連祥媛，孔慧華，潘晉孝*，高文波，王 攀

多通道聯(lián)合的廣義總變分能譜CT重建

連祥媛1,2，孔慧華1,2，潘晉孝1,2*，高文波1,2，王 攀3

1中北大學(xué)理學(xué)院，山西 太原 030051；2信息探測(cè)與處理山西省重點(diǎn)實(shí)驗(yàn)室，山西 太原 030051；3湖南云箭集團(tuán)有限公司，湖南 辰溪 419503

基于光子計(jì)數(shù)探測(cè)器的能譜CT在材料分解、組織表征、病變檢測(cè)等應(yīng)用中具有巨大的潛力。在重建過(guò)程中，通道數(shù)的增加會(huì)造成單通道中光子數(shù)減少，從而導(dǎo)致重建圖像質(zhì)量下降，難以滿足實(shí)際需求。本文從能譜CT重建的角度出發(fā)，將廣義總變分向矢量延伸，利用奇異值的稀疏性，促進(jìn)圖像梯度的線性依賴，提出一種基于核范數(shù)的多通道聯(lián)合廣義總變分的能譜CT重建算法。在圖像重建過(guò)程中，多層共享結(jié)構(gòu)信息，同時(shí)保留獨(dú)特的差異。實(shí)驗(yàn)結(jié)果表明，本文提出的算法在抑制噪聲的同時(shí)，能夠更有效地恢復(fù)圖像細(xì)節(jié)及邊緣信息。

CT重建；能譜CT；廣義總變分；核范數(shù)；多通道聯(lián)合

1 引 言

在基于光子計(jì)數(shù)探測(cè)器(photon-counting detector，PCD)的能譜CT[1]中，PCD技術(shù)消除了電子噪聲，提供了比傳統(tǒng)CT探測(cè)器更高的信噪比[2]。然而，PCD技術(shù)仍然存在兩個(gè)主要的問題，首先，單個(gè)能量通道只包含總光子的一小部分；其次，大部分PCD只能承受有限的計(jì)數(shù)率，所以從PCD獲得的多通道投影通常包含非常強(qiáng)的泊松噪聲。因此，開發(fā)能譜CT重建算法對(duì)于改善臨床應(yīng)用具有重要意義[3]。

壓縮感知理論的出現(xiàn)，使正則化約束項(xiàng)的迭代類重建算法發(fā)揮出巨大的潛力。能譜CT重建中，正則化約束項(xiàng)以先驗(yàn)圖像引導(dǎo)和稀疏性條件為主。先驗(yàn)圖像引導(dǎo)以Yu采用全光譜圖像作為先驗(yàn)圖像的壓縮感知(prior image constrained compressed sensing，PICCS)算法[4]為代表，稀疏性條件[5-8]包括全變分、緊框架、小波和字典學(xué)習(xí)等已被應(yīng)用于CT重建，并取得了不同程度的成功。受空間域信息相關(guān)性的啟發(fā)，Zhang等[9]將總變分與能譜均值相結(jié)合，提高重建圖像的質(zhì)量。Li等[10]提出一種能譜非局部均值的方法，利用圖像相關(guān)性來(lái)抑制噪聲和條紋偽影。Hu等[11]利用張量字典塊的稀疏性表示來(lái)提高重建質(zhì)量。陳佩君等[12]提出一種總變分與傳統(tǒng)張量字典學(xué)習(xí)結(jié)合的重建算法，可進(jìn)一步恢復(fù)圖像的微小結(jié)構(gòu)，有效抑制噪聲。為了更好地利用能譜CT在能量軸方向的信息相關(guān)性，Rigie等[13]將總變分向矢量延伸，提出了總核變分(total nuclear variation，TNV)正則化方法，可以更好地保存圖像特征。Niu等[14]提出了一種alpha發(fā)散約束的廣義總變分(total generalized variation，TGV)方法用于稀疏視角X射線CT圖像重建，有效地消除總變分正則化中經(jīng)常出現(xiàn)的階梯狀和斑片狀偽影。

為了更好地使用空間的結(jié)構(gòu)信息，本文將單通道的TGV向矢量延伸，提出一種基于核函數(shù)的多通道聯(lián)合TGV的能譜重建算法，簡(jiǎn)稱Mutli-NTGV。利用核范數(shù)作為約束，增強(qiáng)通道間耦合，恢復(fù)圖像的結(jié)構(gòu)特征。

2 理論推導(dǎo)

