{T}\right)}^{T}diag(\frac{{\widetilde {B}}_{\left({j}_{{k}_{1}}, 1\right)}-{\widetilde {A}}_{{j}_{{k}_{1}}}{x}_{1}^{\left(k\right)}}{{||{\widetilde {A}}_{{j}_{{k}_{1}}}||}_{2}^{2}}, \dots , \frac{{\widetilde {B}}_{\left({j}_{{k}_{d}}, d\right)}-{\widetilde {A}}_{{j}_{{k}_{d}}}{x}_{d}^{\left(k\right)}}{{||{\widetilde {A}}_{{j}_{{k}_{d}}}||}_{2}^{2}})$

To reduce computational effort, Algorithm 2 is employed to extract certain rows from matrix A corresponding to the index set ${\mathcal{T}}_{{k}_{j}}$ . The sub-matrix formed by these rows is denoted as $\widetilde {A},$ , while the sub-matrix of B corresponding to the index set ${\mathcal{T}}_{k}$ is denoted as $\widetilde {B}$ . The specific configurations of $\widetilde {A}$ and $\widetilde {B}$ are as follows.

$\begin{array}{c}{\widetilde {B}}_{{\mathcal{T}}_{{k}_{j}}}^{T} = \left({B}_{{\mathcal{T}}_{{k}_{{j}_{1}}}}^{T}, {B}_{{\mathcal{T}}_{{k}_{{j}_{2}}}}^{T}, \dots , {B}_{{\mathcal{T}}_{{k}_{{j}_{t}}}}^{T}\right)\end{array}$

(17)

and

$\begin{array}{c}{\widetilde {A}}_{{\mathcal{T}}_{{k}_{j}}}^{T} = \left({A}_{{\mathcal{T}}_{{k}_{1}}}^{T}, {A}_{{\mathcal{T}}_{{k}_{2}}}^{T}, \dots , {A}_{{\mathcal{T}}_{{k}_{t}}}^{T}\right)\end{array}$

(18)

The transposed matrix ${B}_{{\mathcal{T}}_{{k}_{{j}_{i}}}}^{T}$ is represented as $\left({B}_{\left({\mathcal{T}}_{{k}_{{j}_{i}}}, 1\right)}, {B}_{\left({\mathcal{T}}_{{k}_{{j}_{i}}}, 2\right)}, \dots , {B}_{\left({\mathcal{T}}_{{k}_{{j}_{i}}}, d\right)}\right)$ , where ${\mathcal{T}}_{{k}_{{j}_{i}}}$ is an element of the index set ${\mathcal{T}}_{{k}_{j}}$ . Here, ${B}_{\left({\mathcal{T}}_{{k}_{{j}_{i}}}, t\right)}$ denotes the element in the $i$ row and $t$ column of the sub-matrix $\widetilde {B}$ , and $\left|{\mathcal{T}}_{{k}_{j}}\right|$ denotes the number of row indices contained in the index set ${\mathcal{T}}_{{k}_{j}}$ with $\left|{\mathcal{T}}_{{k}_{j}}\right|$ equal to $t$ .

The multi-righthand-side method proposed in this paper is a specialized block approach for solving large-scale righthand side linear equations using a leverage-based pseudo-inverse GBK method. We provide a detailed framework for this method, which involves simultaneous computation of multiple righthand sides, with each iteration involving a matrix $B\in {R}^{m\times t}$ containing $t$ righthand sides, resulting in the selection of $t$ row indices.

This article exclusively discusses the scenario where ${\alpha }_{k}$ equals $1$ . The convergence theory for solving large-scale righthand side linear equation groups with the LFGBK method is presented below:

Theorem 3: Assume that the linear equation system (13) is consistent, and so are its extracted subsystems. The iterative sequence ${X}^{\left(k\right)}$ from $k = 0$ to infinity, generated by Algorithm 4 starting from the initial value ${x}_{0}$ , converges to the least squares solution ${X}^{*} = {A}^{†}B$ . The expected error for the iterative sequence solutions ${X}^{\left(k\right)}$ from $k = 0$ to infinity is:

$\begin{array}{c}E{||{X}^{\left(k+1\right)}-{X}^{\mathrm{*}}||}_{F}^{2}\le \left(1-\frac{{\sigma }_{min}^{2}\left({\widetilde {A}}_{\left({\mathcal{T}}_{{k}_{j}}\right)}\right)}{{||{\widetilde {A}}_{\left({\mathcal{T}}_{{k}_{j}}\right)}||}_{F}^{2}}\right)E{||{X}^{k}-{X}^{\mathrm{*}}||}_{F}^{2}\end{array}$

(19)

