cvpr cvpr2013 cvpr2013-35 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Oren Barkan, Jonathan Weill, Amir Averbuch, Shai Dekel
Abstract: One of the main challenges in Computed Tomography (CT) is how to balance between the amount of radiation the patient is exposed to during scan time and the quality of the CT image. We propose a mathematical model for adaptive CT acquisition whose goal is to reduce dosage levels while maintaining high image quality at the same time. The adaptive algorithm iterates between selective limited acquisition and improved reconstruction, with the goal of applying only the dose level required for sufficient image quality. The theoretical foundation of the algorithm is nonlinear Ridgelet approximation and a discrete form of Ridgelet analysis is used to compute the selective acquisition steps that best capture the image edges. We show experimental results where for the same number of line projections, the adaptive model produces higher image quality, when compared with standard limited angle, non-adaptive acquisition algorithms.
Reference: text
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1 i l Abstract One of the main challenges in Computed Tomography (CT) is how to balance between the amount of radiation the patient is exposed to during scan time and the quality of the CT image. [sent-4, score-0.144]
2 We propose a mathematical model for adaptive CT acquisition whose goal is to reduce dosage levels while maintaining high image quality at the same time. [sent-5, score-0.518]
3 The adaptive algorithm iterates between selective limited acquisition and improved reconstruction, with the goal of applying only the dose level required for sufficient image quality. [sent-6, score-0.633]
4 The theoretical foundation of the algorithm is nonlinear Ridgelet approximation and a discrete form of Ridgelet analysis is used to compute the selective acquisition steps that best capture the image edges. [sent-7, score-0.332]
5 We show experimental results where for the same number of line projections, the adaptive model produces higher image quality, when compared with standard limited angle, non-adaptive acquisition algorithms. [sent-8, score-0.519]
6 Introduction In the last decade, several studies have shown that radiation exposure during CT scanning is a significant factor in raising the total public risk of cancer deaths [1,20,21]. [sent-10, score-0.145]
7 It’s meant to ensure that “… CT dose factors are kept to a point where risk is minimized for maximum diagnostic benefit. [sent-12, score-0.221]
8 ", where the dose can be determined by the product of the CT tube current and the time the patient is exposed to radiation (see [2] for an overview). [sent-15, score-0.324]
9 Currently, there are several state-ofthe-art technologies that attempt to achieve dose reduction. [sent-16, score-0.221]
10 This paper describes an adaptive acquisition model that is superior, in the CT image quality, to existing limited angle, non-adaptive acquisition methods and in theory may allow minimal and optimal dosage levels. [sent-21, score-0.683]
11 The method can be considered a significant generalization of existing twostep adaptive acquisition methods [7,8] and can potentially use the same hardware configurations that are capable of changing their geometric configuration and acquisition protocols on-the-fly [9]. [sent-22, score-0.569]
12 Observe that adaptive acquisition should not be confused with adaptive reconstruction. [sent-23, score-0.487]
13 In the latter, the acquisition model is a non-adaptive uniform sampling scheme, where over a discrete set of pre-determined angles, line projections are computed at equal intervals. [sent-24, score-0.704]
14 In this setup, the adaptive elements, if exist, are part of the post-acquisition reconstruction step. [sent-25, score-0.197]
15 The outline of the algorithm is as follows: First, the system projects the object with an extreme low dose according to a uniform predetermined pattern and reconstructs an initial low quality image. [sent-26, score-0.327]
16 Then, the system iterates by incorporating the newly added line projections in order to obtain a refined approximation of the object's image. [sent-28, score-0.514]
17 The algorithm continues to iterate between estimation of locations of significant features, adaptive acquisition and reconstruction until a convergence criterion is met. [sent-29, score-0.414]
18 The goal is to converge to a high quality reconstruction using a minimal number of rays (line projections). [sent-30, score-0.103]
19 The mathematical theory of [5] quantifies, in the setup of CT, the geometric ‘structure’ of the image and how fast a Ridgelet approximation converges to the image. [sent-32, score-0.149]
