Title

題目

Outlier detection in cardiac diffusion tensor imaging: Shot rejection or robust fitting?

心臟擴散張量成像中的異常值檢測：射擊拒絕還是穩(wěn)健擬合？

01

文獻速遞介紹

心臟擴散張量成像（Cardiac Diffusion Tensor Imaging, cDTI）是一種非侵入性、無需造影劑且無電離輻射的成像方法，通過檢測水分子（各向異性）隨機運動來表征心肌微觀結構（Reese et al., 1995; Scollan et al., 1998）。擴散模型可用于擬合在不同方向和幅度擴散敏感條件下獲取的圖像。通常對每個體素擬合擴散張量模型（Basser et al., 1994），從中可提取平均擴散率（MD）、分數(shù)各向異性（FA）以及組織方向和組織排列的生物標記（如“片層”角E2A）等參數(shù)（Basser and Pierpaoli, 1996; Kung et al., 2011; Ferreira et al., 2014; Nielles-Vallespin et al., 2017）。這些指標已被證明在病理與健康狀態(tài)之間存在差異，可為心肌疾病對組織結構和排列的影響提供深入見解。

與其他器官的擴散張量成像相比，cDTI面臨更多挑戰(zhàn)，主要源于心臟和呼吸運動，其幅度遠大于水分子的擴散運動。盡管序列設計和運動補償方案已取得顯著進展（Stoeck et al., 2016），但通常需要重復成像以提供數(shù)據(jù)冗余并補償較低的信噪比（SNR）。呼吸運動通常通過后期圖像配準進行校正，但除了熱噪聲外，圖像間仍可能存在結構化差異，這種“生理噪聲”包括心臟相位、成像平面相對心臟位置的變化等。異常值處理方法 在cDTI中，受偽影影響的圖像是常見問題，如信號丟失、配準失敗或過強的生理噪聲。目前，受損圖像通常通過人工檢查識別并移除，而自動化方法的研究較少（Ferreira et al., 2020; Coveney et al., 2023）。盡管腦擴散MRI領域中已有豐富的異常值處理方法，如穩(wěn)健擬合，但這些方法在cDTI中的應用仍然有限。單體素異常值檢測（SVOD）：此方法通過迭代擬合結合穩(wěn)健估計器來檢測異常值（如RESTORE算法），對每個體素獨立處理（Chang et al., 2005, 2012）。然而，SVOD可能無法檢測圖像中明顯的偽影。

多體素異常值檢測（MVOD）：此方法利用多個體素的信息來檢測異常值，例如通過圖像間的相似性或模型預測與觀測值之間的均方誤差（Coveney et al., 2023）。MVOD可克服SVOD的局限，尤其在處理復雜偽影區(qū)域時表現(xiàn)更優(yōu)。研究目標與結果 本研究提出了結合穩(wěn)健M估計器的迭代加權最小二乘（IRLS）方法，并將SVOD和MVOD應用于健康志愿者和肥厚型心肌病（HCM）患者的cDTI數(shù)據(jù)。結果表明：穩(wěn)健擬合優(yōu)于射擊拒絕（SR）：穩(wěn)健擬合在MD、FA和E2A指標上生成了更大的組間差異，并具有更高的統(tǒng)計顯著性。MVOD優(yōu)于SVOD：在MD和FA的組間差異上，MVOD表現(xiàn)更優(yōu)，并在合成實驗中更有效地從受損數(shù)據(jù)中恢復擴散指標。SVOD的不足影響有限：盡管SVOD未能識別所有偽影信號，但對擴散張量模型參數(shù)的穩(wěn)健估計并無顯著影響。結論 本研究表明，穩(wěn)健擬合結合SVOD或MVOD可以完全取代射擊拒絕用于cDTI異常值處理，從而提高數(shù)據(jù)質量并增強疾病相關指標的敏感性。