Algorithm 4 MLFGBK Method

1: Let

$A, b, l, {x}_{0}$ be within the range of

$\left({A}^{T}\right)$ , parameter

$\eta$ within

$\left(\mathrm{0, 1}\right)$ , and consider the sequences of step sizes

${\left({\alpha }_{k}\right)}_{k}\ge 0$ and weights

${\left({w}_{k}\right)}_{k}\ge 0$ .
2: Output:

$x$ .
3: Initialization: Algorithm 2 generates a leverage score sampling transformation matrix

$\widetilde {A} = SA\in {\mathbb{R}}^{d\times n}$ and matrix

$\widetilde {b} = Sb$ , where

$d\ll m$ .
4: for

$k = \mathrm{0, 1}, \dots l-1$ do
5: Calculation

${\widetilde {\varepsilon }}_{{k}_{j}} = \eta \underset{1\le i\le m}{\mathrm{max}}\left\{\frac{\left|{B}_{\left(i, j\right)}-{A}_{\left(i\right)}{X}_{j}^{\left(k\right)}\right|}{{||{A}_{\left(i\right)}||}_{2}^{2}}\right\}, j = \mathrm{1, 2}, \dots , d.$
6: Establish the sequence of indicator sets.

${\mathcal{T}}_{{k}_{j}} = \left\{{i}_{{k}_{j}}:{\left|{\widetilde {b}}_{{i}_{{k}_{j}}}-{\widetilde {A}}_{\left({i}_{{k}_{j}}\right)}{x}_{{k}_{j}}\right|}^{2}\ge {\widetilde {\varepsilon }}_{{k}_{j}}{||{\widetilde {A}}_{\left({i}_{{k}_{j}}\right)}||}_{2}^{2}\right\}$ .
7: Calculation

${X}^{k+1} = {X}^{k}+{\left({\widetilde {A}}_{{j}_{{k}_{1}}}^{T}, \dots , {\widetilde {A}}_{{j}_{{k}_{d}}}^{T}\right)}^{T}diag\left\{\frac{{\widetilde {B}}_{\left({j}_{{k}_{1}}, 1\right)}-{\widetilde {A}}_{{j}_{{k}_{1}}}{x}_{1}^{\left(k\right)}}{{||{\widetilde {A}}_{{j}_{{k}_{1}}}||}_{2}^{2}}, \dots , \frac{{\widetilde {B}}_{\left({j}_{{k}_{d}}, d\right)}-{\widetilde {A}}_{{j}_{{k}_{d}}}{x}_{d}^{\left(k\right)}}{{||{\widetilde {A}}_{{j}_{{k}_{d}}}||}_{2}^{2}}\right\}$ .
8: end for

The tilde-decorated $A$ and $B$ , denoted as $\widetilde {A}$ and $\widetilde {B}$ , respectively, represent the sub-matrices formed by the rows corresponding to the index set ${\mathcal{T}}_{{k}_{j}}$ within matrices $A$ and $B$ .

Proof. According to Algorithm 4 and reference ^[39], we have

${x}_{j}^{\left(k+1\right)} = {x}_{j}^{k}+\frac{{\widetilde {B}}_{\left({\mathcal{T}}_{{k}_{j}}, j\right)}-{\widetilde {A}}_{\left({\mathcal{T}}_{{k}_{j}}\right)}{x}_{j}^{\left(k\right)}}{{||{\widetilde {A}}_{\left({\mathcal{T}}_{{k}_{j}}\right)}||}_{2}^{2}}{\widetilde {A}}_{{\mathcal{T}}_{{k}_{j}}}^{T}$

Since ${X}^{*} = {A}^{†}B$ is the least squares solution, we infer that ${x}_{j}^{*} = {A}^{†}{B}_{(:, j)}$ for $j = \mathrm{1, 2}, \dots , d$ , where ${B}_{(:, j)}$ denotes the $j$ column of matrix $B$ .

Further,

$\begin{array}{c}E{||{X}^{\left(k+1\right)}-{X}^{\mathrm{*}}||}_{F}^{2} = E\sum _{j = 1}^{t}{||{x}_{j}^{\left(k+1\right)}-{x}_{j}^{\mathrm{*}}||}_{2}^{2} = \sum _{j = 1}^{t}E{||{x}_{j}^{\left(k+1\right)}-{x}_{j}^{\mathrm{*}}||}_{2}^{2}\end{array}$

(20)

For $j = \mathrm{1, 2}, \dots , d$ , it can be deduced that

${||{x}_{j}^{\left(k+1\right)}-{x}_{j}^{\mathrm{*}}||}_{2}^{2} = {||{x}_{j}^{\left(k\right)}-{x}_{j}^{\mathrm{*}}||}_{2}^{2}-E{||{x}_{j}^{\left(k+1\right)}-{x}_{j}^{k}||}_{2}^{2}$

