HDTV: a high-order directional total variation reconstruction algorithm from sparse and limited-angle data in electron paramagnetic resonance imaging

Yanjun Zhang; Peng Liu; Chenyun Fang; Yarui Xi; Zhiwei Qiao

doi:10.21037/qims-2025-8

Original Article

HDTV: a high-order directional total variation reconstruction algorithm from sparse and limited-angle data in electron paramagnetic resonance imaging

Yanjun Zhang¹ , Peng Liu^1,2 , Chenyun Fang¹ , Yarui Xi^3,4 , Zhiwei Qiao¹

¹School of Computer and Information Technology, Shanxi University, Taiyuan, China; ²Department of Big Data and Intelligent Engineering, Shanxi Institute of Technology, Yangquan, China; ³Key Laboratory of Optoelectronic Technology and Systems, Chongqing University, Chongqing, China; ⁴The Engineering Research Center of Industrial Computed Tomography Nondestructive Testing, Chongqing University, Chongqing, China

Contributions: (I) Conception and design: Y Zhang; (II) Administrative support: Z Qiao; (III) Provision of study materials or patients: Z Qiao; (IV) Collection and assembly of data: Y Zhang; (V) Data analysis and interpretation: Y Zhang; (VI) Manuscript writing: All authors; (VII) Final approval of manuscript: All authors.

Correspondence to: Zhiwei Qiao, PhD. School of Computer and Information Technology, Shanxi University, No. 92, Wucheng Road, Taiyuan 030006, China. Email: zqiao@sxu.edu.cn.

Background: Electron paramagnetic resonance imaging (EPRI)-based oxygen imaging technology enables adaptive radiation therapy, thereby improving tumor control rates. However, the long scanning time limits the development of EPRI. In this study, we endeavored to reduce the scanning time. The general method is sparse reconstruction; if it can be collected in limited-angle range under sparse conditions, the scanning time can be further shortened.

Methods: Based on the abovementioned theory, we performed sparse acquisition based on limited-angle range to further accelerate scanning. Moreover, high-order constraints were introduced into the directional total variation (DTV) algorithm to suppress staircase artifacts, and we proposed a high-order DTV (HDTV) model and derived the Chambolle-Pock (CP) solving algorithm. We aimed to realize three-dimensional (3D) sparse and limited-angle EPRI with high precision and thus accelerate the scanning time.

Results: The correctness of the HDTV-CP algorithm was validated on simulation data and the limited-angle and sparse reconstruction ability was investigated using real data. The results indicate that the HDTV method effectively suppresses limited-angle artifacts, sparse artifacts, and staircase artifacts while preserving the edge and texture features. Our method showed significant improvements compared to the classic TV method. The normalized root mean square error (nRMSE) decreased from 0.34 to 0.16, and the Pearson correlation coefficient (PCC) increased from 0.93 to 0.98 based on 50 views within half the angular range.

Conclusions: For the first time, we combined the limited-angle and sparse problems. The HDTV method may realize 16 times acceleration while ensuring the imaging quality in certain situations. The findings of this study can also extend to the field of limited-angle computed tomography (CT) image reconstruction.

Keywords: Electron paramagnetic resonance imaging (EPRI); sparse reconstruction; limited-angle reconstruction; directional total variation (DTV); high-order

Submitted Jan 02, 2025. Accepted for publication Jun 10, 2025. Published online Aug 15, 2025.

doi: 10.21037/qims-2025-8

Introduction

As the core treatment for tumors, radiation therapy serves a crucial role (1). The oxygen concentration in tumor tissue has been shown to be a key parameter in radiation therapy, with low oxygen areas exhibiting significant resistance to radiation, and high oxygen areas being more sensitive (2,3). Electron paramagnetic resonance imaging (EPRI)-based oxygen imaging technology enables adaptive radiation therapy, thereby improving tumor control rates (4,5). Currently, the long scanning time limits the development of EPRI. For example, a scanning process may take about 7 minutes, during which time, a total of 828 projections of different angles are acquired. To improve the signal-to-noise ratio, the projections of each angle are obtained by repeating acquisition multiple times and averaging the values, which contributes to the overall imaging time being excessively long. The general method of solving this problem of scanning duration is sparse reconstruction. If it can be collected in limited-angle range under sparse, the scanning time can be further shortened. Based on this theory, this work performs sparse acquisition based on limited-angle range to further accelerate scanning. For three-dimensional (3D) pulsed EPRI, the commonly used commercial image reconstruction algorithm is the 3D filtered back projection (FBP) (6) algorithm. However, FBP is not capable of sparse reconstruction and limited-angle reconstruction since it causes severe streaking artifacts in the reconstructed image, resulting in degradation of image quality. Therefore, it is necessary to explore new reconstruction algorithms to realize high-precision sparse limited angle reconstruction.

It is widely acknowledged that limited-angle and sparse reconstruction problems are challenging as they are highly ill-conditioned. Iterative algorithms (7,8) provide effective solutions for these two types of reconstruction problems. Total variation (TV) regularization models (9,10) are extensively adopted in various inverse problem solutions due to their great constraint ability. Nevertheless, TV regularization is an isotropic operator (11). This characteristic results in a decrease on the edge retention capability of non-fuzzy areas. Thus, various algorithms have been developed to optimize the performance of TV models (12-15). For sparse reconstruction, in 2011, Tian et al. proposed edge-preserving TV (EPTV) by introducing penalty weights to enhance the edge features of the image (16); in 2012, Liu et al. proposed adaptive-weighted TV (awTV) by assigning different weights to the gradients in different directions (17). For limited-angle reconstruction (18-20), in 2013, Chen et al. presented an upgraded version of anisotropic TV (ATV) by imposing different weights according to the angle values to balance the TV minimization and data fidelity constraints (21). It is difficult to utilize angular information to assist limited-angle EPRI because the projection angle is determined by both the polar angle $(θ)$ and azimuthal angle $(φ)$ . Many efforts have been made to suppress the artifacts caused by the limited-angle reconstruction (22-25).