20 Our algorithm, whose goal is to acquire an unknown image, regards the adaptive Ridgelet approximation of the image as the ‘optimal’ benchmark and is designed to match its performance. [sent-33, score-0.234]
21 This approach has strong ties with the 222111999533 waveform analysis presented in [10] that allowed the authors to classify singularities and quantify the ‘stability’ of limited angle tomography. [sent-34, score-0.123]
22 Indeed, although in our work we limit the number of line projections, but do not limit the angles, the fundamental understanding of the relationship between a function’s edge singularities and its Radon representation, as explained in [10], is at the core of our algorithm (see Figure 3. [sent-35, score-0.232]
23 We show in the experimental results section that for the same number of line projections, our algorithm yields higher image reconstruction quality, when compared with known limited angle, non-adaptive acquisition algorithms. [sent-37, score-0.446]
24 Section 3 describes in detail our adaptive acquisition algorithm. [sent-39, score-0.352]
25 As we shall see in Section 3, in our case, the sparsity is due to the fact that each row of A is associated with weighted integration over a digital line in the image I and therefore a vector of weights. [sent-110, score-0.2]
26 Each weight value corresponds to a pixel in Iand determined by the amount of intersection between the analytic line and the pixel itself. [sent-111, score-0.143]
27 As a result, only weights that are located in entries associated with the pixels of the digital line have non-zero values. [sent-112, score-0.164]
28 We note that even if we use a more accurate model based interpolation, where the line is given more significant width, the matrix A would remain sparse. [sent-117, score-0.164]
29 In this work we focus on the 2D model and in future work we plan to investigate whether in the 3D case our smaller adaptive sampling set can be stored in memory or computed on-the-fly. [sent-122, score-0.21]
30 In applications, this means that the Ridgelet transform can be computed by the application of the Radon transform at a given angle, followed by 1D fast wavelet transform [15]. [sent-184, score-0.197]
31 We find that Ridgelets are the right mathematical tool in the setup of CT, because the acquisition device is not able to capture, through its sampling process, well localized functionals such as Curvelet coefficients. [sent-185, score-0.357]
32 From approximation theoretical perspective, the mathematical foundation of our adaptive algorithm follows the framework of characterizing the images by the appropriate function smoothness spaces and then providing an estimate for the order of convergence. [sent-187, score-0.272]
33 , of their absolute values and denote the n-term adaptive approximation to f by fn ? [sent-322, score-0.198]
34 1/ 2 , certain assumptions on the input function, not only the convergence of the adaptive approximation is ensured, but its rate is also estimated. [sent-347, score-0.198]
35 The outcome the theory is that the approximation rate of an adaptive Ridgelet approximation depends on the smoothness of the function in a given Ridgelet smoothness space. [sent-348, score-0.319]
36 As we shall see in Section 3, our adaptive acquisition method follows the adaptive Ridgelet approximation to the image Ias a model. [sent-349, score-0.586]
37 , Then we use these coefficients in order to select the next set of line projections that are considered as best candidates to project I with, in the subsequent iteration. [sent-351, score-0.469]
38 We then ask, how many line projections are needed as rows in the matrix A , such that the image of Figure 3. [sent-354, score-0.433]
39 1) contains only 8 rows associated with 8 line projections. [sent-360, score-0.187]
40 This is achieved by selecting the unique four pairs of line projections that 222111999755 are the immediate neighbors of each of the four lines associated with the edges of the white square. [sent-366, score-0.459]
41 1(b) and (c) show the locations of the line projections and the reconstructed image, respectively. [sent-368, score-0.418]
42 The moral of this example, which correlates well with the theory in [5], is that during the acquisition process, we should try to adaptively sample the line projections that are aligned and centered around the edges of the image. [sent-369, score-0.647]
43 Initialization: We create an initial sampling matrix A using a relatively small uniform set of line projections and sample the image I to obtain an initial observations vector y . [sent-374, score-0.523]
44 The number of the initial line projections is determined relative to the image size. [sent-375, score-0.406]