Aastract

摘要

Cardiac diffusion tensor imaging (cDTI) is highly prone to image corruption, yet robust-fitting methods arerarely used. Single voxel outlier detection (SVOD) can overlook corruptions that are visually obvious, perhapscausing reluctance to replace whole-image shot-rejection (SR) despite its own deficiencies. SVOD’s deficienciesmay be relatively unimportant: corrupted signals that are not statistical outliers ay not be detrimental.Multiple voxel outlier detection (MVOD), using a local myocardial neighbourhood, may overcome the shareddeficiencies of SR and SVOD for cDTI while keeping the benefits of both. Here, robust fitting methods usingM-estimators are derived for both non-linear least squares and weighted least squares fitting, and outlierdetection is applied using (i) SVOD; and (ii) SVOD and MVOD. These methods, along with non-robust fittingwith/without SR, are applied to cDTI datasets from healthy volunteers and hypertrophic cardiomyopathypatients. Robust fitting methods produce larger group differences with more statistical significance for MD,FA, and E2A, versus non-robust methods, with MVOD giving the largest group differences for MD and FA.Visual analysis demonstrates the superiority of robust-fitting methods over SR, especially when it is difficultto partition the images into good and bad sets. Synthetic experiments confirm that MVOD gives lowerroot-mean-square-error than SVOD.

心臟擴散張量成像（cardiac Diffusion Tensor Imaging, cDTI）非常容易受到圖像損壞的影響，但穩(wěn)健擬合方法很少被使用。單體素異常值檢測（Single Voxel Outlier Detection, SVOD）可能會忽略一些肉眼明顯的損壞，這可能導致盡管存在不足，仍傾向于使用全圖像射擊拒絕（Shot Rejection, SR）。然而，SVOD的不足可能并不重要：并非統(tǒng)計學上的異常信號未必對結果有顯著影響。多體素異常值檢測（Multiple Voxel Outlier Detection, MVOD），通過利用局部心肌鄰域，可能克服SR和SVOD在cDTI中的共同缺陷，同時保留兩者的優(yōu)勢。本研究中，為非線性最小二乘法和加權最小二乘法擬合推導了使用M估計器的穩(wěn)健擬合方法，并將異常值檢測應用于以下兩種情況：(i) 僅使用SVOD；(ii) 同時使用SVOD和MVOD。這些方法以及非穩(wěn)健擬合（有/無SR）被應用于來自健康志愿者和肥厚型心肌病患者的cDTI數(shù)據(jù)集。

結果表明，穩(wěn)健擬合方法在MD（平均擴散率）、FA（分數(shù)各向異性）和E2A（擴展軸比）方面，與非穩(wěn)健方法相比，能夠產(chǎn)生更大的組間差異，并具有更顯著的統(tǒng)計學意義。MVOD在MD和FA的組間差異中表現(xiàn)出最大優(yōu)勢。視覺分析顯示，當難以將圖像明確劃分為“良好”或“損壞”時，穩(wěn)健擬合方法明顯優(yōu)于SR。合成實驗進一步證實，MVOD比SVOD具有更低的均方根誤差（Root Mean Square Error, RMSE）。

Method

方法

DTI data are a series of 𝑛 images 𝑖 = 1 …𝑛 obtained with diffusionweightings 𝑏𝑖 and directions (unit vectors) 𝐠𝐢 = (𝑔𝑖,𝑥, 𝑔𝑖,𝑦, 𝑔𝑖,𝑧). Considering a single voxel, the noisy signal 𝑦𝑖 observed in image 𝑖 can be relatedto the signal model 𝑓(𝜽*, 𝑏𝑖 , 𝐠𝐢 ), with parameters 𝜽 and the error 𝜖𝑖 as:𝑦𝑖 = 𝑓(𝜽, 𝑏𝑖 , 𝐠𝐢 ) + 𝜖𝑖(1)We focus on the diffusion tensor model of the signal:𝑓(𝜽, 𝑏𝑖 , 𝐠𝐢) = 𝑆0 exp ( ?𝑏𝑖 𝐠𝐢 𝐃 𝐠 𝑇 𝐢 ) = exp ( 𝐱𝑖𝜽 𝑇 )(2)where 𝐱𝑖 = ( 1,?𝑏𝑖 𝑔𝑖,𝑥 2 ,?2𝑏**𝑖 𝑔𝑖,𝑥𝑔𝑖,𝑦,?2𝑏𝑖 𝑔𝑖,𝑥𝑔𝑖,𝑧,?𝑏𝑖 𝑔𝑖,𝑦 2 ,?2𝑏𝑖 𝑔𝑖,𝑦𝑔𝑖,𝑧,?𝑏𝑖 𝑔𝑖,𝑧 2 ) , and model parameters 𝜃 = ( log(𝑆0 ), 𝐷𝑥𝑥, 𝐷𝑥𝑦, 𝐷𝑥𝑧, 𝐷𝑦𝑦,𝐷𝑦𝑧, 𝐷*𝑧𝑧 ) . The methods presented here can apply to other models, suchas the diffusion kurtosis imaging (DKI) model. We denote the standarddeviation of the Gaussian noise in the original complex images as 𝜎.We consider magnitude images, for which the noise distribution can becomplicated (St-Jean et al., 2020); we assume Rician distributed errorfor simplicity (Cárdenas-Blanco et al., 2008), and note that 𝜖**𝑖 resultsfrom both the Gaussian noise in the complex images and the magnitudeoperation.