In 2021, Zhang et al. presented the directional total variation (DTV) (23) method, which exploits the spatial relationship between image pixels to constrain the horizontal and vertical directions of numerical phantoms with limited-angle range respectively, and effectively increases the image edge clarity. The DTV method demonstrates good performance in limited-angle reconstruction since it emphasizes the TV constraints along different directions. In the same year, Chen et al. investigated the performance of DTV in dual-energy computed tomography (CT) (26). In 2022, Zhang et al. explored the reconstruction performance of DTV in the orthogonal arc scanning range (24). However, when DTV is applied to real data in EPRI (27), the reconstructed image shows significant staircase effects in the edge regions. This is because the EPRI images obtained are not piecewise-constant in real data research (28). It brings challenges for sparse and limited-angle EPRI. Against this background, the high-order TV method shows unique merits in suppressing the staircase artifacts (29,30). In 2010, Yang et al. showed that when the region of interest (ROI) is piecewise polynomial, high-order TV minimization methods can effectively cope with this non-constant property and provide accurate solutions (31). In 2012, Hu et al. used high-degree derivatives to reconstruct images from standardized pictures such as Lena with white noise (29). In 2022, Liu et al. applied a higher-order method to remove noise and artifacts in seismic data images (32). In 2023, Xi et al. applied high-order method to CT images (33). They all demonstrated the advantages of higher-order methods in mitigating staircase artifacts. In the next sections of this paper, we call the sparse reconstruction of the limited-angle EPR the sparse-limited-angle EPRI.

The data acquisition for EPRI is regularly conducted at the full angle $(θ \in [0, π / 2], φ \in [0, 2 π])$ , and algorithms have been developed to reconstruct EPRI images based on the full-angular projection data (3,34-36). Recently, there has been increasing interest in EPRI for data acquisition within a limited-angle range (37) because the scanning time can be greatly reduced when data are collected within the limited-angle range. In this study, to significantly improve the imaging efficiency and shorten the scanning time, we perform sparse acquisition based on a limited-angle range. Moreover, we present a high-order directional total variation (HDTV) model for sparse-limited-angle EPRI. We aimed to explore the minimal angular range and the minimal number of sparse-view projections within the minimal angular range to realize EPRI with high precision and thus accelerate the scans. The contributions of this paper include:

To our knowledge, we are the first to combine the limited-angle and sparse problems to further accelerate scanning. The HDTV method may realize 16 times acceleration while ensuring the imaging quality.
The image reconstruction problem is characterized as an optimization program constrained by HDTV, and we derive an algorithm based on the Chambolle-Pock (CP) (38) algorithm framework to solve the HDTV minimization model, which we call the HDTV-CP algorithm. Specifically, the HDTV-CP algorithm introduces high-order constraint information into the DTV algorithm to suppress staircase artifacts, and to help maintain the texture characteristics and edge information of the image.
Compared with existing sparse reconstruction and limited-angle reconstruction methods [TV (39), DTV (23)], our proposed HDTV method effectively suppresses limited-angle artifacts, sparse artifacts, and staircase artifacts while preserving the edge and texture features in sparse-limited-angle EPRI.

We present our work in the following structure: in “Methods” section, we first describe the scanning configuration and data acquisition of sparse-limited-angle EPRI. Then, the HDTV-CP algorithm is derived. Next, we discuss the reconstruction parameters. In “Results” section, firstly, the correctness of the HDTV-CP algorithm is verified, subsequently, the performance of the algorithm is tested on real data. “Discussion” section discusses the TV limit and analyzes the advantages and limitations of our approach.

“Conclusions” section summarizes the article.

Methods

This section begins by detailing the scanning configuration and data acquisition of the sparse-limited-angle EPRI. Subsequently, we present the mathematical formulation of the HDTV model, and develop the corresponding HDTV-CP reconstruction algorithm. Finally, some key reconstruction parameters are introduced.

Scanning configuration and data acquisition

We performed sparse-limited-angle EPRI by using the JIVA-25 EPR oxygen imager from O2M Technologies (Chicago, IL, USA). The projection point of EPRI presents a hemispherical structure. As shown in Figure 1, the coordinates (x, y, z) of projection point P are defined by the polar coordinates $(θ, φ)$ , where $θ \in [0, π / 2]$ and $φ \in [0, 2 π]$ .

Figure 1 The projection distribution for EPR oxygen imager. The coordinates (x, y, z) of projection point P are defined by the polar coordinates

(θ, φ)

. EPR, electron paramagnetic resonance.

The full-angle is relative and refers to the number of projection points required for complete reconstruction of EPR images. The number of projection points of full-angle was set at 828 in our study. We formed uniformly distributed projection points using a maximally spaced projection sequencing (MSPS) technique (40). Sparse-angle EPRI is the uniform reduction of projection points at complete polar and azimuthal angles and ensures that a constant stereo angle relationship is maintained between the projection angles. The effectiveness of sparse reconstruction has been shown in CT, magnetic resonance imaging (MRI), and EPRI. Limited-angle EPRI reduces the angular range of polar angle or azimuthal angle; sparse-limited-angle EPRI is the uniform reduction of projection points while reducing the polar angle or azimuthal angle. This paper takes the angular range of reducing the azimuthal angle as an example. The illustration of the full-angle, sparse-angle, limited-angle, and sparse-limited-angle is illustrated in Figure 2.

Figure 2 EPRI projection point diagram. The illustration of the full-angle (A), sparse-angle (B), limited-angle (C) and sparse-limited-angle (D). The polar angle

(θ)

and azimuthal angle

(φ)

of (A-D) are: (A,B):

θ \in [0, π / 2], φ \in [0, 2 π]

; (C,D):

θ \in [0, π / 2]

,

φ \in [0, π]

. It is noted that the Figure 2 is only used for showing the relationship between them, and it do not represent the actual number of projection points of the EPRI. EPRI, electron paramagnetic resonance imaging.