Then

$\begin{array}{l} E{||{x}_{j}^{\left(k+1\right)}-{x}_{j}^{k}||}_{2}^{2} = \frac{{\left|{\widetilde {B}}_{\left({\mathcal{T}}_{{k}_{j}}, j\right)}-{\widetilde {A}}_{\left({\mathcal{T}}_{{k}_{j}}\right)}{x}_{j}^{\left(k\right)}\right|}^{2}}{{||{\widetilde {A}}_{\left({\mathcal{T}}_{{k}_{j}}\right)}||}_{2}^{2}} = \underset{{\mathcal{T}}_{{k}_{{j}_{i}}}\in {\mathcal{T}}_{{k}_{j}}}{\mathrm{max}}\left\{\frac{{\left|{\widetilde {B}}_{\left({\mathcal{T}}_{{k}_{{j}_{i}}}, j\right)}-{\widetilde {A}}_{\left({\mathcal{T}}_{{k}_{{j}_{i}}}\right)}{x}_{j}^{\left(k\right)}\right|}^{2}}{{||{\widetilde {A}}_{\left({\mathcal{T}}_{{k}_{{j}_{i}}}\right)}||}_{2}^{2}}\right\} \end{array}$

The specific details of the term $E{||{x}_{j}^{\left(k+1\right)}-{x}_{j}^{k}||}_{2}^{2}$ are elaborated in the proof of Theorem 3.2 in ^[39].

Due to

$\begin{array}{l} \;\; \underset{{\mathcal{T}}_{{k}_{{j}_{i}}}\in {\mathcal{T}}_{{k}_{j}}}{\mathrm{max}}\left\{\frac{{\left|{\widetilde {B}}_{\left({\mathcal{T}}_{{k}_{{j}_{i}}}, j\right)}-{\widetilde {A}}_{\left({\mathcal{T}}_{{k}_{{j}_{i}}}\right)}{x}_{j}^{\left(k\right)}\right|}^{2}}{{||{\widetilde {A}}_{\left({\mathcal{T}}_{{k}_{{j}_{i}}}\right)}||}_{2}^{2}}\right\}\\ = \frac{\underset{{\mathcal{T}}_{{k}_{{j}_{i}}}\in {\mathcal{T}}_{{k}_{j}}}{\mathrm{max}}\left\{\frac{{\left|{\widetilde {B}}_{\left({\mathcal{T}}_{{k}_{{j}_{i}}}, j\right)}-{\widetilde {A}}_{\left({\mathcal{T}}_{{k}_{{j}_{i}}}\right)}{x}_{j}^{\left(k\right)}\right|}^{2}}{{||{\widetilde {A}}_{\left({\mathcal{T}}_{{k}_{{j}_{i}}}\right)}||}_{2}^{2}}\right\}}{{||{A}_{k}{x}_{j}^{\left(k\right)}-{\widetilde {B}}_{\left(:, j\right)}||}_{2}^{2}}{||{A}_{k}{x}_{j}^{\left(k\right)}-{\widetilde {B}}_{\left(:, j\right)}||}_{2}^{2}\\ = \frac{\underset{{\mathcal{T}}_{{k}_{{j}_{i}}}\in {\mathcal{T}}_{{k}_{j}}}{\mathrm{max}}\left\{\frac{{\left|{\widetilde {B}}_{\left({\mathcal{T}}_{{k}_{{j}_{i}}}, j\right)}-{\widetilde {A}}_{\left({\mathcal{T}}_{{k}_{{j}_{i}}}\right)}{x}_{j}^{\left(k\right)}\right|}^{2}}{{||{\widetilde {A}}_{\left({\mathcal{T}}_{{k}_{{j}_{i}}}\right)}||}_{2}^{2}}\right\}{||{A}_{k}{x}_{j}^{\left(k\right)}-{\widetilde {B}}_{\left(:, j\right)}||}_{2}^{2}}{\sum _{i = 1}^{t}{||{\widetilde {A}}_{\left({\mathcal{T}}_{{k}_{{j}_{i}}}\right)}||}_{2}^{2}\frac{{||{\widetilde {B}}_{\left({\mathcal{T}}_{{k}_{{j}_{i}}}, j\right)}-{A}_{{\mathcal{T}}_{{k}_{{j}_{i}}}}{x}_{j}^{\left(k\right)}||}_{2}^{2}}{{||{\widetilde {A}}_{\left({\mathcal{T}}_{{k}_{{j}_{i}}}\right)}||}_{2}^{2}}}\\ \ge \frac{{||{\widetilde {A}}_{\left({\mathcal{T}}_{{k}_{j}}\right)}{x}_{j}^{\left(k\right)}-{\widetilde {B}}_{\left(:, j\right)}||}_{2}^{2}}{\sum _{i = 1}^{t}{||{\widetilde {A}}_{\left({\mathcal{T}}_{{k}_{{j}_{i}}}\right)}||}_{2}^{2}}\\ \ge \frac{{\sigma }_{min}^{2}\left({\widetilde {A}}_{\left({\mathcal{T}}_{{k}_{j}}\right)}\right)}{{||{\widetilde {A}}_{\left({\mathcal{T}}_{{k}_{j}}\right)}||}_{2}^{2}}{||{x}_{j}^{\left(k\right)}-{x}_{j}^{*}||}_{2}^{2} \end{array}$

