International Journal of Computer Assisted Radiology and Surgery (2018) 13:1871–1880https://doi.org/10.1007/s11548-018-1840-5

ORIG INAL ART ICLE

Using the variogram for vector outlier screening: applicationto feature-based image registration

Jie Luo1,2 · Sarah Frisken1 · Ines Machado1 ·Miaomiao Zhang3 · Steve Pieper1 · Polina Golland4 ·Matthew Toews5 · Prashin Unadkat1 · Alireza Sedghi1 · Haoyin Zhou1 · Alireza Mehrtash1 · Frank Preiswerk1 ·Cheng-Chieh Cheng1 · Alexandra Golby1 ·Masashi Sugiyama2,6 ·William M. Wells III1,4

Received: 12 January 2018 / Accepted: 27 July 2018 / Published online: 10 August 2018© CARS 2018

AbstractPurpose Matching points that are derived from features or landmarks in image data is a key step in some medical imagingapplications. Since most robust point matching algorithms claim to be able to deal with outliers, users may place highconfidence in the matching result and use it without further examination. However, for tasks such as feature-based registrationin image-guided neurosurgery, even a few mismatches, in the form of invalid displacement vectors, could cause seriousconsequences. As a result, having an effective tool by which operators can manually screen all matches for outliers couldsubstantially benefit the outcome of those applications.Methods We introduce a novel variogram-based outlier screening method for vectors. The variogram is a powerful geostatis-tical tool for characterizing the spatial dependence of stochastic processes. Since the spatial correlation of invalid displacementvectors, which are considered as vector outliers, tends to behave differently than normal displacement vectors, they can beefficiently identified on the variogram.Results We validate the proposed method on 9 sets of clinically acquired ultrasound data. In the experiment, potentialoutliers are flagged on the variogram by one operator and further evaluated by 8 experienced medical imaging researchers.The matching quality of those potential outliers is approximately 1.5 lower, on a scale from 1 (bad) to 5 (good), than validdisplacement vectors.Conclusion The variogram is a simple yet informative tool. While being used extensively in geostatistical analysis, it has notreceived enough attention in the medical imaging field. We believe there is a good deal of potential for clinically applyingthe proposed outlier screening method. By way of this paper, we also expect researchers to find variogram useful in othermedical applications that involve motion vectors analyses.

Keywords Vector outlier screening · Variogram · Feature-based registration · Neurosurgery

B Jie Luojluo5@bwh.harvard.edu

1 Brigham and Women’s Hospital, Harvard Medical School, 75Francis St, Boston, MA 02115, USA

2 Graduate School of Frontier Sciences, The University ofTokyo, 5 Chome-1-5 Kashiwanoha, Kashiwa, Chiba277-8561, Japan

3 Computer Science and Engineering Department, LehighUniversity, 19 Memorial Drive West, Bethlehem, PA 18015,USA

4 Computer Science and Artificial Intelligence Laboratory,Massachusetts Institute of Technology, 32 Vassar St,Cambridge, MA 02139, USA

Introduction

Matching points that are derived from features or landmarksin image data is a key step in many medical imaging appli-cations, such as 3D reconstruction and image registration forsurgical navigation. Since most robust point matching algo-rithms claim to be able to deal with outliers [1–5], usersmay place high confidence in the matching result and use itwithout further examination. However, for tasks where pre-

5 Ecole de Technologie Superieure, 1100 Rue Notre-DameOuest, Montreal H3C 1K3, Canada

6 Center for Advanced Intelligence Project, RIKEN, 1-4-1Nihonbashi, Chuo-ku, Tokyo 103-0027, Japan

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cision is of paramount importance, even a few mismatchescan cause serious consequences. As a result, having an effec-tive tool by which operators can manually screen all matchesfor outliers could substantially benefit the outcome of thoseapplications.

Image-guided neurosurgery is a task that requires pre-cise localization of tumor boundaries [6]. Conventionally,surgeons use commercial image-guidedneuronavigation sys-tems (IGNSs) to map the preoperative image data to anintraoperative patient coordinate system so as to get anupdated view of the brain [7,8]. Unfortunately, intraopera-tive brain deformation, also known as brain shift, invalidatesthis mapping and limits the trustworthiness of using preop-erative images in intraoperative surgical navigation [9,10].In order to compensate the brain shift and increase the accu-racy of neurosurgery, non-rigid image registration [11–14]is sometimes adopted to spatially align the preoperative andintraoperative images.