Optimization model (HDTV)

For discrete EPR image data $u \in R^{N_{x} \cdot N_{y} \cdot N_{z}}$ , our proposed HDTV minimization model is formulated as:

$\begin{array}{l} u^{*} = \arg \min_{u} \frac{1}{2} {‖ g - A u ‖}_{2}^{2} \\ s .t . {‖ u ‖}_{H T V_{x}} \leq t_{x}, {‖ u ‖}_{H T V_{y}} \leq t_{y}, {‖ u ‖}_{H T V_{z}} \leq t_{z} \end{array}$ [1]

${‖ u ‖}_{H T V_{x}} = {‖ (| D_{x} u |) ‖}_{1}$ [2]

${‖ u ‖}_{H T V_{y}} = {‖ (| D_{y} u |) ‖}_{1}$ [3]

${‖ u ‖}_{H T V_{z}} = {‖ (| D_{z} u |) ‖}_{1}$ [4]

where, u^* denotes the final solution; g denotes actual measurements; model data is represented by Au; and the basic unit of the discrete image u is defined as voxel. A represents system matrix, in which A_i_,_j represents the intersection area between the j-th voxel and the i-th measurement; ${‖ • ‖}_{2}^{2}$ denotes square of l₂ norm. For different limited-angle ranges, the values of t_x, t_y, and t_z need to be adjusted accordingly to meet the demand for the reconstruction quality of sparse-limited-angle EPRI. The high-order TV (HTV) norm is defined as Eq. [2] to Eq. [4], where, D_xu, D_yu, and D_zu denote the second-order discrete gradient, and it can be expressed as:

${(D_{x} u)}_{i, j, k} = u_{i - 1, j, k} + u_{i + 1, j, k} - 2 u_{i, j, k}$ [5]

${(D_{y} u)}_{i, j, k} = u_{i, j - 1, k} + u_{i, j + 1, k} - 2 u_{i, j, k}$ [6]

${(D_{z} u)}_{i, j, k} = u_{i, j, k - 1} + u_{i, j, k + 1} - 2 u_{i, j, k}$ [7]

where, D_xu is set to 0 when i=1 or i=N_x, D_yu is set to 0 when j=1 or j=N_y, D_zu is set to 0 when k=1 or k=N_z. N_x, N_y, and N_z denote voxel amounts along the x-axis, y-axis, and z-axis, respectively. In 3D sparse-limited-angle EPR image reconstruction, the HTV constraint may play a crucial role in effectively narrowing the range of feasible solution sets, in view of the limitation of the angular range and the sparsity of the projection points.

Solving algorithm (HDTV-CP)

To balance the data fidelity term and the high-order TV constraint, this study introduces $λ$ , $v_{x}$ , $v_{y}$ , and $v_{z}$ , as shown in Table 1. Based on this, we proposed an improved HDTV optimization model.

$\begin{array}{l} u^{*} = \arg \min_{u} \frac{λ}{2} {‖ g - A u ‖}_{2}^{2} \\ s .t . v_{x} {‖ u ‖}_{H T V_{x}} \leq v_{x} t_{x}, v_{y} {‖ u ‖}_{H T V_{y}} \leq v_{y} t_{y}, v_{z} {‖ u ‖}_{H T V_{z}} \leq v_{z} t_{z} \end{array}$ [8]

Table 1

Implementation of HDTV-CP algorithm pseudocode [see (41) for calculations of $P r o j e c t O n t o l_{1} B a l l$ ]

Input: g,

A

,

λ

,

b

,

t_{1}

,

t_{2}

,

t_{3}

1:

v_{x} = b {‖ A ‖}_{S V} / {‖ D_{x} ‖}_{S V}; v_{y} = b {‖ A ‖}_{S V} / {‖ D_{y} ‖}_{S V}; v_{z} = b {‖ A ‖}_{S V} / {‖ D_{z} ‖}_{S V}

;

L = {‖ A; v_{x} D_{x}; v_{y} D_{y}; v_{z} D_{z} ‖}_{S V}

;

t_{x} = t_{1} D_{x}

;

t_{y} = t_{2} D_{y}

;

t_{z} = t_{3} D_{z}

2:

σ = 1 / L

;

τ = 1 / L

;

θ = 1

;

n = 0

3: Initialize:

P^{(0)}

,

q_{1}^{(0)}

,

q_{2}^{(0)}

,

q_{3}^{(0)}

,

u^{(0)}

and

{\bar{u}}^{(0)}

to zeros

4: repeat

5:

P^{(n + 1)} = {P^{(n)} + σ [A {\bar{u}}^{(n)} - g]} / (1 + σ / λ)

6:

a_{1} = q_{1}^{(n)} + σ v_{x} D_{x} {\bar{u}}^{(n)}

s_{1} = P r o j e c t O n t o l_{1} B a l l_{v_{x} t_{x}} ({| a_{1} |}_{m a g} / σ)

q_{1}^{(n + 1)} = a_{1} (1_{I} - σ s_{1} / {| a_{1} |}_{m a g})

7:

a_{2} = q_{2}^{(n)} + σ v_{y} D_{y} {\bar{u}}^{(n)}

s_{2} = P r o j e c t O n t o l_{1} B a l l_{v_{y} t_{y}} ({| a_{2} |}_{m a g} / σ)

q_{2}^{(n + 1)} = a_{2} (1_{I} - σ s_{2} / {| a_{2} |}_{m a g})

8:

a_{3} = q_{3}^{(n)} + σ v_{z} D_{z} {\bar{u}}^{(n)}

s_{3} = P r o j e c t O n t o l_{1} B a l l_{v_{z} t_{z}} ({| a_{3} |}_{m a g} / σ)

q_{3}^{(n + 1)} = a_{3} (1_{I} - σ s_{3} / {| a_{3} |}_{m a g})

9:

u^{(n + 1)} = u^{(n)} - τ [A^{T} P^{(n + 1)} + v_{x} D_{x}^{T} q_{1}^{(n + 1)} + v_{y} D_{y}^{T} q_{2}^{(n + 1)} + v_{z} D_{z}^{T} q_{3}^{(n + 1)}]

10:

{\bar{u}}^{(n + 1)} = u^{(n + 1)} + θ [u^{(n + 1)} - u^{(n)}]

11:

n = n + 1

12: until the convergence conditions are satisfied

Output: The convergence solution

u^{(c o n v)}

HDTV-CP, high-order directional total variation Chambolle-Pock.