(21)

In this context, ${\widetilde {B}}_{\left(:, j\right)}$ denotes the $j$ column of the matrix $\widetilde {B}$ .

$\begin{array}{r}E{||{X}_{}^{\left(k+1\right)}-{X}_{j}^{k}||}_{2}^{2} = {||{x}_{j}^{\left(k\right)}-{x}_{j}^{\mathrm{*}}||}_{2}^{2}-E{||{x}_{j}^{\left(k+1\right)}-{x}_{j}^{k}||}_{2}^{2}\\ \le \left(1-\frac{{\sigma }_{min}^{2}\left({\widetilde {A}}_{\left({\mathcal{T}}_{{k}_{j}}\right)}\right)}{{||{\widetilde {A}}_{\left({\mathcal{T}}_{{k}_{j}}\right)}||}_{2}^{2}}\right){||{x}_{j}^{\left(k\right)}-{x}_{j}^{\mathrm{*}}||}_{2}^{2}\end{array}$

(22)

Thus, by combining Eqs (20) and (22), we can derive inequality (19).

The proof of Theorem 3 reveals that the convergence rate of Algorithm 4 is related to the sub-matrix $\widetilde {A}$ sampled during each leveraged iteration.

5. Numerical experiment

In this section, we evaluate the effectiveness of large-scale image restoration equations using several algorithms through comparative examples: the GBK method ^[29], the semi-stochastic block Kaczmarz method with simple random sampling for single righthand sides ^[40], the LFGBK method for single righthand sides, the semi-stochastic block Kaczmarz method with simple random sampling for multiple righthand sides ^[40], and the LFGBK method for multiple righthand sides. All experiments are implemented in MATLAB, with metrics such as peak signal-to-noise ratio (PSNR), structural similarity index (SSIM), mean squared error (MSE), and computational time (CPU in seconds) reported. Following the approach of ^[27], the PSNR, SSIM, MSE, and CPU metrics reflect the average iterations and computational time needed for 50 repeated calculations. During all computations, we initialize the matrix $x = zeros(n, m)$ and set the righthand side $B = AX$ , where $X$ represents images from MATLAB's Cameraman, Phantom, and Mri datasets, each sized $100\times 100$ . Matrix $A$ is constructed with elements corresponding to the product of the number of X-ray beams (4), the range of scanning angles t (0° to 179°), and the image pixels. The stopping criterion is set as $RSE = \frac{{||{X}_{k}-{X}_{*}||}_{2}^{2}}{{||{X}_{*}||}_{2}^{2}}\le {10}^{-6}$ . In practice, the condition $O({n}^{2}/\delta {\theta }^{2})$ is quite stringent, but in many real-world computations, a sketching factor of $d = {n}^{2}$ yields satisfactory results. We consider matrices $A\in {\mathbb{R}}^{(4*180*n, m)}$ of two types: type a, generated by radon(), and type b, generated by randn().

Based on the numerical results from Tables 1 and 2, when variable A is generated using radon, we can draw the following conclusions: ⅰ) The GBK method, single righthand side SRBK (single RHS SRBK) method, single RHS LFGBK method, multiple righthand side SRBK (multiple RHS SRBK) method, and multiple RHS LFGBK method all demonstrate effectiveness in solving equation systems for image restoration. ⅱ) The GBK method, single RHS SRBK method, single RHS LFGBK method, multiple RHS SRBK method, and multiple RHS LFGBK method show similar restoration quality; however, the single RHS LFGBK and multiple RHS LFGBK methods significantly outperform the GBK, Single RHS SRBK, and Multiple RHS SRBK methods in terms of time efficiency. ⅲ) The LFGBK method exhibits superior acceleration effects at η = 0.9 compared to η = 0.8. ⅳ) The MLFGBK method displays more pronounced acceleration effects at η = 0.9 than at η = 0.8.