45 256 , we measured 8 equally spaced line integrals at eight uniformly spaced angles, which generates a total of 64 initial measurements that are about 0. [sent-377, score-0.318]
46 TV Minimization: The inputs to this step are: an updated sampling matrix A (with new additional rows that correspond to the newly acquired line projections) , an observations vector y and the previous approximation as the initial guess. [sent-384, score-0.391]
47 This speeds up this step in the algorithm, but in some cases, its effect on the next analysis step implies that more line projections are needed to be acquired in order to achieve the same reconstruction quality. [sent-392, score-0.48]
48 In any case, our adaptive acquisition process terminates U? [sent-393, score-0.352]
49 % if one of the following conditions: or number of rows in A # L , holds, where % is a predetermined threshold and L is a limit on the total number of line projections that is permitted to be acquired. [sent-408, score-0.452]
50 of Ridgelet coefficients by the application of Radon transform followed by the application of wavelet transform, as shown in (2. [sent-411, score-0.187]
51 In practice, we realize that if we choose the number of angles to be a quarter of the image length, then our sampling scheme is sufficiently dense for high quality reconstruction, but not too much as to lead to subsequent unnecessary acquisition. [sent-414, score-0.14]
52 256 , we compute the Ridgelet coefficients for only 64 uniformly spaced angles, ? [sent-416, score-0.136]
53 For our experimental results, we applied the univariate discrete Haar wavelet [15] transform at each of the 64 angles to the 256 computed line projections. [sent-424, score-0.316]
54 In this case, the discrete sampling of Ridgelet coefficients is controlled by the pairs ( ? [sent-450, score-0.151]
55 Adaptive Sampling of New Line Projections: The analysis of the Ridgelet coefficients , computed at step 3, enables us to decide who are the new line projections that are added to A as new rows. [sent-472, score-0.469]
56 222111999866 we choose these line projections to be associated with the M largest Ridgelet coefficients that have not yet been marked as sampled by the algorithm. [sent-478, score-0.516]
57 The goal of the selected line projections is to approximate (2. [sent-479, score-0.389]
58 3, we see an illustration of the support of the Haar Ridgelet function (dashed lines) and the associated two line projections (inner lines) within its support. [sent-486, score-0.41]
59 Now, we look closer at the implication of using only two line projections to approximate the value of the Haar Ridgelet. [sent-488, score-0.389]
60 In this case, the two values of the line projections that we acquire are RI ? [sent-500, score-0.425]
61 Hence, the ATA algorithm can be summarized as follows: ATA (A , , I , M , L , ,% ) M, Input: A - Initial sampling matrix , I Input image, M Number of the Ridgelet coefficients subset considered in each iteration. [sent-525, score-0.153]
62 Find the M Ridgelet coefficients that have the largest absolute values that have not been sampled yet. [sent-560, score-0.106]
63 For each of the newly found Ridgelet coefficients: add new rows to A associated with line projections, whose sampling approximates the value of the Ridgelet coefficient on the image I . [sent-563, score-0.282]
64 :Line tgralsthawer acquiredprasignfcant Ridgelet coefficient: The external dashed lines correspond to the support of the Ridgelet and the inner lines are the sampled line projections In Figure 3. [sent-582, score-0.479]
65 4 (a) and (b) show the uniform acquisition pattern described in the initialization step and the resulted first approximation image , respectively. [sent-588, score-0.307]
66 4 (c), (e) and (g) show the newly sampled line projections associated with the next M largest unsampled Ridgelet coefficients in iterations 0, 1 and 2. [sent-591, score-0.587]
67 4 (b)), the algorithm finds from the Ridgelet analysis that it should first acquire line projections associated with Ridgelet coefficients from coarse resolution as seen in Figs. [sent-605, score-0.526]
68 4 (f)), Ridgelet coefficients from finer resolution become significant and the line projections associated with them are acquired as seen in Fig. [sent-610, score-0.519]
69 In summary, the ATA algorithm attempts to acquire only line projections that are around and aligned with edge singularities that are ordered by resolution. [sent-613, score-0.476]
70 4: Adaptive acquisition of the Ellipse image: Iterations of newly added projection lines and approximations U? [sent-618, score-0.292]
