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Original article
Volumetric MRI Classification for Alzheimer’s Diseases
Based on Kernel Density Estimation of Local Features
YAN Hao, WANG Hu, WANG Yong-hui and ZHANG Yu-mei
School of Foreign Languages, Xidian University, Xi’an 710071, China (Yan H)
School of Psychology, Shaanxi Normal University, Xi’an 710062, China (Yan H and
Wang YH)
Institute of Automation, Chinese Academy of Sciences, Beijing, 100190, China
(Wang H)
Department of Neurology, Beijing Tiantan Hospital affiliated with Capital Medical
University, Beijing 100050, China (Zhang YM)
Correspondence to: Dr. Zhang Yumei, Department of Neurology, Beijing Tiantan
Hospital affiliated with Capital Medical University, Beijing 100050, China (Tel:
010-67098316. Fax: 010-67098316. E-mail: [email protected])
This work was supported by the Fundamental Research Funds for the Central
University, the National Natural Science Foundation of China (grant numbers
81071217, 31171073); the Beijing Nova program (grant number Z111101054511116);
the National Science and Technology Major Project of China (2011BAI08B02), and
Beijing Natural Science Foundation (grant number 4122082), Major Project of
National Social Science Foundation (11&ZD186). There is no conflict of interests.
Keywords: Classification; Inter-subject variability; Local feature; Kernel density
estimation; Alzheimer’s disease
Background Classification of Alzheimer’s disease (AD) from magnetic resonance
images is challenged by the lack of effective and reliable biomarkers due to
inter-subject variability. This paper presents a classification method for AD based on
kernel density estimation of local features.
Methods First, a large number of local features are extracted from stable image
blobs to represent various anatomical patterns for potential effective biomarkers.
Based on distinctive descriptors and locations, the local features are robustly clustered
in order to identify correspondences of the same underlying patterns. Then, the kernel
density estimation is used to estimate distribution parameters of the correspondences
by weighting contributions according to their distances. Thus, biomarkers could be
reliably quantified by reducing the effects of further away correspondences which are
more likely noises from inter-subject variability. Finally, the Bayes classifier is
applied on the distribution parameters for the classification of AD.
Results Experiments were performed on different divisions of a publicly available
database to investigate the accuracy and the effects of age and AD severity. Our
method achieved an equal error classification rate of 0.85 for subject aged 60-80 years
exhibiting mild AD, and outperformed a recent local-feature based work regardless of
both effects.
Conclusions We proposed a volumetric brain MRI classification method for
neurodegenerative disease based on statistics of local features using kernel density
estimation. The results demonstrate that the method may be potentially useful for the
computer-aided diagnosis in clinical settings.
INTRODUCTION
Increasing neuroimaging researches have proved that certain neuroanatomical
structures may be preferentially modified by particular cognitive skills, genes or
diseases1,2. Morphological analysis of medical images is therefore used in a variety of
research and clinical studies that detect and quantify spatially complex and often
subtle abnormal imaging patterns of pathology. In neurodegenerative diseases (i.e. the
Alzheimer’s disease, AD), the pattern of brain pathology evolves as the disease
progresses-starting mainly in the hippocampus and entorhinal cortex, and
subsequently spreading throughout most of the temporal lobe and the posterior
cingulate, finally involving extensive brain regions3,4. It is inferred that some
neurodegenerative changes starts much earlier before symptomatic disease are
observable. Thus, a recent upsurge of interests has made in developing both
diagnostic and prognostic biomarkers that can predict which individuals are relatively
more likely to progress clinically.
Quantifying inter-individual anatomical variability within a population is very
important to the medical imaging, as it becomes the prerequisite to obtain reliable
statistical results. Several methods have been developed to deal with the one-to-one
correspondence by adopting inter-subject registration or image modeling5. The
majority of them intended to quantify appearance and geometry between subjects or
between a model and a subject6,7. However, the assumption of one-to-one
correspondence can only be operative at a local scale, and cannot effectively represent
multiple and distinct modes of appearance even in the same local anatomical area8.