Figure 1. Reconstructed images when A is generated by radon().

DownLoad: Full-Size Img PowerPoint

Table 1. Numerical experimental results of the GBK method, SSRBK method, and SLFGBK method when

$A$ is generated by radon().

		GBK		SSRBK		SLFGBK
data	Metrics	$\eta =0.8$	$\eta =0.9$	$\eta =0.8$	$\eta =0.9$	$\eta =0.8$	$\eta =0.9$
Cameraman	PSNR	19.595	17.996	19.076	17.640	17.595	17.519
	SSIM	0.999	0.998	0.999	0.998	0.999	0.999
	MSE	0.011	0.015	0.012	0.017	0.017	0.017
	CPU	234.78	335.37	55.00	94.755	11.581	7.543
Phantom	PSNR	73.636	72.694	73.639	73.655	72.377	72.430
	SSIM	1.000	1.000	1.000	1.000	1.000	1.000
	MSE	0	0	0	0	0	0
	CPU	266.67	279.43	56.186	75.168	12.496	8.556
Mri	PSNR	29.319	28.936	28.068	29.023	28.057	28.112
	SSIM	0.969	0.975	0.924	0.919	0.962	0.978
	MSE	0.001	0.001	0.001	0.001	0.001	0.001
	CPU	259.86	375.48	60.963	162.06	19.340	15.563

| Show Table

DownLoad: CSV

Table 2. Numerical experimental results of the SSRBK method, MSRBK method, and MLFGBK method when

$A$ is generated by radon().

		SLFGBK		MSRBK		MLFGBK
data	Metrics	$\eta =0.8$	$\eta =0.9$	$\eta =0.8$	$\eta =0.9$	$\eta =0.8$	$\eta =0.9$
Cameraman	PSNR	17.595	17.519	17.611	17.732	17.894	17.876
	SSIM	0.999	0.999	0.999	0.999	0.999	0.999
	MSE	0.017	0.017	0.017	0.016	0.016	0.016
	CPU	11.581	7.543	18.027	14.207	3.196	2.880
Phantom	PSNR	72.377	72.430	72.492	72.532	72.486	72.760
	SSIM	1.000	1.000	1.000	1.0000	1.000	1.000
	MSE	0	0	0	0	0	0
	CPU	12.496	8.556	43.106	38.075	20.874	20.023
Mri	PSNR	28.057	28.112	28.081	28.234	28.161	28.506
	SSIM	0.962	0.978	0.901	0.899	0.895	0.891
	MSE	0.001	0.001	0.001	0.001	0.001	0.001
	CPU	19.340	15.563	42.535	37.575	18.237	17.539

| Show Table

DownLoad: CSV

Based on the numerical results from Tables 3 and 4, when variable A is generated using randn, we can draw the following conclusions: ⅰ) The GBK method, single RHS SRBK method, single RHS LFGBK method, multiple RHS SRBK method, and multiple RHS LFGBK method all demonstrate effectiveness in solving equation systems for image restoration. ⅱ) The GBK method, single RHS SRBK method, single RHS LFGBK method, multiple RHS SRBK method, and multiple RHS LFGBK method show similar restoration quality; however, the Single RHS LFGBK and Multiple RHS LFGBK methods significantly outperform the GBK, single RHS SRBK, and multiple RHS SRBK methods in terms of time efficiency. ⅲ) The LFGBK method exhibits superior acceleration effects at η = 0.9 compared to η = 0.8. ⅳ) The MLFGBK method displays more pronounced acceleration effects at η = 0.9 than at η = 0.8.

Figure 2. Reconstructed images when A is generated by randn().

DownLoad: Full-Size Img PowerPoint

Table 3. Numerical experimental results of the GBK method, SSRBK method, and SLFGBK method when

$A$ is generated by randn().

		GBK		SSRBK		SLFGBK
data	Metrics	$\eta =0.8$	$\eta =0.9$	$\eta =0.8$	$\eta =0.9$	$\eta =0.8$	$\eta =0.9$
Cameraman	PSNR	20.122	19.195	18.419	17.594	17.519	17.552
	SSIM	0.999	0.998	0.998	0.998	0.999	0.999
	MSE	0.009	0.012	0.014	0.017	0.017	0.017
	CPU	3.651	6.916	1.433	3.596	1.698	0.723
Phantom	PSNR	73.904	72.475	74.040	73.704	72.619	72.464
	SSIM	1.000	1.000	1.000	1.000	1.000	1.000
	MSE	0	0	0	0	0	0
	CPU	68.434	129.151	19.739	35.465	7.776	6.878
Mri	PSNR	30.615	29.072	29.289	28.899	28.080	28.169
	SSIM	0.912	0.900	0.893	0.884	0.892	0.905
	MSE	0	0.001	0.001	0.001	0.001	0.001
	CPU	63.758	122.983	28.132	31.338	9.451	8.641

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DownLoad: CSV

Table 4. Numerical experimental results of the SSRBK method, MSRBK method, and MLFGBK method when

$A$ is generated by randn().