Because of tumor resection, the same structure may notexist on both preoperative and intraoperative images; hence,feature-based registration (FBR) becomes a promising strat-egy due to its robustness in registering images with missingcorrespondence [15]. FBR consists of three steps. In thefeature extraction step, distinctive local image features areautomatically extracted and identified as key-points on pre-operative and intraoperative images. In the feature matchingstep, a matcher searches for a corresponding intraoperativekey-point for each key-point on the preoperative image. Fromevery matched key-point pair, a displacement vector can beobtained to indicate the movement of the preoperative key-point due to the brain shift process. In the last step of FBR, thealgorithm generates a dense deformation field for the entirepreoperative image from the displacement vectors and uses itto map the preoperative image to the intraoperative space. Ifinvalid displacement vectors, which falsely reflect the tissuemovement under brain shift, are obtained from mismatchedkey-point pairs, itwill negatively affect the registration result.

In this paper, we introduce a simple yet informativevariogram-based outlier screening method for vectors anddemonstrate its usefulness in the context of feature-basedregistration. While variograms are extensively used in geo-statistics to capture the spatial dependence of stochasticprocesses [16], they have not yet received much attentionin the medical imaging field. Since the spatial correlationof invalid displacement vectors, which are considered vectoroutliers, tends to behave differently than valid displacementvectors, they can be identified on the variogram cloud andflagged for further examinations. The paper is organized asfollows: In “Summarizing variogram in FBR” section, weexplain how to estimate and understand the variogram in theFBR context. In “Variogram and vector-outliers removal”section, we demonstrate how to identify outliers on the

variogram. Finally, experimental results and conclusion arepresented in “Experiments” section.

Summarizing variogram in FBR

The variogram is a powerful tool that is originated in geo-statistics. In this section, we summarize the variogram in thecontext of FBR.

In the feature matching step of FBR, we obtain a displace-ment vector from everymatched key-point pair. For example,as shown in Fig. 1a, b, assume kp is the coordinate of anextracted preoperative key-point and ki is the coordinate ofits corresponding intraoperative counterpart. After overlay-ing both images in Fig.1c, the displacement vector d fromthe preoperative key-point pointing to the intraoperative key-point can be calculated as ki − kp.

All displacement vectors form a vector field, which is sup-posed to indicate the intraoperative brain shift. We attemptto model the spatial dependence of the vector field and usethe dependency as a criterion for identifying displacementvector outliers.

Random process model

Brain shift is caused by the interaction of physical, surgicaland biological factors. Due to the complexity and incompleteunderstanding of the phenomenon, we model the deforma-tion of the brain as a stochastic process. In our model, letx ∈ R

d be a generic data location in the d-dimensionalEuclidean space. Suppose Z(x) is a random quantity at loca-tion x; its realization z(x) is a displacement vector like theone in Fig. 1d. Given an index set D ⊂ R

d , we can modelthe vector field as

{Z(x) : x ∈ D}. (1)

For brain shift, the deformation at a particular location islikely to have influence in all directions. Capturing the spa-tial correlation of deformation is essential for understandingthe phenomenon. The definition of the variogram comes nat-urally as we quantify the spatial dependence of Z(x).

The variogram and empirical variogram

The pairwise dependence between displacement vectors ina random field is the variance of the difference between thevalues at two locations across the field [16]:

2γ (x1, x2) := var(Z(x1) − Z(x2)). (2)

We make the mild assumption that the random field has aconstant mean and is intrinsically stationary. Intrinsic sta-

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Fig. 1 a Extracted key-point on the preoperative image; b extracted key-point on the intraoperative image; c overlaying two images; d displacementvector d for the preoperative key-point

Fig. 2 a Two key-points x1 andx2; b displacement differencesbetween z(x1) and z(x2), andthe lag h

tionarity states that the variance of difference is the samebetween any two points that are separated by the same dis-tance. Adopting a terminology from time series analysis, werepresent the separation between two spatial locations by alag vector h = x2 − x1 and rewrite (2) as

2γ (x1, x2) = E[((Z(x1) − Z(x1 + h))2]= 2γ (h).

(3)

γ (h), also known as the variogram (sometimes called thesemivariogram) of a randomfieldZ(x), is defined as one-halfthe average squared difference between the value at locationsseparated by distance h [17]. The variogram function γ (h)

depends only on h, and it measures the correlation as a func-tion of distance.

Notice that γ (h) is a theoretical function. In practice, wedo not have the displacement vector for every location toestimate γ (h), so we use its empirical alternative [18]. Thisdefinition will be clarified in “Summarizing variogram inFBR” section.

γ̂ (h ± δ) := 1

2|N (h ± δ)|∑

(i, j)∈N (h±δ)

‖z(xi ) − z(x j )‖2.