In this study, the CP algorithm framework is adopted for the optimization problem shown in Eq. [8]. The specific implementation steps are as follows:

Construction of the optimization framework: $x * = \arg \min_{x} {F (y) + G (x)} s . t . y = K x$ .
Computation of conjugate functions.
Calculation of the corresponding proximal mappings ( ${prox}_{σ} [F^{*}]$ , ${prox}_{τ} [G]$ ).
Substitution of the obtained proximal mappings into Table 2.

Table 2

N-iteration framework of the CP algorithm

1.

L = {‖ K ‖}_{S V}

;

σ = 1 / L

;

τ = 1 / L

;

θ = 1

;

n = 0

2.

x^{(0)} = 0

;

{\bar{x}}^{(0)} = 0

;

y^{(0)} = 0

3. repeat

4.

y^{(n + 1)} = {prox}_{σ} [F^{*}] [y^{(n)} + σ K {\bar{x}}^{(n)}]

5.

x^{(n + 1)} = {prox}_{τ} [G] [x^{(n)} + τ K^{T} y^{(n + 1)}]

6.

{\bar{x}}^{(n + 1)} = x^{(n + 1)} + θ [x^{(n + 1)} - x^{(n)}]

7.

n = n + 1

8. until

n \geq N

CP, Chambolle-Pock.

Table 2 shows the computational framework of the CP algorithm, where, ${‖ \cdot ‖}_{S V}$ denotes maximum singular value, and the superscript T represents the transpose operator. Under the CP algorithm framework, Eq. [8] can be converted to:

$u * = \arg \min_{u} {\frac{λ}{2} {‖ g - A u ‖}_{2}^{2} + δ_{l_{1} B a l l (v_{x} t_{x})} ({| v_{x} D_{H T V_{x}} u |}_{m a g}) + δ_{l_{1} B a l l (v_{y} t_{y})} ({| v_{y} D_{H T V_{y}} u |}_{m a g}) + δ_{l_{1} B a l l (v_{z} t_{z})} ({| v_{z} D_{H T V_{z}} u |}_{m a g})}$ [9]

Here, $δ_{l_{1} B a l l}$ denotes the indicator function, defined as follows:

$δ_{l_{1} B a l l (a)} (x) = {\begin{cases} \begin{matrix} 0, & {‖ x ‖}_{1} \leq a \end{matrix} \\ \begin{matrix} \infty, & {‖ x ‖}_{1} > a \end{matrix} \end{cases}$ [10]

This indicator function can be expressed as follows: when vector x is located within l₁-sphere with radius $a$ , the function value is 0; otherwise, the function value tends to infinity.

$Define x = u, p = A u, q_{1} = v_{x} D_{H T V_{x}} u, q_{2} = v_{y} D_{H T V_{y}} u, q_{3} = v_{z} D_{H T V_{z}} u$ [11]

$K = {\begin{matrix} A \\ v_{x} D_{H T V_{x}} \\ v_{y} D_{H T V_{y}} \\ v_{z} D_{H T V_{z}} \end{matrix}}, y = {\begin{matrix} p \\ q_{1} \\ q_{2} \\ q_{3} \end{matrix}}$ [12]

Therefore,

$F (y) = F_{1} (p) + F_{2} (q_{1}) + F_{3} (q_{2}) + F_{4} (q_{3})$ [13]

$F_{1} (p) = \frac{λ}{2} {‖ g - A u ‖}_{2}^{2}$ [14]

$F_{2} (q_{1}) = δ_{l_{1} B a l l (v_{x} t_{x})} ({| q_{1} |}_{m a g})$ [15]

$F_{3} (q_{2}) = δ_{l_{1} B a l l (v_{y} t_{y})} ({| q_{2} |}_{m a g})$ [16]

$F_{4} (q_{3}) = δ_{l_{1} B a l l (v_{z} t_{z})} ({| q_{3} |}_{m a g})$ [17]

$G (x) = 0$ [18]

Based on the CP algorithm iterative framework, the core computational task of the next stage of the algorithm is to solve the conjugate of the given convex functions $F_{1} (P)$ , $F_{2} (q_{1})$ , $F_{3} (q_{2})$ , and $F_{4} (q_{3})$ :

$F_{1}^{*} (p) = \frac{1}{2 λ} {‖ p ‖}_{2}^{2} + g^{T} p$ [19]

$F_{2}^{*} (q_{1}) = v_{x} t_{x} {‖ {| q_{1} |}_{m a g} ‖}_{\infty}$ [20]

$F_{3}^{*} (q_{2}) = v_{y} t_{y} {‖ {| q_{2} |}_{m a g} ‖}_{\infty}$ [21]

$F_{4}^{*} (q_{3}) = v_{z} t_{z} {‖ {| q_{3} |}_{m a g} ‖}_{\infty}$ [22]

Here, ${‖ \cdot ‖}_{\infty}$ denotes infinite norm. Next, we obtain proximal mappings for $F_{1}^{*} (P)$ , $F_{2}^{*} (q_{1})$ , $F_{3}^{*} (q_{2})$ , $F_{4}^{*} (q_{3})$ , and $G (x)$ .

${Prox}_{σ} [F_{1}^{*}] (p) = (p - σ g) / (1 + σ / λ)$ [23]

${Prox}_{σ} [F_{2}^{*}] (q_{1}) = q_{1} {1_{I} - [σ P r o j e c t O n t o l_{1} B a l l_{v_{x} t_{x}} ({| q_{1} |}_{m a g} / σ)] / {| q_{1} |}_{m a g}}$ [24]

${Prox}_{σ} [F_{3}^{*}] (q_{2}) = q_{2} {1_{I} - [σ P r o j e c t O n t o l_{1} B a l l_{v_{y} t_{y}} ({| q_{2} |}_{m a g} / σ)] / {| q_{2} |}_{m a g}}$ [25]

${Prox}_{σ} [F_{4}^{*}] (q_{3}) = q_{3} {1_{I} - [σ P r o j e c t O n t o l_{1} B a l l_{v_{z} t_{z}} ({| q_{3} |}_{m a g} / σ)] / {| q_{3} |}_{m a g}}$ [26]

${Prox}_{τ} [G] (x) = x$ [27]

The HDTV-CP algorithm example shown in Table 1 can be obtained by substituting Eq. [23] to Eq. [27] into Table 2.