DTI數(shù)據(jù)是一系列包含n張圖像（𝑖 = 1 …𝑛）的集合，這些圖像具有不同的擴散加權系數(shù) 𝑏 和方向（單位向量）𝐠𝐢 = (𝑔𝑖,𝑥, 𝑔𝑖,𝑦, 𝑔𝑖,𝑧）?？紤]單個體素，在圖像𝑖中觀測到的帶噪信號 𝑦𝑖 與信號模型 𝑓(𝜽*, 𝑏𝑖* , 𝐠𝐢 ) 的關系可以表示為：𝑦𝑖 = 𝑓(𝜽, 𝑏𝑖, 𝐠𝑖) + 𝜖𝑖 \tag{1}其中，參數(shù)為 𝜽，誤差為 𝜖**𝑖。我們重點研究信號的擴散張量模型，其形式為：𝑓(𝜽, 𝑏𝑖, 𝐠𝑖) = 𝑆0 \exp( -𝑏𝑖 \𝐠𝑖 \𝐃 \𝐠𝑖^𝑇 ) = \exp( \𝐱_𝑖 \𝜽^𝑇 ) \tag{2}其中，𝐱𝑖=(1,?𝑏𝑖𝑔𝑖,𝑥2,?2𝑏𝑖𝑔𝑖,𝑥𝑔𝑖,𝑦,?2𝑏𝑖𝑔𝑖,𝑥𝑔𝑖,𝑧,?𝑏𝑖𝑔𝑖,𝑦2,?2𝑏𝑖𝑔𝑖,𝑦𝑔𝑖,𝑧,?𝑏𝑖𝑔𝑖,𝑧2)𝐱𝑖 = \big( 1, -𝑏𝑖 𝑔{𝑖,𝑥}^2, -2𝑏𝑖 𝑔{𝑖,𝑥}𝑔{𝑖,𝑦}, -2𝑏𝑖 𝑔{𝑖,𝑥}𝑔{𝑖,𝑧}, -𝑏𝑖 𝑔{𝑖,𝑦}^2, -2𝑏𝑖 𝑔{𝑖,𝑦}𝑔{𝑖,𝑧}, -𝑏𝑖 𝑔{𝑖,𝑧}^2 \big)

模型參數(shù)為 \𝜃 = \big( \log(𝑆0), 𝐷{𝑥𝑥}, 𝐷{𝑥𝑦}, 𝐷{𝑥𝑧}, 𝐷{𝑦𝑦}, 𝐷{𝑦𝑧}, 𝐷_{𝑧𝑧} \big)。本文所述方法同樣適用于其他模型，例如擴散峰度成像（Diffusion Kurtosis Imaging, DKI）模型。我們將原始復圖像中高斯噪聲的標準差記為 𝜎。對于幅度圖像，其噪聲分布可能較為復雜（St-Jean et al., 2020）；為了簡化，假設誤差服從Rician分布（Cárdenas-Blanco et al., 2008）。需要注意的是，𝜖**𝑖 既來自復圖像中的高斯噪聲，也來自幅度運算的影響。

Conclusion

結論

In this work, we have attempted to answer the question of whetherrobust-estimation can replace shot-rejection in cardiac diffusion tensorimaging, and whether the deficiencies of single voxel outlier detection are important for recovering correct diffusion tensor metrics. Wehave presented robust fitting with M-estimators followed by singlevoxel-outlier-detection and multiple-voxel-outlier-detection. Our results demonstrate that MVOD is more robust than SVOD, particularlyfor large numbers of corrupted images and low SNR. Nonetheless, theimprovement of MVOD over SVOD seems relatively minor for cardiacDTI, even as SVOD gives large improvements over shot-rejection,suggesting that researchers need not worry that SVOD misses signalsthat would be identified by shot-rejection, even if MVOD could identifythese signals. We recommend cDTI to start using robust-estimation inplace of shot-rejection.