2.1 單通道TGV正則化

2.2 基于核范數(shù)的多通道TGV正則化

當(dāng)兩幅圖像擁有相同的曲線時(shí)，兩幅圖像具有相同方向的梯度，反之亦然[16]。如果各通道圖像的所有梯度向量是平行的或反平行的，那么雅可比矩陣的秩將是1，因此只有一個(gè)非零奇異值。在此基礎(chǔ)上，若各通道間圖像梯度平行，會(huì)使得核范數(shù)最小，將TGV推向矢量場(chǎng)形式如下：

類似地：

其中：

矢量場(chǎng)的離散梯度算子為

對(duì)稱化梯度算子為

若：

則：

3 算法模型

3.1 基于核范數(shù)的多通道聯(lián)合TGV能譜CT重建算法

在能譜CT的不同能量通道下，圖像的梯度信息是相似的。為了更好地克服基于導(dǎo)數(shù)的正則化方法的缺陷，有效地利用通道間的結(jié)構(gòu)化信息。提出一種基于核范數(shù)的多通道聯(lián)合TGV能譜CT重建算法，其目標(biāo)函數(shù)可以表述為以下凸極小化問題:

3.2 模型求解

式(6)中含有兩個(gè)變量，可以采用交替迭代的方法優(yōu)化，將上述式子分裂成兩個(gè)子問題：

重新整理正則項(xiàng)的形式，將式(9)的最小化問題表述為

采用一階原始—對(duì)偶算法解決上述鞍點(diǎn)問題。

具體的迭代步驟為

在此過(guò)程中，青少年價(jià)值觀與道德判斷將會(huì)得到進(jìn)一步提升，出現(xiàn)攻擊行為的可能性也會(huì)大大降低。與此同時(shí)，群體觀與自我人際價(jià)值感之間的關(guān)系主要呈現(xiàn)負(fù)相關(guān)關(guān)系。主要是指越是認(rèn)為自我觀重要的個(gè)體，個(gè)體存在的人際價(jià)值感愈加淡薄，相反則反之?？梢哉f(shuō)，學(xué)校方面必須加強(qiáng)對(duì)青少年價(jià)值觀與道德判斷方面的教育能力，盡量規(guī)避青少年攻擊行為的出現(xiàn)。為了進(jìn)一步提升青少年價(jià)值觀與道德判斷能力，教師可以適當(dāng)提升關(guān)于這方面的教育能力，確保青少年規(guī)范自身行為。

其中：離散梯度算子的逆算子為

對(duì)稱化算子的逆算子為

能譜的Mutil-NTGV算法步驟：

4 實(shí)驗(yàn)結(jié)果

4.1 實(shí)驗(yàn)設(shè)置

為驗(yàn)證本文算法的有效性，采用濾波反投影重建算法(FBP)、聯(lián)合代數(shù)重建算法(SART)、總變分正則項(xiàng)的迭代重建算法(TV)、基于張量的字典學(xué)習(xí)算法(TDL)、廣義總變分正則化的迭代重建算法(TGV)為比較算法。所用算法都是用MATLAB和C++的混合模式實(shí)現(xiàn)的，接口在MATLAB中實(shí)現(xiàn)，所有的大規(guī)模計(jì)算部分在C++中實(shí)現(xiàn)，并通過(guò)MEX函數(shù)進(jìn)行編譯。本文采用峰值信噪比(PSNR)、歸一化均方根誤差(NRMSE)與結(jié)構(gòu)相似度指數(shù)(SSIM)定量評(píng)價(jià)各方法性能，驗(yàn)證算法的有效性。

4.2 小鼠胸腔仿真實(shí)驗(yàn)

圖1為FBP、SART、TV、TDL、TGV與Multi-NTGV算法迭代30次的重建效果圖。

從圖1可以看出，F(xiàn)BP、SART重建的結(jié)果中噪聲很強(qiáng)，TDL、TGV與Mutli-NTGV都可以不同程度地減少各個(gè)能譜通道下噪聲的影響。在重建效果的對(duì)比中，可以明顯地看出TV在平滑噪聲的過(guò)程中，容易產(chǎn)生階梯狀偽影，導(dǎo)致重建效果不佳。TDL平滑效果較好，但無(wú)法區(qū)別噪聲與細(xì)節(jié)，局部去噪效果不佳。TGV很好地克服了TV中所產(chǎn)生的階梯偽影的狀況，保存更多的細(xì)節(jié)，但在邊緣部分會(huì)產(chǎn)生一點(diǎn)弱化的效果。本文提出的方法利用了通道間的結(jié)構(gòu)化信息，不僅有效地抑制了噪聲，而且對(duì)于微小的結(jié)構(gòu)也保持得很好，邊緣更加清晰。