		SLFGBK		MSRBK		MLFGBK
data	Metrics	$\eta =0.8$	$\eta =0.9$	$\eta =0.8$	$\eta =0.9$	$\eta =0.8$	$\eta =0.9$
Cameraman	PSNR	17.519	17.552	17.748	17.762	17.860	17.805
	SSIM	0.999	0.999	0.999	0.999	0.999	0.999
	MSE	0.017	0.017	0.016	0.016	0.016	0.016
	CPU	1.698	0.723	14.582	11.879	2.005	1.672
Phantom	PSNR	72.619	72.464	72.398	72.629	72.695	72.353
	SSIM	1.000	1.000	1.000	1.000	1.000	1.000
	MSE	0	0	0	0	0	0
	CPU	7.776	6.878	47.513	45.431	13.824	13.757
Mri	PSNR	28.080	28.169	28.241	28.093	28.241	28.328
	SSIM	0.892	0.905	0.904	0.904	0.902	0.897
	MSE	0.001	0.001	0.001	0.001	0.001	0.001
	CPU	9.451	8.641	54.802	38.773	11.533	10.332

| Show Table

DownLoad: CSV

6. Conclusions

This study aims to explore a new image reconstruction algorithm, the LFGBK method, which utilizes a greedy strategy. The core of this method lies in transforming the image reconstruction problem into solving a linear system problem with multiple righthand sides. In traditional Kaczmarz algorithms, the iterative process of gradually approaching the solution is often inefficient and susceptible to the influence of initial value selection. However, the LFGBK method introduces leverage score sampling and extends from solving single righthand side linear equations to solving multiple righthand side linear equations, which is crucial for handling large-scale problems due to the time-consuming and resource-intensive nature of pseudo-inverse computation on large matrices. By avoiding pseudo-inverse calculation, the LFGBK method significantly improves the algorithm's computational efficiency. To validate the effectiveness of the proposed algorithm, this study conducts in-depth research through theoretical analysis and simulation experiments. The theoretical analysis confirms the convergence of the LFGBK method, ensuring the reliability and stability of the algorithm. The simulation experiments, compared with traditional Filtered Back Projection (FBP) methods, demonstrate the advantages of LFGBK in terms of image reconstruction quality. The experimental results show that the LFGBK method significantly preserves image details and reduces noise, proving its practicality and superiority in image reconstruction tasks.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The authors declare there is no conflict of interest.