在本研究中，我們探討了穩(wěn)健估計是否可以替代射擊拒絕（shot-rejection）用于心臟擴散張量成像（cardiac DTI），以及單體素異常值檢測（SVOD）的不足是否對恢復正確的擴散張量指標有重要影響。我們提出了結合M估計器的穩(wěn)健擬合方法，并輔以單體素異常值檢測和多體素異常值檢測（MVOD）。

研究結果表明，MVOD比SVOD更穩(wěn)健，特別是在損壞圖像數(shù)量較多和信噪比（SNR）較低的情況下。然而，對于心臟DTI而言，MVOD相較于SVOD的改進相對較小，而SVOD相較于射擊拒絕則表現(xiàn)出顯著改進。這表明研究者無需擔心SVOD會遺漏射擊拒絕能夠識別的信號，即使這些信號可以被MVOD檢測到。

因此，我們建議在心臟DTI中使用穩(wěn)健估計以替代射擊拒絕。

Results

結果

3.1. Application to datasets

Fig. 2 shows global MD, FA, and absolute E2A, as fit to the originaldatasets, for the methods explained in Section 2.4. Table 1 showsthe group differences of the mean and median, and the 𝑝-value, forthe methods in Fig. 2. Table 1 shows that the group differences between Volunteers and HCM Patients was largest for MVOD methodsfor both MD and FA, and the significance was also higher (the 𝑝-valuewas lower). The SVOD methods showed the second largest differencebetween the groups for MD and FA, except for RWLS where shotrejection (WLS_SR) showed a slightly larger difference for mean MD(although the values are nearly identical) — note that difference ofmedians was nontheless much bigger for RWLS MD than for WLS_SRMD, and RWLS had nearly 6 times decrease in 𝑝-value compared toWLS_SR. It is visually clear that robust-fitting methods seem to reducethe spread of metric values in the healthy volunteers, especially for WLSmethods. Importantly for WLS, the 𝑝-value for MD does not pass thestandard significance test of 𝑝 < 0.05 at least without shot rejection,and the robust-fitting methods are much more significant still. For E2A,either SVOD or MVOD gave the largest group differences — there areseveral ties in the 𝑝-value due to the nature of the Wilcoxon signedrank test. Care should be taken to interpret the box-plots (dependingon whether points are classed as ‘fliers’, the box and whiskers will bedrawn differently, which can lead to discontinuous changes in the boxsizes). For WLS and WLS_SR methods, the box plot for volunteer E2Aseems narrower than for the robust methods, but in fact this is becausethe robust methods do not produce as many flier (out-lying) values ofE2A, so in this respect the ‘spread’ is reduced for robust methods.Figs. 3 and 4 show MD, FA, E2A and HA (helix angle) maps foran example volunteer and HCM patient respectively. These exampleswere the subjects with the largest MD values for the non-robust WLSmethods. Furthermore, even attempts at manual shot-rejection wereextremely difficult in these cases, since it was difficult to separate goodfrom bad images: even the better images appeared to contain large shotto-shot variations and corruptions at the sub-image level. Note that,we have shown all the myocardium in Figs. 3 and 4, although regionsof distortion and isolated artefacts were excluded from global metriccalculations, as explained in Appendix B.Fig. 3 shows a marked decrease in MD across the myocardium forrobust fitting compared to non-robust fitting (with or without shotrejection), especially around the septum. This decrease is slightly largerfor MVOD than SVOD, especially in the basal slice. The effects on FA areequivalent but in the opposite direction (an increase). The transmuralvariation of HA from right-handed (red) to left-handed (blue) is alsosuperior in MVOD, with the blue fibres in the bottom left of the midslice only convincingly recovered with MVOD. Additionally, SVOD andMVOD both show significantly more convincing transmural variationfrom red (right-handed) to green (circumferential) to blue (left-handed)in the septum of the basal slice, where the non-robust (with or withoutSR) methods show an extended region of red in the same area.Fig. 4 shows a strong increase in (magnitude of) E2A in the septumfor robust methods, particularly in the apical and basal slices. Thesechanges are also accompanied by a complete recovery of HA by robustfitting methods, whereas there are obvious corruptions of HA in bothnon-robust methods (including shot-rejection). Especially important isthe recovery of right-handed (red) HA in the septum of the basal slice,for both robust methods, whereas these right-handed fibres appearto be missing for not-robust methods. SVOD and MVOD appear todecrease MD substantially, moreso for MVOD than for SVOD.We reviewed differences in fitting methods for every subject byeye. Overall, the visual differences in diffusion measure maps betweenSVOD and MVOD seemed quite minor for most subjects. This seemsconsistent with Fig. 2 and Table 1: while MVOD modifies the results further in the ‘same direction’ in which SVOD improves uponshot-rejection, the improvement upon SVOD would appear relativelyminor in comparison to the improvement that SVOD makes over shotrejection.The RMSE of the predicted versus observed signals, correspondingto Figs. 3 and 4, are shown in Fig. 5. What is especially clear isthat while shot-rejection can reduce the RMSE of the fit to somedegree, the robust-estimation with SVOD and MVOD do substantiallybetter, leaving a nearly uniform (and lower) RMSE except in regions ofisolated artefact. Note that, after robust-fitting, these artefact regionsare extremely easy to identify from RMSE, suggesting that robust-fittingis able to turn RMSE into a useful metric for determining the qualityof the fit in a way that becomes independant of outliers and corruptionin some images, therefore leaving only the effects of artefacts thatpermeate the entire image series. Even after robust fitting, there aresome regions of elevated MD in Figs. 3 and 4 that remain, but these correspond to significantly higher RMSE than the rest of the myocardium,suggesting that these are likely to be artefacts not real features. Notethat MVOD has lower and more uniform RMSE than SVOD