Reconstruction parameters

Our optimization method mainly contains model parameters such as system matrix A, TV limits ( $t_{x}$ , $t_{y}$ , $t_{z}$ ), projection approach, and algorithm parameters such as $σ$ , $τ$ , $θ$ , $λ$ , $v_{x}$ , $v_{y}$ , and $v_{z}$ . We adopt isotropic cubic voxels as the basic unit to achieve real-time calculation of A (41). The values of $t_{x}$ , $t_{y}$ , and $t_{z}$ are related to the imaging target. TV limits can be dynamically adjusted by adjustment coefficients $t_{1}$ , $t_{2}$ , and $t_{3}$ , as shown in Table 1. For the projection approach, we adopt the approach presented in previous work (42,43). Algorithm parameters do not alter the HDTV model’s solution set but optimize the HDTV-CP algorithm’s convergence path, thereby accelerating convergence. ${‖ • ‖}_{S V}$ denotes maximum singular value for matrix. For details on the calculation, see (38).

Results

For this part, we first verify the correctness of the HDTV-CP algorithm. Next, data definitions and quantitative metrics are introduced. Finally, the performance of the proposed algorithm is investigated on limited-angle and sparse-limited-angle data.

Algorithm correctness verification

From a mathematical perspective, image reconstruction is classified as an ill-posed inverse problem. Its algorithm verification must meet two criteria: visual consistency and theoretical convergence. This paper uses three quantitative metrics: normalized root mean square error (nRMSE), normalized data error (NDE), and normalized TV error (NTVE), which are defined below.

$nRMSE [u^{(n)}] = {‖ u^{(n)} - u^{[t r u t h]} ‖}_{2} / {‖ u^{[t r u t h]} ‖}_{2}$ [28]

$NDE [u^{(n)}] = {‖ g - A u^{(n)} ‖}_{2} / {‖ g ‖}_{2}$ [29]

$NTVE [u^{(n)}] = | ({‖ u^{(n)} ‖}_{T V} - {‖ u^{[t r u t h]} ‖}_{T V}) | / {‖ u^{[t r u t h]} ‖}_{T V}$ [30]

Among them, nRMSE measures the normalized difference of reconstructed image $u^{(n)}$ and true image $u^{[truth]}$ ; NDE evaluates the difference of model data $A u^{(n)}$ and measurement data g; and NTVE calculates the difference of their TV values. The conditions for determining algorithm convergence are expressed below.

$nRMSE [u^{(n)}] \leq 10^{- 3}$ [31]

$NDE [u^{(n)}] \leq 10^{- 3}$ [32]

$NTVE [u^{(n)}] \leq 10^{- 3}$ [33]

This study used a 32×32×32 3D six-sphere numerical phantom for verification. Figure 3A displays the original central transverse slice, and Figure 3B displays the reconstruction results, with both exhibiting identical visual characteristics. The profiles in Figure 3C,3D perfectly align with the true image. Figure 4 demonstrates that the nRMSE, NDE, and NTVE values all exceed the predefined convergence thresholds, confirming the correctness of the HDTV-CP algorithm.

Figure 3 Verification results. (A) The central transverse slice of the original image, (B) the corresponding reconstruction result. (C,D) The horizontal center profile and 3/4 profile along the yellow and red lines in (A). For (A) and (B), both the x-axis and y-axis represent voxel position. For (C) and (D), the x-axis represents voxel position, and the y-axis represents voxel value. HDTV, high-order directional total variation.

Figure 4 The reconstruction evaluation metrics of the HDTV-CP algorithm under the 828 views: (A) nRMSE, (B) NDE, (C) NTVE. HDTV, high-order directional total variation; HDTV-CP, high-order directional total variation Chambolle-Pock; NDE, normalized data error; nRMSE, normalized root mean square error; NTVE, normalized total variation error.

Real-data study

This study adopts EPR oxygen imaging (main magnetic field 250 G, resonance frequency 720 MHz) to conduct experiments on physical phantoms. It includes two 1 mL ampoules and one 5 mL ampoule, filling with 2×10⁻³ mol/L OX063-d24 trityl solution (see Figure 5). We researched EPRI over limited-angles (R₁-R₇) qualitatively and quantitatively, and selected the minimum angle range allowed by the HDTV-CP algorithm. Based on this, we found the minimum number of sparse projections in the minimum angle range. For the big phantom which consisted of a 5 mL ampoule, this study adopted equal solid angle method to obtain 828 projection angles, with 80 measurement points for each angle, generated 80×828 simulated projection data based on the pixel-driven approach in references (42,43), and performed uniform sampling using MSPS technology (40). In addition, four sparse projection datasets were constructed, specifically: 50 views, 100 views, 150 views, and 200 views. For small phantoms which consisted of two 1-mL ampoule, we generated 64×828 simulated projection data.

Figure 5 Experimental equipment. The EPR imager (A) and the physical phantoms (B). EPR, electron paramagnetic resonance.

For our experiments, TVcDM-CP (39) and DTV-CP (23) were taken as comparison algorithms. The image quality of the reconstructed images was quantitatively evaluated using nRMSE and Pearson correlation coefficient (PCC), and the image quality of HDTV-CP was qualitatively analyzed visually.

Data definitions

Definitions of the angle range for limited-angle EPRI to facilitate analysis and discussion of the reconstructed images are displayed in Table 3. $R$ represents the full-angle range data which has a polar angle $θ$ of $(0 °, 90 °)$ and an azimuthal angle $φ$ of $(0 °, 360 °)$ . The full angle range data is expressed in the form of $(θ, φ)$ with $R : [(0 °, 90 °), (0 °, 360 °)]$ . $R_{1} - R_{7}$ are limited-angle range data.