References

[1]	Zhang J, Shi H, Wang Y, et al. (2018) Effects of limit‐feeding diets with different forage‐to‐concentrate ratios on nutrient intake, rumination, ruminal fermentation, digestibility, blood parameters and growth in Holstein heifers. Anim Sci J 89: 527-536.
[2]	Celi P, Cowieson AJ, Fru-Nji F, et al. (2017) Gastrointestinal functionality in animal nutrition and health: new opportunities for sustainable animal production. Anim Feed Sci Technol 234: 88-100.
[3]	Leeson S, Atteh JO (1995) Utilization of fats and fatty acids by turkey poults. Poult Sci 74: 2003-2010.
[4]	Doreau M, Chilliard Y (1997) Digestion and metabolism of dietary fat in farm animals. Br J Nutr 78: S15-S35.
[5]	Lebedev SV, Gavrish IA, Shejda EV, et al. (2019) Effect of various fats on digestibility of nutrients in diet of calves Conference on Innovations in Agricultural and Rural development. IOP Conf Series: Earth Environ Sci 341: 012066.
[6]	Keshri A, Roy D, Kumar V, et al. (2019) Impact of different chromium sources on physiological responses, blood biochemicals and endocrine status of heat stress in dairy calves. Biol Rhythm Res 1-12.
[7]	Wysocka D, Snarska A, Sobiech P (2019) Copper-an essential micronutrient for calves and adult cattle. J Elem 24:101-110.
[8]	Chen G, Gao Z, Chu W, et al. (2018) Effects of chromium picolinate on fat deposition, activity and genetic expression of lipid metabolism-related enzymes in 21 day old Ross broilers. Asian-Austral J Anim Sci 31: 569-575.
[9]	Zhou B, Wang H, Luo G, et al. (2013) Effect of dietary yeast chromium and L-carnitine on lipid metabolism of sheep. Biol Trace Elem Res 155: 221-227.
[10]	Mertz W (1993) Chromium in human nutrition: a review. J Nutr 123: 626-633.
[11]	Dalólio FS, Albino LFT, Silva JN, et al. (2018) Dietary chromium supplementation for heat-stressed broilers. Worlds Poult Sci J 74: 101-116.
[12]	Vincent JB (2000) Quest for the molecular mechanism of chromium action and its relationship to diabetes. Nutr Rev 58: 67-72.
[13]	Mikulski D, Jankowski J, Zduńczyk Z, et al. (2009) The effect of selenium source on performance, carcass traits, oxidative status of the organism, and meat quality of turkeys. J Anim Feed Sci 18: 518-530.
[14]	Wang MQ, Xu ZR, Zha LY, et al. (2007) Effects of chromium nanocomposite supplementation on blood metabolites, endocrine parameters and immune traits in finishing pigs. Anim Feed Sci Technol 139: 69-80.
[15]	Mooney KW, Cromwell GL (1995) Effects of dietary chromium picolinate supplementation on growth, carcass characteristics, and accretion rates of carcass tissues in growing-finishing swine. J Anim Sci 73:3351-3157.
[16]	Mooney KW, Cromwell GL (1997) Efficacy of chromium picolinate and chromium chloride as potential carcass modifiers in swine. J Anim Sci 75: 2661-2671.
[17]	Jamilian M., Modarres SZ, Siavashani MA, et al. (2018) The influences of chromium supplementation on glycemic control, markers of cardio-metabolic risk, and oxidative stress in infertile polycystic ovary syndrome women candidate for in vitro fertilization: a randomized, double-blind, placebo-controlled trial. Biol Trace Elem Res 185: 48-55.
[18]	Lebedev SV, Kvan OV, Gubajdullina IZ, et al. (2018) Effect of chromium nanoparticles on digestive enzymes activity and morphological and biochemical parameters of calf blood. Anim Husb Fodder Prod 101: 136-142.
[19]	Kośla T, Lasocka I, Kołnierzak M (2019) Chromium Mammals and Birds as Bioindicators of Trace Element Contaminations in Terrestrial Environments. In: Kalisinska E (Ed), Springer, Cham, 57-124.
[20]	Kozloski GV, de Moraes Flores EM, Martins AF (1998) Use of chromium oxide in digestibility studies: variations of the results as a function of the measurement method. J Sci Food Agric 76: 373-376.
[21]	Salnikow K, Zhitkovich A (2008) Genetic and epigenetic mechanisms in metal carcinogenesis and cocarcinogenesis: nickel, arsenic, and chromium. Chem Res Toxicol 21: 28-44.
[22]	Mertz W, Roginski E (1963) The effect of trivalent chromium on galactose entryin rat epididymal fat tissue. J Biol Chem 238: 868-872.
[23]	Stearns DM, Wise Sr JP, Patierno SR, et al. (1995) Chromium (III) picolinate produces chromosome damage in Chinese hamster ovary cells. FASEB J 9: 1643-1648.
[24]	Laschinsky N, Kottwitz K, Freund B, et al. (2012) Bioavailability of chromium (III)-supplements in rats and humans. Biometals 25: 1051-1060.
[25]	Ban C, Park SJ, Lim S, et al. (2015) Improving flavonoid bioaccessibility using an edible oil-based lipid nanoparticle for oral delivery. J Agricl Food Chem 63: 5266-5272.
[26]	Zha LY, Wang MQ, Xu ZR, et al. (2007) Efficacy of chromium (III) supplementation on growth, body composition, serum parameters, and tissue chromium in rats. Biol Trace Elem Res 119: 42-50.
[27]	Seryh MM (1959) Study of the exocrine secretion of the pancreas in calves in relation to age and feeding. Izv Timiryazevsk S-kh Akad 6: 161-172.
