Interior Tomography Using 1D Generalized Total Variation. Part I: Mathematical Foundation

Ward, John Paul; Lee, Minji; Ye, Jong Chul; Unser, Michaël

doi:10.1137/140982428

Cited by 19 publications

(26 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this section, we provide the mathematical preliminaries and review the main idea of our previous work [32].…”

Section: Mathematical Preliminariesmentioning

confidence: 99%

“…Accordingly, a required interior tomography formulation finds an appropriate regularization term that suppresses the signal belonging to the null space of the truncated Hilbert transform by exploiting the infinite differentiability of f Nul . Instead, our previous work [32] introduced a generalized 1D TV penalty minimization as a simplification of multidimensional TV in [13,34,35]. For completeness, we repeat the main ideas of [32].…”

Section: 3mentioning

confidence: 99%

“…Instead, our previous work [32] introduced a generalized 1D TV penalty minimization as a simplification of multidimensional TV in [13,34,35]. For completeness, we repeat the main ideas of [32]. Specifically, we consider a regularization with respect to a Fourier multiplier operator L that is defined on L 2 (R) and satisfies two conditions.…”

Section: 3mentioning

confidence: 99%

“…In our previous work [32], we revisited the classical interior problem, where projections at each view extend only to the shadow of a circular region completely interior to the subject being scanned, and the object model is continuous, and the view angle and detector sampling are also continuous. In particular, we were also interested in reconstructing images that are piecewise smooth; i.e., the domain where the image is defined can be decomposed into a finite number of subdomains such that the image is a smooth function on each piece.…”

mentioning

confidence: 99%

“…However, our approach generalized the ideas developed in [13,34,35] with a simplified one-dimensional (1D) formulation on the chord lines, and as a result, we were able to perfectly reconstruct new classes of functions. In particular, we showed that we can reconstruct any image that is a generalized L-spline along a collection of chord lines passing through the ROI [32]. For example, we can reconstruct images whose 1D restrictions are nonpolynomial exponential B-splines, which were not covered in [13,34,35].…”

mentioning

confidence: 99%

See 4 more Smart Citations

Interior Tomography Using 1D Generalized Total Variation. Part II: Multiscale Implementation

Lee¹,

Han²,

Ward³

et al. 2015

SIAM J. Imaging Sci.

Self Cite

View full text Add to dashboard Cite

Abstract. To address the classic interior tomography problem where projections at each view extend only to the shadow of a circular region completely interior to the subject being scanned, previously we showed that the exact recovery of two-and three-dimensional piecewise smooth images is guaranteed using a one-dimensional generalized total variation seminorm penalty which allows a much faster reconstruction. To further accelerate the algorithm up to a level for clinical use, this paper proposes a novel multiscale reconstruction method by exploiting the Bedrosian identity of the Hilbert transform. More specifically, we show that the high frequency parts of the one-dimensional signals can be quickly recovered analytically with the Hilbert transform because of the Bedrosian identity. This implies that computationally expensive iterative reconstruction need only be applied to low resolution images in the downsampled domain, which significantly reduces the computational burden. Moreover, even for incomplete trajectories such as circular cone-beam geometry, we demonstrate that the proposed multiscale interior tomography approach can be combined with a novel spectral blending method in order to mitigate cone-beam artifacts from missing frequency regions. We show the efficacy of the proposed multiscale algorithm using circular fan-beam, helical cone-beam data, and circular conebeam geometry. With a graphics processing unit implementation, we demonstrate that the speed of the algorithm can be significantly accelerated up to the level for clinical use for various acquisition geometries.

show abstract

“…In this section, we provide the mathematical preliminaries and review the main idea of our previous work [32].…”

Section: Mathematical Preliminariesmentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Interior Tomography Using 1D Generalized Total Variation. Part II: Multiscale Implementation

Lee¹,

Han²,

Ward³

et al. 2015

SIAM J. Imaging Sci.