Table 3

Definition of angular ranges. Neighboring ranges share identical polar angles but differ in azimuth by 15

R_{index} : (θ, φ)

R_{1} : [(0 °, 90 °), (0 °, 180 °)]

R_{2} : [(0 °, 90 °), (0 °, 195 °)]

R_{3} : [(0 °, 90 °), (0 °, 210 °)]

R_{4} : [(0 °, 90 °), (0 °, 225 °)]

R_{5} : [(0 °, 90 °), (0 °, 240 °)]

R_{6} : [(0 °, 90 °), (0 °, 255 °)]

R_{7} : [(0 °, 90 °), (0 °, 270 °)]

R : [(0 °, 90 °), (0 °, 360 °)]

Quantitative metrics

In the real data experiment, nRMSE and PCC are selected as quantitative indicators for evaluating different reconstruction algorithms in EPRI. It is expressed mathematically below.

$nRMSE [u^{(*)}] = {‖ u^{(*)} - u^{[r e f]} ‖}_{2} / {‖ u^{[r e f]} ‖}_{2}$ [34]

$PCC [u^{(*)}] = | Cov [u^{(*)}, u^{[r e f]}] | / {σ [u^{(*)}] σ (u^{[r e f]})}$ [35]

In the quantitative evaluation, $u^{(*)}$ represents the reconstructed image based on limited-angle and sparse-limited-angle data, and $u^{[r e f]}$ represents the reference image using the HDTV-CP algorithm with complete angular sampling. $Cov (\cdot)$ represents covariance. $σ (\cdot)$ denotes the standard deviation. These two metrics provide an objective quantitative standard for the similarity between the reconstructed image and the reference image.

Selection of the reference image

In real data research, since we do not have ground-truth images, we need to choose reference images. As shown in Figure 6, by comparing the results of FBP and HDTV reconstruction, we found that the HDTV algorithm can significantly improve edge clarity. Therefore, full-angle HDTV-CP reconstructed images were used as references.

Figure 6 The FBP and HDTV reconstruction results obtained by using full angle. The display window ranges from [0, 0.001]. The red box indicates the main area of the image. FBP, filtered back projection; HDTV, high-order directional total variation.

The results of the limited-angle EPRI

To qualitatively evaluate reconstruction performance, we visually compared slice-images generated by different algorithms from limited-angle data. As shown in Figure 7, the image edges reconstructed by the TV-CP algorithm are severely missing, and the profile appears too smooth. TV lacks the ability to reconstruct the image in limited-angle data. The right upper corner of the image edges reconstructed by the DTV-CP algorithm is missing at R₁–R₄. However, the HDTV-CP algorithm reconstructed the image edges completely at the R₁ angle. This result can be interpreted as follows: the edges are the regions where the EPR signal changes dramatically, and the HDTV method takes into account the second-order derivatives of the image, which enables it to more accurately represent the local variations of the image, thus preserving more details of the edges during the EPR image reconstruction process. In addition, the HDTV method is able to control the smoothness of the image more carefully by taking into account the high order derivatives of the image, suppressing the staircase effect, and making the reconstructed image more natural.

Figure 7 The reconstructed object’s slice image at y=28. The reconstruction results for five limited-angle range R₁–R₅ are shown from left to right. R represents the image reconstructed from the full angle data. The display window is [0, 0.0021]. The red boxes indicate the enlarged image. DTV, directional total variation; HDTV, high-order directional total variation; TV, total variation.

Figure 8 shows slice images at x=27. The ERP image reconstructed by the HDTV-CP algorithm had no obvious staircase artifacts in the interior of the image, and the image edges were more complete. The visual difference between the EPR images reconstructed from the limited-angle data R₁–R₅ and the EPR images reconstructed from the full angle R was relatively small when the HDTV-CP algorithm was applied. In addition, we found that z-shaped artifacts, represented as the places circled by red ellipses, existed in the interior of the image for TV, DTV, and HDTV. When the angle was small, the artifacts were not obvious. This result can be explained as follows: when the angular range increases, the overall quality of the image notably improves, making the subtle artifacts that are already existent more visible. It does not mean that the artifacts themselves increased; rather, our perception increased.

Figure 8 The reconstructed object’s slice image at x=27. The reconstruction results for five limited-angle range R₁–R₅ are shown from left to right. R represents the image reconstructed from the full angle data. The display window is [0, 0.0022]. The red arrows indicate the area of change. The red circle indicates the area of change. DTV, directional total variation; HDTV, high-order directional total variation; TV, total variation.

In the quantitative analysis, nRMSE and PCC were employed for comparing the reconstruction accuracy of TV, DTV, and HDTV within five limited-angle ranges. Figure 9 shows that the nRMSE and PCC of all the EPR limited-angle images reconstructed by the HDTV minimization method are optimal compared with the two methods of TV and DTV. Figures 10,11 show the reconstructed images on two small phantoms. It can be seen from the figures that the HDTV method is superior and can reconstruct the image completely at R₇. Figure 12 displays the value of nRMSE and PCC. It is revealed that the HDTV-CP algorithm has better texture features and edge information preservation in the limited-angle EPRI.

Figure 9 The nRMSE (A) and PCC (B) for the limited-angular range R₁–R₅ are calculated by TV, DTV, and HDTV methods respectively. DTV, directional total variation; HDTV, high-order directional total variation; nRMSE, normalized root mean square error; PCC, Pearson correlation coefficient; TV, total variation.

Figure 10 The reconstructed object’s slice image at y=39. The reconstruction results for four limited-angle range R₁, R₃, R₅, and R₇ are shown from left to right. The display window is [0, 0.001]. The red box indicates the main area of the image. DTV, directional total variation; HDTV, high-order directional total variation; TV, total variation.

Figure 11 The reconstructed object’s slice image at x=17. The reconstruction results for four limited-angle range R₁, R₃, R₅, and R₇ are shown from left to right. The display window is [0, 0.0015]. The red box indicates the main area of the image. DTV, directional total variation; HDTV, high-order directional total variation; TV, total variation.

Figure 12 The nRMSE (A) and PCC (B) for the limited-angular range R₁, R₃, R₅ and R₇ are calculated by TV, DTV, and HDTV methods respectively. DTV, directional total variation; HDTV, high-order directional total variation; nRMSE, normalized root mean square error; PCC, Pearson correlation coefficient; TV, total variation.