(4)

Estimating the empirical variogram in FBR

In FBR, there are several ways to represent ‖z(xi ) − z(x j )‖and h. In this paper, we use the following representations. InFig. 2a, on a preoperative magnetic resonance image (MRI),

there are two key-points whose coordinates are x1 and x2.z(x1) and z(x2) are their associated displacement vectors.As shown in Fig. 2b, ‖z(x1) − z(x2)‖ is equal to the norm ofthe vector z(x2) − z(x1), and lag h is the Euclidean distancebetween the two key-points ‖x2 − x1‖.

Estimating an empirical variogram can be analogous tobuilding a histogram based on samples from a continuousdistribution. The strategy is to group pairs of points that havesimilar h into the same bin and to approximate the variogramby combining information from all of the bins. Here we illus-trate the 4 steps of estimating an empirical variogram usingFig. 3.

(a) Construct the variogram cloud by plotting ‖z(xi ) −z(x j )‖2 for all key-point pairs with their lag hi j .

(b) Introduce a variable δ as the tolerance range for lag h,and divide the variogram cloud into bins whose width isset to 2δ.

(c) Calculate the mean γ̂ (hbink ± δ), which are marked asblue in Fig. 3c, for every bin using Eq. (4). |N (hbink±δ)|represents the number of key-point pairs of bin k.

(d) Plot all γ̂ (hbink ± δ) with their hbini to obtain the empir-ical variogram.

The lag tolerance δ should be neither too large nor toosmall, since it may veil the short-distance correlations orproduce empty bins. Typically, δ is determined in ad hocfashion. When δ = 0, the empirical variogram γ̂ (h±δ) is anunbiased estimator of the theoretical variogram γ (h) [16].Notice that the proposed variogram-based outlier screening

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Fig. 3 a Variogram cloud; b binning the variogram cloud with the bin width 2δ; c computing the mean for every bin; d plotting γ̂ (hbink ± δ) andhbink

Fig. 4 a Variogram cloud for a set of US data; b empirical variogram for the same US data

method primarily uses the variogramcloud; hence, the choiceof δ has no impact on its performance.

In Fig. 4, we display a variogram cloud and the empiricalvariogram estimated from a set of preoperative and intraop-erative ultrasound (US) data. For these US data, we wereable to find 71 pairs of matched key-points, and the δ used toestimate the empirical variogram is 0.5mm. Here the x-axisrepresents lag h and the y-axis is the value difference amongpairs. The empirical variogramdoes indicate that nearby key-points tend to have more similar displacement vectors thanthose that are far apart. In the rest of this article, we use thevariogram and empirical variogram interchangeably.

Understanding variograms

In variogram-based diagnostics, it is often necessary to fit amodel, that is a continuous function or curve, to the empiricalvariogram. Although the methodology for fitting a model tothe empirical variogram is beyond the scope of this paper,we use a fitted model to explain some important parametersof a variogram.

Figure 5a is an example of a fitted model. A fitted modelis commonly described by the following characteristics:

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Fig. 5 a Example of a fitted variogram model; b a continuous model fitted to the empirical variogram of the US data that have 71 key-point pairs

Nugget When two key-points are extremely close, theyshould have very similar displacement vectors. Inother words, at lag h = 0, the variogram valueshould be 0. However, at a small separation dis-tance, the variogram sometimes exhibits a nuggeteffect, which is some value greater than 0.

Sill The value at which the model first flattens out.Range It is the lag distance for which the sill is reached.

Presumably, only samples that are separated by dis-tances closer than the range are spatially correlated.

In Fig. 5b, we fit a continuous model to the variogramin Fig. 4b. We can see insignificant nugget effect and anapproximate 8.8mm sill value. The range is around 21mm,which indicates that the spatial influence for a particular key-point to its neighbors is 21mm.

Variogram and vector outliers removal

In FBR, the feature matching algorithm, e.g., [15], mayidentify feature correspondences between preoperative andintraoperative images solely based on the local image appear-ance similarity. While the algorithm is claimed to be robustagainst outliers, similar image patches and artifacts at non-corresponding locations of the brain will sometimes beextracted and matched. These mismatched key-point pairsproduce invalid displacement vectors, as outliers, that fail toreflect the actual brain deformation. Since all displacementvectors will be used to generate the dense deformation field,we want a simple yet informative tool so that operators canefficiently screen all displacement vectors for outliers.