Self Cite

View full text Add to dashboard Cite

show abstract

One network to solve all ROIs: Deep learning CT for any ROI using differentiated backprojection

Han

2019

Medical Physics

Self Cite

View full text Add to dashboard Cite

Purpose Computed tomography for the reconstruction of region of interest (ROI) has advantages in reducing the x‐ray dose and the use of a small detector. However, standard analytic reconstruction methods such as filtered back projection (FBP) suffer from severe cupping artifacts, and existing model‐based iterative reconstruction methods require extensive computations. Recently, we proposed a deep neural network to learn the cupping artifacts, but the network was not generalized well for different ROIs due to the singularities in the corrupted images. Therefore, there is an increasing demand for a neural network that works well for any ROI size. Method Two types of neural networks are designed. The first type learns ROI size‐specific cupping artifacts from FBP images, whereas the second type network is for the inversion of the truncated Hilbert transform from the truncated differentiated backprojection (DBP) data. Their generalizabilities for different ROI sizes, pixel sizes, detector pitch and starting angles for a short scan are then investigated. Results Experimental results show that the new type of neural networks significantly outperform existing iterative methods for all ROI sizes despite significantly lower runtime complexity. In addition, performance improvement is consistent across different acquisition scenarios. Conclusions Since the proposed method consistently surpasses existing methods, it can be used as a general CT reconstruction engine for many practical applications without compromising possible detector truncation.

show abstract

Deep‐Interior: A new pathway to interior tomographic image reconstruction via a weighted backprojection and deep learning

Zhang,

Chen

2023

Medical Physics

View full text Add to dashboard Cite

BackgroundIn recent years, deep learning strategies have been combined with either the filtered backprojection or iterative methods or the direct projection‐to‐image by deep learning only to reconstruct images. Some of these methods can be applied to address the interior reconstruction problems for centered regions of interest (ROIs) with fixed sizes. Developing a method to enable interior tomography with arbitrarily located ROIs with nearly arbitrary ROI sizes inside a scanning field of view (FOV) remains an open question.PurposeTo develop a new pathway to enable interior tomographic reconstruction for arbitrarily located ROIs with arbitrary sizes using a single trained deep neural network model.MethodsThe method consists of two steps. First, an analytical weighted backprojection reconstruction algorithm was developed to perform domain transform from divergent fan‐beam projection data to an intermediate image feature space, , for an arbitrary size ROI at an arbitrary location inside the FOV. Second, a supervised learning technique was developed to train a deep neural network architecture to perform deconvolution to obtain the true image from the new feature space . This two‐step method is referred to as Deep‐Interior for convenience. Both numerical simulations and experimental studies were performed to validate the proposed Deep‐Interior method.ResultsThe results showed that ROIs as small as a diameter of 5 cm could be accurately reconstructed (similarity index 0.985 ± 0.018 on internal testing data and 0.940 ± 0.025 on external testing data) at arbitrary locations within an imaging object covering a wide variety of anatomical structures of different body parts. Besides, ROIs of arbitrary size can be reconstructed by stitching small ROIs without additional training.ConclusionThe developed Deep‐Interior framework can enable interior tomographic reconstruction from divergent fan‐beam projections for short‐scan and super‐short‐scan acquisitions for small ROIs (with a diameter larger than 5 cm) at an arbitrary location inside the scanning FOV with high quantitative reconstruction accuracy.

show abstract

Interior Tomography Using 1D Generalized Total Variation. Part I: Mathematical Foundation

Cited by 19 publications

References 15 publications

Interior Tomography Using 1D Generalized Total Variation. Part II: Multiscale Implementation

Interior Tomography Using 1D Generalized Total Variation. Part II: Multiscale Implementation

One network to solve all ROIs: Deep learning CT for any ROI using differentiated backprojection

Deep‐Interior: A new pathway to interior tomographic image reconstruction via a weighted backprojection and deep learning

Contact Info

Product

Resources

About