五種算法重建過(guò)程中的NRMSE、PSNR與SSIM指標(biāo)如圖2所示，F(xiàn)BP重建效果的評(píng)價(jià)指標(biāo)如表1所示。由圖2可以看出，在前三個(gè)通道中，本文所提出的算法明顯優(yōu)于其他算法，在第四通道中，雖然Mutli-NTGV算法與TDL、TGV的數(shù)量性評(píng)價(jià)結(jié)果相近，但是從重建圖效果可以看出Mutli-NTGV方法優(yōu)于其他方法。實(shí)驗(yàn)結(jié)果表明，本文所提的方法在各個(gè)通道中，會(huì)保持較大的優(yōu)勢(shì)，極大地提升了重建圖像的質(zhì)量。

4.3 臨床小鼠實(shí)驗(yàn)

為了進(jìn)一步驗(yàn)證所提出方法的有效性，采用了MARS(Medipix All Resolution System)微型CT上采集的來(lái)自真實(shí)臨床前小鼠的投影。電壓設(shè)置為120 kV，電流為175 mA。從照射源到系統(tǒng)原點(diǎn)的距離為158 mm，到探測(cè)器的距離為255 mm。在整個(gè)掃描范圍內(nèi)均勻收集了13個(gè)能量通道的371個(gè)投影視圖。探測(cè)器一行1024個(gè)元素，單位長(zhǎng)度為55mm。

圖3為各算法對(duì)臨床前小鼠的重建效果圖，從上到下依次展示的是第1、3、5和7通道的效果圖。實(shí)際小鼠切片實(shí)驗(yàn)中，各算法重建結(jié)果與小鼠胸腔仿真實(shí)驗(yàn)類似，F(xiàn)BP與SART重建效果中含有大量的噪聲，其余方法都在一定程度上達(dá)到了去噪的效果，從實(shí)驗(yàn)結(jié)果可以明顯看出，在各個(gè)重建算法中，Mutli-NTGV在臨床小鼠的重建效果較好。

圖1 小鼠模型的重建結(jié)果。從左到右的方法依次為FBP、SART、TV、TDL、TGV與Mutli-NTGV，從上到下依次為1到4通道

圖2 小鼠模型重建效果的數(shù)量性評(píng)價(jià)指標(biāo)。 從左到右行依次為NRMSE、PSNR、SSIM，從(a)到(d)行依次為1到4通道

表1 FBP重建的數(shù)量性評(píng)價(jià)指標(biāo)

圖3 臨床小鼠模型的重建結(jié)果。從左到右的方法依次為FBP、SART、TV、TDL、TGV與Mutli-NTGV,從上到下依次為1, 3, 5, 7通道

5 結(jié) 論

本文針對(duì)能譜CT，提出了一種基于核范數(shù)的多通道聯(lián)合廣義總變分能譜CT重建算法，將TGV推向矢量化，采用逐像素的更新方式，使用多通道聯(lián)合的二階廣義總變分作為專用的正則化函數(shù)在圖像重建步驟中耦合多個(gè)通道，耦合分別在一階和二階導(dǎo)數(shù)的水平上用核范數(shù)和F范數(shù)約束來(lái)實(shí)現(xiàn)，促進(jìn)了多個(gè)圖像通道的梯度的線性依賴，從而使邊緣對(duì)齊。該方法在噪聲條件下，可以保持更多的細(xì)節(jié)與更清晰的邊緣。仿真數(shù)據(jù)與實(shí)際數(shù)據(jù)的運(yùn)行結(jié)果表明，本文提出的算法具有較好的魯棒性。但是，本文算法中參數(shù)較多，采用經(jīng)驗(yàn)選取，比較耗費(fèi)時(shí)間，后續(xù)研究中可以研究采用更靈活的參數(shù)選取模型。

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Joint multi-channel total generalized variational algorithm for spectral CT reconstruction

Lian Xiangyuan1,2, Kong Huihua1,2,Pan Jinxiao1,2*, Gao Wenbo1,2, Wang Pan3

1School of Science, North University of China, Taiyuan, Shanxi 030051, China;2Shanxi Key Laboratory of Signal Capturing & Processing, North University of China, Taiyuan, Shanxi 030051, China;3Hunan Vsngusrd Group. Co. Ltd, Chenxi, Hunan 419503, China