3.1. 數(shù)據(jù)集的應用

圖2展示了基于第2.4節(jié)中方法的原始數(shù)據(jù)集的全局MD（平均擴散率）、FA（分數(shù)各向異性）和E2A（片層角）的結果。表1則展示了不同方法在健康志愿者和肥厚型心肌病（HCM）患者之間的組間差異，包括均值、中位數(shù)和p值。

表1顯示，對于MD和FA，MVOD方法的組間差異最大且顯著性更高（p值更低）。SVOD在MD和FA方面的組間差異次于MVOD，但對于RWLS方法，射擊拒絕（WLS_SR）在MD均值上的差異略大（盡管差異幾乎相同）。然而，RWLS在MD中位數(shù)的差異顯著更大，其p值相較于WLS_SR減少了近6倍。在健康志愿者的測量值中，穩(wěn)健擬合方法明顯減少了指標值的離散程度，尤其是WLS方法中。對于E2A，SVOD或MVOD通常表現(xiàn)出最大的組間差異，但由于Wilcoxon符號秩檢驗的性質，部分p值存在相同情況。箱線圖的解釋需要注意：基于是否將點分類為異常值，箱線圖的箱體和須狀線的繪制方式可能不同，從而導致箱體尺寸的非連續(xù)變化。在WLS和WLS_SR方法中，志愿者E2A的箱線圖看起來比穩(wěn)健方法更窄，但這是因為穩(wěn)健方法減少了異常值的數(shù)量，從而降低了“離散性”。圖3和圖4分別展示了健康志愿者和HCM患者的MD、FA、E2A和HA（螺旋角）圖。這些案例是非穩(wěn)健WLS方法中MD值最大的個體。對于這些案例，即使進行人工射擊拒絕，仍然很難區(qū)分“好圖像”和“壞圖像”，因為即使較好的圖像也在子圖像級別表現(xiàn)出顯著的偽影和變化。

圖3 相比非穩(wěn)健擬合方法（無論是否使用射擊拒絕），穩(wěn)健擬合方法顯著降低了心肌中的MD，尤其是在室間隔區(qū)域；MVOD在降低MD方面略優(yōu)于SVOD，尤其是在基底切片。FA的影響相反（顯著增加）。MVOD在螺旋角的跨壁變化中表現(xiàn)更優(yōu)，從右旋（紅色）到左旋（藍色）的過渡更清晰，尤其是中切片的左下角區(qū)域僅通過MVOD才能明顯恢復。此外，在基底切片的室間隔中，SVOD和MVOD顯示了更明顯的跨壁變化，而非穩(wěn)健方法顯示的紅色區(qū)域更廣。圖4 穩(wěn)健方法在室間隔中顯著增加了E2A的幅值，特別是在心尖和基底切片。這些變化伴隨著穩(wěn)健擬合方法對HA的完全恢復，而非穩(wěn)健方法（包括射擊拒絕）在HA中存在明顯的偽影。尤其重要的是，基底切片中室間隔區(qū)域的右旋HA（紅色）通過穩(wěn)健方法得以恢復，而非穩(wěn)健方法未能表現(xiàn)出這些右旋纖維。此外，SVOD和MVOD顯著降低了MD，且MVOD的效果優(yōu)于SVOD。