The results of sparse reconstruction for the limited-angle EPRI

For the big phantom which consisted of a 5-mL ampoule, we chose R₁ as the smallest limited-angle to explore its sparse reconstruction performance. For the small phantoms which consisted of two 1-mL ampoules, we chose R₇ as the smallest limited-angle. As shown in Figure 13, the reconstructed EPR image using the TV method suffered from serious missing edges and over-smooth profiles. TV lacks the ability to sparse reconstruct images in EPR limited-angle data. The upper right corner of the EPR image edges reconstructed by the DTV-CP algorithm was missing at 50 to 200 views. However, the HDTV-CP algorithm reconstructed the image edges completely at 50 views, achieving the same effect as the limited-angle data R₁.

Figure 13 The reconstructed object’s slice image at y=28. The reconstruction results on 50 views, 100 views, 150 views, and 200 views are shown from left to right. R₁ represents the image reconstructed from limited angle of

θ \in (0 °, 90 °)

and

φ \in (0 °, 180 °)

. The display window is [0, 0.0021]. The red boxes indicate the enlarged image. DTV, directional total variation; HDTV, high-order directional total variation; TV, total variation.

Figure 14 shows slice images at x=27. It is clear from Figure 14 that the edges of a reconstructed image by the TV method were incomplete and too smooth and there was a staircase artifact in the interior of the image reconstructed by the DTV method. However, the HDTV method eliminated both the smoothness of TV and the staircase artifacts of DTV. The edges of the reconstructed EPR images by the HDTV method were more complete and natural. For the big phantom, Figure 15 further shows the comparison of spatial profiles for the TV, DTV, and HDTV methods on 50 views. It can be seen from figure that the edges of the HDTV reconstructed images are closer to the edges of the reference image. Figures 16,17 display the result on two small phantoms. It can be seen from figures that the HDTV method is superior and can reconstruct the image completely at 100 views. Figure 18 displays the comparison of spatial profiles for the TV, DTV, and HDTV methods on 100 views. The experiments show that the HDTV-CP algorithm effectively suppresses limited-angle artifacts, sparse artifacts, and staircase artifacts while preserving the edge and texture features, making the reconstructed image more natural.

Figure 14 The reconstructed object’s slice image at x=27. The reconstruction results on 50 views, 100 views, 150 views, and 200 views are shown from left to right. R₁ represents the image reconstructed from limited angle of

θ \in (0 °, 90 °)

and

φ \in (0 °, 180 °)

. The display window is [0, 0.00225]. The red arrows indicate the area of change. DTV, directional total variation; HDTV, high-order directional total variation; TV, total variation.

Figure 15 Spatial profiles of images reconstructed by TV algorithm (green), DTV algorithm (blue), HDTV algorithm (red) and reference image (black) at the position of x=27 and z=35. The reference image is the full-angle reconstructed image by HDTV method. DTV, directional total variation; HDTV, high-order directional total variation; TV, total variation.

Figure 16 The reconstructed object’s slice image at y=39. The reconstruction results on 50 views, 100 views, 150 views, and 200 views are shown from left to right. R₇ represents the image reconstructed from limited angle of

θ \in (0 °, 90 °)

and

φ \in (0 °, 270 °)

. The display window is [0, 0.001]. The red box indicates the main area of the image. DTV, directional total variation; HDTV, high-order directional total variation; TV, total variation.

Figure 17 The reconstructed object’s slice image at x=17. The reconstruction results on 50 views, 100 views, 150 views and 200 views are shown from left to right. R₇ represents the image reconstructed from limited angle of

θ \in (0 °, 90 °)

and

φ \in (0 °, 270 °)

. The display window is [0, 0.0015]. The red box indicates the main area of the image. DTV, directional total variation; HDTV, high-order directional total variation; TV, total variation.

Figure 18 Spatial profiles of images reconstructed by TV algorithm (green), DTV algorithm (blue), HDTV algorithm (red) and reference image (black) at the position of x=16 and y=39. The reference image is the full-angle reconstructed image by HDTV method. DTV, directional total variation; HDTV, high-order directional total variation; TV, total variation.

Discussion

This section first discusses the impact of reconstruction parameters on the reconstructed image, and then provides an in-depth analysis of the advantages and limitations of HDTV-CP algorithm. Finally, we analyze the potential applications in CT image reconstruction.

The effect of reconstruction parameters on the reconstructed image

Similar to all of the image reconstruction based on optimization, the model parameters play a crucial role to the final solution. Here, we mainly discuss the effect of TV bounds on the solution. TV bounds affect the solution of the model. For TV methods, there is only one TV bound. It can easily introduce noise when its value is too large, and it can cause the image to be overly smooth when its value is too small. For the HDTV-CP algorithm, there are three TV bound values. It requires searching the optimal value in 3D space. The approach used in this paper is as follows.

Firstly, the image is reconstructed from 828 projections by 3D FBP algorithm to obtain its gradient values $D_{x}$ , $D_{y}$ , and $D_{z}$ along x, y, and z directions. Secondly, $t_{x}$ , $t_{y}$ , and $t_{z}$ are adjusted according to the angle range, where, $t_{x} = t_{1} D_{x}$ , $t_{y} = t_{2} D_{y}$ , and $t_{z} = t_{3} D_{z}$ . We can obtain accurate images by adjusting $t_{1}$ , $t_{2}$ , and $t_{3}$ , for example, using 0.25, 0.45, 0.65, 0.9, and 1.2, which is the best trade-off between smoothness and shape preservation. When we obtain these optimal model parameters, namely, the three TV bounds, we can use them in the sparse reconstruction of the limited-angle EPRI. The method we used to adjust $t_{1}$ , $t_{2}$ , and $t_{3}$ is as follows: First, we set the values of $t_{1}$ , $t_{2}$ , and $t_{3}$ to 0.45, then kept $t_{1}$ and $t_{3}$ fixed and adjust the value of $t_{2}$ . The approximate interval is found by setting $t_{2}$ to different values, as shown in Figure 19. Then, the value of $t_{2}$ is fine-tuned within the interval. When the value of $t_{2}$ is adjusted, $t_{1}$ and $t_{2}$ remain fixed and the value of $t_{3}$ is adjusted. Finally, the value of $t_{1}$ is adjusted. We take the adjustment of $t_{2}$ as an example to observe its effect on EPRI at the limited-angle range of R₁. As shown in Figure 20, when the value of $t_{2}$ is too small, the image is incomplete; when the value of $t_{2}$ is too large, the image edges easily introduce artifacts. Figure 21 displays the difference image. It can be seen from figure that the artifacts are less when $t_{2}$ is equal to 0.45, revealing that the optimal value of $t_{2}$ is around 0.45.