[28]	Hashemi SM, Loh TC, Foo HL, et al. (2014) Small intestine morphology, growth performance and nutrient digestibility of young broilers affected by different levels of dietary putrescine. J Anim Poult Sci 3: 95-104.
[29]	Association of Official Analytical Chemist (AOAC) (1980) Official method of analysis of AOAC International. 16th ed. Arlington: Association of Official Analytical Chemist, Washington DC, USA. Available from: https://archive.org/stream/gov.law.aoac.methods.1980/aoac.methods.1980_djvu.txt.
[30]	Smith BW, Roe JH (1949) A photometric method for the determination of α-amylase in blood and urine, with use of the starch-iodine color. J Biol Chem 179: 53-59.
[31]	Reimerdes EH, Klostermeyer H (1976) Determination of proteolytic activities on casein substrates. Methods Enzymol 45: 26-28.
[32]	Jenkins TC (1993) Lipid metabolism in the rumen. J Dairy Sci 76: 3851-3863.
[33]	Pechova A, Pavlata L (2007) Chromium as an essential nutrient: a review. Vet Med 52: 1-18.
[34]	Parra O, Ojeda A, Combellas J, et al. (1999) Blood metabolites and their relationship with production variables in dual-purpose cows in Venezuela. Prev Vet Med 38: 133-145.
[35]	Iskra R, Vlizlo V, Fedoruk R (2016) Role of chromium (III) in the nutrition of pigs and cattle. Agric Sci Pract 3: 56-62.
[36]	Saeed AA, Sandhu MA, Khilji MS, et al. (2017) Effects of dietary chromium supplementation on muscle and bone mineral interaction in broiler chicken. J Trace Elem Med Biol 42: 25-29.
[37]	Di Bona KR, Love S, Rhodes NR, et al. (2011) Chromium is not an essential trace element for mammals: effects of a "low-chromium" diet. JBIC: J Biol Inorg Chem 16:381-390.
[38]	Rao SR, Prakash B, Raju MV, et al. (2016) Effect of supplement-ing organic forms of zinc, selenium and chromium on performance, anti-oxidant and immune responses in broiler chicken reared in tropical summer. Biol Trace Elem Res 172: 511-520.
[39]	Piotrowska A, Pilch W, Tota L, et al. (2018) Biological significance of chromium III for the human organism. Med Pr 69: 211-223.
[40]	White PE, Vincent JB (2019) Systematic review of the effects of chromium (III) on chickens. Biol Trace Elem Res 188: 99-126.
[41]	Janes AN, Weekes TEC, Armstrong DG (1985) Carbohydrase activity in the pancreatic tissue and small intestine mucosa of sheep fed dried-grass or ground maizebased diets. J Agric Sci 104: 435-443.
[42]	Ścibior A, Zaporowska H, Wolińska A, et al. (2010) Antioxidant enzyme activity and lipid peroxidation in the blood of rats co-treated with vanadium (V⁺⁵) and chromium (Cr⁺³). Cell Biol Toxicol 26: 509-526.
[43]	Evans E, Buchanan-Smith JG, Macleod GK (1975) Postprandial patterns of plasma glucose, insulin and volatile fatty acids in ruminants fed low-and high-roughage diets. J Anim Sci 41:1474-1479.
[44]	Lien TF, Horng YM, Yang KH (1999) Performance, serum characteristics, carcase traits and lipid metabolism of broilers as affected by supplement of chromium picolinate. Br Poult Sci 40: 357-363.
[45]	Weiser M (1984) Calcium. In: Solomons NW, Rosenberg IH. (Eds), Absorption and Malabsorbtion of Mineral Nutrients. New York: Alan R. Liss Inc., 39-42.
[46]	Zemel MB, Shi H, Greer B, et al. (2000) Regulation of adiposity by dietary calcium. FASEB J 14: 1132-1138.
[47]	Ubarretxena-Belandia I, Boots JWP, Verhei HM, et al. (1998) Role of the cofactor calcium in the activation of outer membrane phospholipase A. Biochemistry 37:16011-16018.
[48]	Hayirli A, Bremmer DR, Bertics SJ, et al. (2001) Effect of chromium supplementation on production and metabolic parameters in periparturient dairy cows. J Dairy Sci 84: 1218-1230.
[49]	Pinchasov Y, Nir I (1992) Effect of dietary polyunsaturated fatty acid concentration on performance, fat deposition, and carcass fatty acid composition in broiler chickens. Poult Sci 71: 1504-1512.
[50]	Ferramosca A, Conte A, Burri L, et al. (2012) A krill oil supplemented diet suppresses hepatic steatosis in high-fat fed rats. PloS one 7: e38797.
[51]	Walker CG, Loos RJF, Olson AD, et al. (2011) Genetic predisposition influences plasma lipids of participants on habitual diet, but not the response to reductions in dietary intake of saturated fatty acids. Atherosclerosis 215: 421-427.
[52]	Jami E, Israel A, Kotser A, et al. (2013) Exploring the bovine rumen bacterial community from birth to adulthood. ISME J 7:1069-1079.
[53]	Malmuthuge N (2017) Understanding host-microbial interactions in rumen: searching the best opportunity for microbiota manipulation. J Anim Sci Biotechnol 8: 8.
[54]	Zened A, Troegeler‐Meynadier A, Nicot MC, et al. (2011) Starch and oil in the donor cow diet and starch in substrate differently affect the in vitro ruminal biohydrogenation of linoleic and linolenic acids. J Dairy Sci 64:5634-5645.

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