RMSE分析 圖5展示了預測信號與觀測信號之間的均方根誤差（RMSE）。射擊拒絕能夠在一定程度上降低RMSE，但穩(wěn)健擬合結合SVOD和MVOD顯著降低了RMSE，除孤立偽影區(qū)域外，其余區(qū)域RMSE接近均勻。穩(wěn)健擬合后的偽影區(qū)域在RMSE中易于識別，表明穩(wěn)健擬合方法可以將RMSE轉化為一種有效的擬合質量評估指標，能夠獨立于異常值和圖像偽影的影響，僅反映貫穿整個圖像序列的偽影效應。

盡管穩(wěn)健擬合后仍存在MD升高的區(qū)域，但這些區(qū)域的RMSE明顯高于其他心肌區(qū)域，表明它們可能是偽影而非真實特征。MVOD的RMSE比SVOD更低且更均勻，進一步證明了其優(yōu)越性。

Figure

圖

Fig. 1. Examples from 6 datasets (1 per row in each sub-figure): (a) examples where shot-rejection has (correctly) identified a corrupted image, but SVOD has not identified allmyocardial voxels in the corrupted image; (b) examples where shot-rejection has not identified a corrupted image. The ‘reference’ image shows a typical ‘good image’, while the‘a(chǎn)ccepted’ or ‘rejected’ image columns show a different image with the same diffusion weighting for the same subject. The myocardial segmentation is also shown. For SVOD andMVOD columns, black/white voxels indicate outlier/non-outlier signals respectively

圖1. 來自6個數(shù)據(jù)集的示例（每個子圖中的每行對應一個數(shù)據(jù)集）： (a) 射擊拒絕（shot-rejection）方法正確識別了損壞圖像，但單體素異常值檢測（SVOD）未能識別損壞圖像中所有的心肌體素； (b) 射擊拒絕未能識別損壞圖像的示例?！皡⒖紙D像”（reference）展示了一個典型的“良好圖像”，而“接受”（accepted）或“拒絕”（rejected）列顯示了對同一受試者在相同擴散加權下獲取的另一幅圖像。圖中同時顯示了心肌分割結果。對于SVOD和MVOD列，黑色/白色體素分別表示異常值/非異常值信號。

Fig. 2. Diffusion measures MD (×10?3 mm2/s), FA, and absolute E2A (degrees) for volunteers and HCM patients for different fitting methods.

圖2. 不同擬合方法下志愿者與HCM患者的擴散測量值：MD（×10?3 mm2/s）、FA和絕對E2A（單位：度）。

Fig. 3. Example healthy volunteer (highest MD from non-robust WLS fitting in the HV group of Fig. 2). First row: non-robust NLLS; Second row: non-robust NLLS after shot-rejection;Third row: robust NLLS with SVOD; Fourth row: robust NLLS with SVOD and MVOD (10 voxel neighbourhood).

圖3. 健康志愿者示例（來自圖2中HV組非穩(wěn)健WLS擬合的最高MD值）。 第一行：非穩(wěn)健NLLS（非線性最小二乘法）。 第二行：射擊拒絕后的非穩(wěn)健NLLS。 第三行：結合SVOD的穩(wěn)健NLLS。 第四行：結合SVOD和MVOD（10體素鄰域）的穩(wěn)健NLLS。

Fig. 4. Example HCM patient (highest MD from non-robust WLS fitting in the HCM group of Fig. 2). First row: non-robust NLLS; Second row: non-robust NLLS after shot-rejection;

Third row: robust NLLS with SVOD; Fourth row: robust NLLS with SVOD and MVOD (10 voxel neighbourhood).