In “Summarizing variogram in FBR” section, we men-tioned that the variogram cloud can capture the spatialdependence of displacement vectors. Since the spatial cor-relation of vector outliers tends to behave differently thanvalid displacement vectors, they can be spotted on the vari-ogram. There are two key features for the variogram-basedoutlier screening: 1 It is effective and easy to use. Operators

can quickly spot and flag potential outliers on the variogramcloud. 2 The variogram is also very informative. Operatorsalso can grasp the quality/trend of the vector field by observ-ing variogram patterns.

Global and local outliers in FBR

Outliers are extreme values that deviate from other observa-tions in the data. In FBR, there are generally two types ofoutliers [20]:

1. A global outlier is a sample point that has a very high or avery lowvalue compared to all the values in a dataset. Forexample, due to the symmetry of cerebral hemispheres,similar intensity patterns exist on the opposite half of thebrain for MRI images. If these patterns are mismatchedas corresponding key-point pairs, they produce globaloutliers. These global outliers are clearly incorrect andshould be removed.

2. A local outlier is a sample key-point that has a valuewithin the normal range for the entire dataset, but isunusually high or low compared to its neighboringpoints. In FBR, some displacement vectors may seemto be local outliers at first, yet they capture the actuallocal deformations that are induced by the brain shiftphenomenon. These key-points are vital for understand-ing and correcting the brain shift and may be the mostsignificant points in the registration. Therefore, it isunnecessary to remove all local outliers, but to suppressthem to a certain degree or perhaps to flag them for man-ual intervention.

Outliers in the variogram cloud

Here we explain how to look for global and local outliers ina variogram cloud by the following two examples.

In a synthetic vector field shown in Fig. 6a, we have 25valid displacement vectors and a manually added global out-

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Fig. 6 a Displacement vectors and a red global outlier; b the global outlier identified in the variogram cloud

Fig. 7 a Displacement vectors and a red local outlier; b local outlier identified in the variogram cloud

lier. Since the displacement vector of that global outlier isdrastically different from other key-points, all pairings ofpoints involving that global outlier, regardless of the lagdistance h, have high ‖z(xi )−z(x j )‖2 values. From the vari-ogram cloud in Fig. 6b, we can see two main strata of points.All the higher values, which are marked in red, come frompairings with the manually added global outlier. On the otherhand, the lower strata are composed of pairings among therest of valid displacement vectors.

In another synthetic vector field shown in Fig. 7a, we have25 valid displacement vectors and a manually added localoutlier. The local outlier does not have high values acrossall distance in the variogram cloud. However, the variogramvalues of its close neighbors are relatively high. In Fig. 7b,all values from pairings with the local outlier are markedin red. On the bottom-left of the variogram cloud, we cannotice a small cluster of abnormally high values as expected.Sometimes, even the local outlier is hard to notice in thedisplacement vector field, it is easy to spot in the variogramcloud.

Given the characteristics of global and local outliers,screening using the variogram is very straightforward. Sincethe variogram provides a distinct visualization for outliers,

users can get an overview of potential outliers by simplylooking at the patterns on the top and bottom-left corner ofthe variogram cloud. If abnormal variogramvalues are found,it is easy to trace back to the key-point matches for furtherexaminations.

Experiments

In this section, we experimentally evaluate the proposedmethod.

Outlier screening

In the experiments we tested the proposed variogram-basedoutlier screening method on a clinically acquired US dataset.This 9-patient dataset contains US images taken from 3different stages of neurosurgery: before opening the dura(predura), after opening the dura (postdura) and during thetumor resection (intraoperative). We registered the intraop-erative image to the predura image. In case the intraoperativeimage is not available,we register the postdura image instead.

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Table 1 Experimental results of the proposed outlier screening method

Case 1 2 3 4 5 6 7 8 9

Data Intra Intra Postd Postd Intra Intra Intra Postd Intra

Matches 123 71 84 14 49 98 12 37 64

#OL 1 2 1 0 2 1 0 3 2

OLS1 −1.3 −1.4 −2.3 n/a −0.9 −2.3 n/a −0.8 −2.0

OLS2 n/a −1.9 n/a n/a −2.2 n/a n/a −0.4 −2.2

OLS3 n/a n/a n/a n/a n/a n/a n/a −1.8 n/a

Fig. 8 Variogram for all potential outliers that were flagged by the operator in a total of 9 cases

After the feature matching, an operator quickly looked forabnormal patterns on the top and bottom-left corner of thevariogram cloud and flagged the matches that are in corre-spondence with those patterns. All flagged matches, mixedwith some valid matches, are further sent to 8 experiencedmedical imaging researchers for rating.