71 Experimental Results In this section we compare the ATA algorithm with known limited angle (non-adaptive) methods and also examine the quality of the estimate for the significant Ridgelet coefficients of the image I produced by our algorithm. [sent-622, score-0.193]
72 We show that for a given number of line projections measured on the image I ATA produces a , significantly better approximation to I The experiments . [sent-623, score-0.452]
73 m line projections (regardless of the target limit), which are m equally spaced line integrals over the angles 0, ? [sent-647, score-0.681]
74 Non Adaptive Equally Spaced (NAS): We used equally spaced rotations and a fixed number of line projections at each angle such that the total number of line projections matched the prescribed budget. [sent-655, score-0.905]
75 Specifically, m / 2 (equally spaced) line projections were acquired over the angles 0, ? [sent-657, score-0.465]
76 In this mode, we uniformly select lines in the Fourier domain of the image and use Fourier coefficients on these lines as the entries of the sampling matrix A. [sent-667, score-0.217]
77 Specifically, m Fourier coefficients were taken on the lines associated with the angles 0, ? [sent-668, score-0.18]
78 Adaptive Tomography Acquisition (ATA): Our proposed adaptive algorithm (see Section 3). [sent-677, score-0.135]
79 2 we see that the ATA algorithm achieves perfect reconstruction using the smallest number of line projections, while the uniform limited angle, non-adaptive acquisition algorithms (NAS Figure 4. [sent-685, score-0.473]
80 1: PSNR comparison between non-adaptive and adaptive acquisition methods for the reconstruction of the 'Shepp-Logan' image. [sent-686, score-0.414]
81 3 shows a comparison between PSNR values of methods 2-5 for different numbers of line projections for the 'Shepp-Logan' image. [sent-689, score-0.389]
82 Next, we show results with simulated low dose as in [3]. [sent-691, score-0.285]
83 For a selected parameter of incident photon count ! [sent-692, score-0.138]
84 4 we see a comparison of ATA and NAS using dose simulation for the Shepp-Logan image. [sent-712, score-0.221]
85 We see that the image quality produced by ATA is higher for a smaller number of line projections. [sent-713, score-0.184]
86 5 shows a clear advantage of ATA over the compared methods on the 'Zubal Head' image under a dose simulation. [sent-715, score-0.221]
87 6, we see a plot of PSNR reconstruction values at various simulated dose levels for the Shepp Logan image. [sent-717, score-0.365]
88 2: PSNR comparison between non-adaptive and adaptive acquisition methods for the reconstruction of the 'Zubal-head' image. [sent-719, score-0.414]
89 100 0 500 1000 1500 Number of line projections 2000 2500 Figure 4. [sent-720, score-0.389]
90 We note that currently the running times of the ATA algorithm, simulated on Matlab, are about 7-10 slower than the non-adaptive methods (NAS, NAF) for the same number of line projections. [sent-722, score-0.207]
91 This relates to the choice of M , the number of new line projections introduced at each iteration. [sent-723, score-0.389]
92 For a given number of line projections L , the choice M ? [sent-724, score-0.389]
93 4: PSNR comparison between ATA and NAS for the reconstruction of the 'Shepp-Logan' image at simulated incident photon count ! [sent-727, score-0.264]
94 5: PSNR comparison between ATA and NAS for the reconstruction of the 'Zubal-head' image at simulated incident photon count ! [sent-731, score-0.264]
95 6: PSNR comparison between the ATA algorithm and NAS after their application to the 'Shepp-Logan' image for various simulated incident photon counts . [sent-734, score-0.179]
96 Conclusion and Future Work In this paper we propose a mathematical model for adaptive CT acquisition whose theoretical goal is to radically reduce dosage levels, while maintaining high quality reconstruction. [sent-745, score-0.516]
97 We presented numerical simulations that demonstrate the potential of the mathematical model of adaptive acquisition and compared our results to known limited angle, non adaptive acquisition methods. [sent-746, score-0.811]
98 We plan to explore an efficient updating scheme that allows the current approximated image to be modified solely by the newly acquired line projections and by the previous approximated image. [sent-752, score-0.505]
99 Furthermore, it should be interesting to test other TV solvers such as [19] and see if they (or a modified version of them) are better suited to the adaptive scheme proposed in this paper. [sent-754, score-0.18]
100 “MBIR aims to outshine ASIR for sharpness, CT dose reduction”, article in AuntMinnie, 18th May, 2010. [sent-781, score-0.221]
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