圖4. HCM患者示例（來自圖2中HCM組非穩(wěn)健WLS擬合的最高MD值）。 第一行：非穩(wěn)健NLLS（非線性最小二乘法）。 第二行：射擊拒絕后的非穩(wěn)健NLLS。 第三行：結合SVOD的穩(wěn)健NLLS。 第四行：結合SVOD和MVOD（10體素鄰域）的穩(wěn)健NLLS。

Fig. 5. RMSE of fitting residuals, excluding identified outliers or excluded images (asapplied). RMSE is lower and more uniform for robust fitting methods, being lower forMVOD than SVOD, with remaining high RMSE regions seeming to indicate patches ofpersistent artefact across the image series.

圖5. 擬合殘差的均方根誤差（RMSE），不包括已識別的異常值或被排除的圖像（如適用）。 穩(wěn)健擬合方法的RMSE更低且更均勻，其中MVOD的RMSE低于SVOD。殘留的高RMSE區(qū)域似乎表明圖像序列中存在持續(xù)性偽影的區(qū)域。

Fig. 6. Global diffusion measures MD (×10?3 mm2/s), FA, and absolute E2A (degrees) for synthetically corrupted data. Each sub-plot corresponds to a different corruption type.Different percentages of images have been corrupted, and different SNRs have been used to generate Rician error. The dotted line extending across each plot is the ‘‘syntheticground truth’’ from which synthetic data (with noise and corruptions) was generated, whereas the darker solid lines are the mean of non-robust fits excluding the images thatwere artificially corrupted. The box-and-whisker plots represent 10 synthetic datasets per corruption configuration (for narrow box-plots, the median line has been removed forclarity). Different colours correspond to different fitting methods, shown in the legend. For each corruption configuration, NLLS and WLS methods are shown to the left and rightagainst a differently shaded background

圖6. 合成損壞數(shù)據(jù)的全局擴散測量結果：MD（×10?3 mm2/s）、FA和絕對E2A（單位：度）。每個子圖對應一種不同的損壞類型。

圖中顯示了不同比例的圖像被損壞，以及使用不同信噪比（SNR）生成的Rician誤差。虛線表示“合成真值”，即用于生成帶噪聲和損壞的合成數(shù)據(jù)的參考值；深色實線表示排除人為損壞圖像后的非穩(wěn)健擬合均值。

箱線圖展示了每種損壞配置下的10個合成數(shù)據(jù)集（對于較窄的箱線圖，為了更清晰移除了中位線）。不同顏色表示圖例中標注的不同擬合方法。對于每種損壞配置，NLLS方法和WLS方法分別顯示在左側和右側，并用不同的陰影背景加以區(qū)分。

Fig. 7. RMSE of global diffusion measures of MD (×10?3 mm2/s), FA, and absolute E2A (degrees) for synthetically corrupted data. Each sub-plot corresponds to a different corruptiontype. Different percentages of images have been corrupted, and different SNRs have been used to generate Rician error. The RMSE is calculated against the mean of non-robustfits excluding the images that were artificially corrupted. The box-and-whisker plots represent the RMSE scores of all subjects (for narrow box-plots, the median line has beenremoved for clarity). Different colours correspond to different fitting methods, shown in the legend. For each corruption configuration, NLLS and WLS methods are shown to theleft and right against a differently shaded background.

圖7. 合成損壞數(shù)據(jù)中全局擴散測量（MD ×10?3 mm2/s，FA和絕對E2A，以度為單位）的RMSE（均方根誤差）。每個子圖對應一種不同的損壞類型。

不同百分比的圖像被損壞，并使用不同信噪比（SNR）生成Rician誤差。RMSE是針對排除人為損壞圖像后的非穩(wěn)健擬合均值計算的。箱線圖表示所有受試者的RMSE得分（對于較窄的箱線圖，為了清晰性移除了中位線）。不同顏色表示圖例中標注的不同擬合方法。對于每種損壞配置，NLLS方法和WLS方法分別顯示在左側和右側，并用不同的陰影背景加以區(qū)分。

Table

表

Table 1Table of difference of group means, medians, and p-values, for MD, FA, absolute E2A, for several tested methods. Within each methodcategory (either NLLS or WLS) the largest difference of means and medians are shown in red, and the lowest 𝑝-value is shown in blue.

表1 MD（平均擴散率）、FA（分數(shù)各向異性）和絕對E2A的組均值差異、中位數(shù)差異及p值的表格比較。 在每種方法類別（NLLS或WLS）中，均值和中位數(shù)差異最大的結果用紅色標注，p值最低的結果用藍色標注。