The reconstruction results of the mouse model by Mutli-NTGV

Overview:Spectral computed tomography (CT) based on photon-counting detectors, has great potential in material decomposition, tissue characterization, lesion detection, and other applications. During the reconstruction, the increase of the number of channels will reduce the photon number in a single channel, resulting in the decline of the quality of the reconstructed image, which is difficult to meet the actual needs. To improve the quality of image reconstruction, this paper proposes a joint multi-channel total generalized variational based on the unclear norm for spectral CT reconstruction. Firstly, in the reconstruction for spectral CT, the image structure of each channel is highly similar, and the reconstruction of a single channel will ignore the structural information of each channel. Second, gradient information contains a lot of structured information and features of the image. When two images have the same curve, the two images have the same direction gradient and the converse is also true. In order to better utilize the image’s structural information between channels, the new regularization function is applied to spectral CT reconstruction. The research shows that if the edges of the two images are aligned, the two images have the same gradient. The image gradients between channels are parallel, which will minimize the nuclear norm. The algorithm will extend total generalized variation to the vector, with the aim of overcoming defects of existing derivative-based regularization. The paper proposed a joint multi-channel total generalized variational for spectral CT reconstruction, employing a vectorial second-order total generalized variation function as joint regularization. The method adopts pixel-by-pixel updating in the image reconstruction, and the multi-channel image coupling is realized by kernel norm and F-norm constraints at the level of first and second derivatives. The nuclear norm and frobenius norm coupling promote joint sparsity of the edge sets and dependence of the gradients. Joint multi-channel total generalized variational is used to promote the linear dependence of the multi-channel image’s gradient so that the image edges of each channel are aligned. The structural information of the multi-channel image is shared during the image reconstruction process while unique differences are preserved. The experiment was done on a numerical mouse thorax phantom and clinical mouse data. The quantitative results of peak signal to noise ratio (PSNR), normalized root mean square error (NRMSE) and structure similarity index (SSIM) show that the proposed algorithm greatly improves the image quality. Experimental results show that the proposed algorithm can effectively recover image details and marginal information while suppressing noise.

Lian X Y, Kong H H,Pan J X,Joint multi-channel total generalized variational algorithm for spectral CT reconstruction[J]., 2021, 48(9): 210211; DOI:10.12086/oee.2021.210211

Joint multi-channel total generalized variational algorithm for spectral CT reconstruction

Lian Xiangyuan1,2, Kong Huihua1,2,Pan Jinxiao1,2*, Gao Wenbo1,2, Wang Pan3

1School of Science, North University of China, Taiyuan, Shanxi 030051, China;2Shanxi Key Laboratory of Signal Capturing & Processing, North University of China, Taiyuan, Shanxi 030051, China;3Hunan Vsngusrd Group. Co. Ltd, Chenxi, Hunan 419503, China

Spectral computed tomography (CT) based on photon-counting detectors, has great potential in material decomposition, tissue characterization, lesion detection, and other applications. During the reconstruction, the increase of the number of channels will reduce the photon number in a single channel, resulting in the decline of the quality of the reconstructed image, which is difficult to meet the actual needs. To improve the quality of image reconstruction, joint multi-channel total generalized variational based on the unclear norm for spectral CT reconstruction was proposed in this paper. The algorithm will extend total generalized variation to the vector, and the sparsity of singular values is used to promote the linear dependence of the image gradient. The structural information of the multi-channel image is shared during the image reconstruction process while unique differences are preserved. Experimental results show that the proposed algorithm can effectively recover image details and marginal information while suppressing noise.

CT reconstruction; spectral CT; total generalized variation; nuclear norm; joint multi-channel

連祥媛，孔慧華，潘晉孝，等. 多通道聯(lián)合的廣義總變分能譜CT重建[J]. 光電工程，2021，48(9): 210211

Lian X Y, Kong H H,Pan J X,Joint multi-channel total generalized variational algorithm for spectral CT reconstruction[J]., 2021, 48(9): 210211

10.12086/oee.2021.210211

TP391

A

2021-06-23；

2021-08-26

國(guó)家自然科學(xué)基金資助項(xiàng)目(61801437，61871351，61971381)

連祥媛(1994-)，女，碩士研究生，主要從事圖像重建與圖像處理方面的研究。E-mail：1393550566@qq.com

潘晉孝(1966-)，男，博士，教授，主要從事信息處理與圖像重建方面的研究。E-mail：panjx@nuc.edu.cn

National Natural Science Foundation of China (61801437, 61871351, 61971381)

* E-mail: panjx@nuc.edu.cn