Figure 19 The method of determining the TV bound. Take the example of determining

t_{2}

as an illustration. First,

t_{1}

and

t_{3}

are fixed. Then, nRMSE values are calculated when

t_{2}

is 0.25, 0.45, 0.65, 0.9, and 1.2. The full-angle image is used as the reference. Then the approximate range of

t_{2}

is determined according to the nRMSE values. Finally, the value of

t_{2}

is selected by fine-tuning within this range and incorporating visual inspection. nRMSE, normalized root mean square error; TV, total variation.

Figure 20 The effect of the TV bounds on the reconstructed image. The slice-images at y=28 of the reconstructed objects (row 1). The display window is [0, 0.0021]. The slice-images at x=27 of the reconstructed objects (row 2). The display window is [0, 0.0022]. The red arrows indicate the area of change. TV, total variation.

Figure 21 Difference image among reconstructed and reference images when

t_{2}

is equal to 0.25, 0.45, 0.65, 0.9, and 1.2 respectively. The reconstructed object’s slice image at y=28. The display window is [0, 0.0021]. The red box indicates the main area of the image. The red arrow indicates the area of change.

Advantages and limitations

For sparse-limited-angle EPRI, the HDTV minimization method proposed in this paper applies different constraint strengths to the three directions of the EPR image and introduces high-order TV constraints to adjust the transition between pixel values more finely, preserving more image details. Our proposed HDTV method effectively suppresses limited-angle artifacts, sparse artifacts, and staircase artifacts while preserving the edge and texture features. Hence, the algorithm has superior reconstruction performance compared to TV and DTV. This paper is the first to combine the limited-angle and sparse problems to further accelerate scanning. The HDTV method may realize 16 times acceleration while ensuring the imaging quality.

Similar to other iterative methods, the HTV bounds value affects the reconstruction accuracy of the image. It is still difficult to quickly determine the optimal value of the HTV limit of the HDTV-CP algorithm.

Potential applications in CT image reconstruction

Our work focuses on scanning strategies and an optimization model for limited angle EPRI. The scanning strategy and the optimization model we proposed can be easily applied to limited angle CT imaging. Firstly, the accelerated scanning strategy which combines the limited-angle and sparse problems to further accelerate scanning proposed in our paper can be directly applied. Secondly, for two-dimensional (2D) CT imaging, the HDTV method can be applied to CT imaging by changing the constraints in our HDTV optimization model: only the x and y directions are constrained.

Conclusions

In this paper, we have proposed a HDTV minimization method for achieving 3D EPRI from sparse-limited-angle data. HDTV method introduces high-order constraint information into the DTV algorithm to suppress staircase artifacts. The HDTV minimization model was designed, and the corresponding CP solution algorithm was derived, namely, the HDTV-CP algorithm, and the correctness of the algorithm was verified. The limited-angle range along the XOY plane was acquired, and the minimum limited-angle range and the minimum number of sparse projection points were comprehensively studied. The results of the experiments revealed that the minimum number of sparse views based on the minimum angular range of R₁ was 50. The discussion analyzed the impact of the bounds on the quality of EPRI and the method of selecting HTV limits. The experimental results showed that the HDTV minimization algorithm can more accurately reconstruct images from sparse-limited-angle data compared to TV and DTV, thus enabling accelerated scanning in EPRI. This paper is the first to combine the limited-angle and sparse problems to further accelerate scanning. The HDTV method may realize 16 times acceleration while ensuring the imaging quality in certain situations. The performance of this method needs to be validated using more complex physical phantoms and mouse tumor data, and parameter selection methods should be optimized to improve reconstruction efficiency and accuracy.

Acknowledgments

None.

Footnote

Data Sharing Statement: Available at https://qims.amegroups.com/article/view/10.21037/qims-2025-8/dss

Funding: This work was supported in part by National Natural Science Foundation of China (No. 62071281), and Local Science and Technology Development Fund Project Guided by the Central Government (No. YDZJSX2021A003).

Conflicts of Interest: All authors have completed the ICMJE uniform disclosure form (available at https://qims.amegroups.com/article/view/10.21037/qims-2025-8/coif). The authors have no conflicts of interest to declare.

Ethical Statement: The authors are accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved.

Open Access Statement: This is an Open Access article distributed in accordance with the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International License (CC BY-NC-ND 4.0), which permits the non-commercial replication and distribution of the article with the strict proviso that no changes or edits are made and the original work is properly cited (including links to both the formal publication through the relevant DOI and the license). See: https://creativecommons.org/licenses/by-nc-nd/4.0/.

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Cite this article as: Zhang Y, Liu P, Fang C, Xi Y, Qiao Z. HDTV: a high-order directional total variation reconstruction algorithm from sparse and limited-angle data in electron paramagnetic resonance imaging. Quant Imaging Med Surg 2025;15(9):8471-8490. doi: 10.21037/qims-2025-8

HDTV: a high-order directional total variation reconstruction algorithm from sparse and limited-angle data in electron paramagnetic resonance imaging

Introduction

Methods

Scanning configuration and data acquisition

Optimization model (HDTV)

Solving algorithm (HDTV-CP)

Table 1

Table 2

Reconstruction parameters

Results

Algorithm correctness verification

Real-data study

Data definitions

Table 3

Quantitative metrics

Selection of the reference image

The results of the limited-angle EPRI

The results of sparse reconstruction for the limited-angle EPRI

Discussion

The effect of reconstruction parameters on the reconstructed image

Advantages and limitations

Potential applications in CT image reconstruction

Conclusions

Acknowledgments

Footnote

References

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