In the rating process, without knowing how many poten-tial outliers are in the group, raters assign a score from 1(bad) to 5 (good) to every match. After a zero-mean normal-ization for each rater, we calculated the score difference ofevery potential outlier to themean score of valid matches andrecorded it as outlier score (OLS). The number of matches,number of potential outliers (#OL) suggested by the vari-ogram, and the OLS of each potential outlier are presented

in Table 1. Here OLS1 means the OLS for potential outlier1, etc.

The feature matching algorithm we use is claimed to berobust [15], yet despite the two cases which have less than15 displacement vectors, we were able to flag at least onepotential outlier using the variogram. Given the fact that allflagged potential outliers for cases 1–3 and cases 5–6 and9 have substantially lower scores than normal matches, it isreasonable to conclude that adding an outlier screening stepafter the feature matching is beneficial.

Figure 8 shows the variogram for all potential outliers thatwere flagged by the operator. In Fig. 9, we show the axial,sagittal and coronal views of some found outlier matchestogether.

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Fig. 9 Screenshots of axial (top-left), sagittal (bottom-left) and coronal (top-right) views of 3 outliers found by the variogram. Note the matchingquality in the axial slice of case 1, the sagittal slice of cases 2 and 3 are considerably bad

Fig. 10 a Variogram cloud for the MRI data that have 386 matched key-point pairs; b the corresponding variogram

Outlier suppression

In case there are too many matched key-point pairs wherea manual screening is too time-consuming, we can combinethe variogram with other statistical tools, such as Tukey’sfences [21], to suppress outliers in a more systematic fash-ion.

The interquartile range (IQR) is the distance between thelower quartile Q1 and the upper quartile Q3. Tukey’s fencescan be stated as

[Q1 − k · IQR, Q3 + k · IQR]. (5)

Equation (5) is called an inner fence with k = 1.5 and anouter fence with k = 3. Conventionally, a value between theinner and outer fences is a possible outlier. A value beyondthe outer fence can be considered as an extreme outlier [21].

The FBR was mainly developed for registering US toUS data. However, for US–US registration, the number ofmatched key-point pairs found is usually less than 100. To

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demonstrate how to use the variogram and Tukey’s fences tosuppress outliers in massive vector fields, we used a pair ofpreoperative and intraoperative MRI images. After the fea-ture matching, we found 386 matched key-point pairs, inwhich outliers are likely to present. Figure 10a, b shows itsvariogram cloud and variogram, respectively. The half-binsize used here is δ = 1.5mm.

Based on the reasoning stated in “Variogram and vector-outliers removal” section, we use the inner fence to filterout all potential global outliers for bins whose 15mm <

h � 80mm, and adopt the more conservative outer fence tosuppress local outliers for h � 15mm.

In FBR, a standard way to measure the registration erroris by the sum of (squared) Euclidean distance between allmatched key-point pairs (SSD). The transformation matrixin affine registration is derived by minimizing the SSD ina least square fashion. In case there are bad matches, thealgorithm, influenced by vector outliers, tends to obtain sub-optimal affine transformation matrix that leads to large SSD.In the preliminary experiment, after the outlier suppression,the SSD of feature-based affine registration dropped from1.66 to 1.25mm, which we believe is a good indicator forhaving less outliers.

Conclusion

Matching points that are derived from features or landmarksin image data is a key step in some medical applications.For applications that demand a high precision, such asfeature-based preop-to-intraop registration, adding an out-lier screening step could make it less prone to mismatchesand improve the overall result. In this paper, we proposedto use the variogram to estimate the spatial dependence ofdisplacement vectors. Since the spatial correlation of vectoroutliers behave differently than valid displacement vectors,they can be efficiently identified on the variogramandflaggedfor further examinations.

The variogram is a simple yet informative tool. Whilebeing used extensively in geostatistical analysis, the var-iogram has not received much attention in the medicalimaging field. We believe there is a good deal of potentialfor clinically applying the variogram. For instance, with aninteractive user interface, such as 3D Slicer [19], cliniciansor their assistants can easily spot and flag outliers on the var-iogram for manual intervention. By way of this paper, wealso expect researchers to find the variogram useful in othermedical applications that involve motion vectors analyses.

Acknowledgements This work was supported by National Institute ofHealth Grants P41-EB015898-09 and P41-EB015902. This work wasalso supported by the International Research Center for Neurointelli-gence (WPI-IRCN) at the University of Tokyo Institute for AdvancedStudy.

Compliance with ethical standards

Conflict of interest The authors declare that they have no conflict ofinterest.

Ethical approval All procedures performed in studies involving humanparticipants were in accordance with the ethical standards of the institu-tional and/or national research committee andwith the 1964Declarationof Helsinki and its later amendments or comparable ethical standards.

Informed consent Informed consent was obtained from all individualparticipants included in the study.

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