Provided that anatomical variability exists in brain morphology of different subjects,
particularly in highly variable areas with different cortical folding patterns, the
one-to-one correspondence is hard to obtain. However, local features provide a
solution to effectively address the situation where one-to-one inter-subject
correspondence does not exist in all subjects9-11. One-to-one correspondence indicates
that every corresponding unit for morphometric analysis, such as the voxel or the
regional element, can be identified in all subjects to represent the same anatomical
structures. It is different from the inter-subject variability. The inter-subject variability
means that the underlying anatomical structures vary in geometry and appearance
from one subject to another.Based on the scales-space theory, anatomical structures
can be modeled and identified by a large number of distinctive local and
scale-invariant features, other than at arbitrarily global or voxel-level12. Local features
are distinctive and informative so as to represent group-related morphology patterns.
It can also obviate nonlinear inter-subject registration since features at different
coordinates could also be well-matched, and local group-informative image patterns
could be largely preserved. Registration error still exists due to inter-subject
variability, particularly in the highly variable cortices, and it is difficult to guarantee
that images are not being over-aligned. Furthermore, it doesn’t require segmenting
images into tissues and regions, which is generally time-consuming and may
introduce some residual components.
Probabilistic models based on correspondences of local features have been
constructed, and their distribution parameters are used to quantify informativeness of
anatomical patterns with respect to groups in recent studies12. The method has shown
good performance to identify group-related structures with different occurrence
likelihoods. However, it could not explain what kind of specific morphological
differences has happened to the structures, e.g., atrophy or enlargement as showed in
traditional morphometry method. The distribution parameters for classification are not
reliable and accurate enough due to the following reasons. First, only the frequencies
of correspondences are utilized to estimate distribution parameters. Such kind of
immature estimation neglects the information of distances between local features
which would be useful to eliminate noises from inter-subject anatomical variability.
Moreover, its estimator has jumps at the edge and zero derivatives everywhere else13.
As a result, noises which may be significant at the edge of similar anatomical patterns
lead to the reduction of reliability of the distribution parameters. Second, nearby
model features from training subjects are used to estimate the distribution parameters
for local features from a test subject. However, a local feature and its nearby model
feature may sometimes arise from different underlying anatomical patterns due to
variability, thus the accuracy of the estimation is affected. Although correspondences
are identified according to the distinctiveness of local features, two type of error may
still occur due to inter-subject variability. False positives (FP) occur when local
features arising from different underlying anatomical structures are accepted as
correspondences, and false negatives (FN) occur when local features arising from the
same structure are rejected as non-correspondences.
In this paper, we present a new classification method for AD based on kernel
density estimation of local features from volumetric MR images. The current study
builds upon previous work12, but emphasizes the accuracy and reliability of the
probabilistic modeling for quantification of the informativeness of anatomical patterns
with respect to groups. The modeling procedure first identifies clusters of
correspondences from a large number of local features using robust measures of
geometric and appearance similarity. Then the clusters are used to estimate the
probabilistic densities of the local feature via the kernel density estimation (KDE)14.
KDE is a nonparametric technique to estimate the probability density function without
any assumptions of the underlying distribution15,16. Its estimator not only includes the
frequency information in the clusters, but also relates with their distances which are
weighted by a specified kernel function. With the kernel function, the estimator
becomes continuous, and its smoothing degree could be adjusted to deal with different
noise levels from the inter-subjects anatomical variability. As such, we could improve
the accuracy and reliability of the probabilistic modeling, and further enhance the
performance of the classification method.
METHODS
Subjects
In order to validate the performance of the proposed classification method, we
tested the method on a large, freely available, cross-sectional dataset from OASIS
project18 (http://www.oasis-brains.org/). The dataset includes MRI data from 100
suspected Alzheimer’s disease (AD) subjects and 98 normal control (NC) subjects.
The probable AD subjects are diagnosed clinically from very mild to moderate
dementia characterized using the Clinical Dementia Rating (CDR) scales18. All
subjects in the dataset are right-handed, with approximately equal age distributions for
both NC and AD subjects ranging from 60 years to 96 years with means of 76/77
years.
The dataset was divided into three different divisions according to both ages and
CDRs the same as that used in12. Therefore the classification performance with the
effect of age and the severity of clinical diagnosis could be evaluated. The details
about the three divisions were listed as follows: 1) Subjects aged 60–80 years,
CDR=1 (66 NC, 20 AD); 2) Subjects aged 60–96 years, CDR=1 (98 NC, 28 AD); 3)
Subjects aged 60–80 years, CDR=0.5 and 1 (66 NC, 70 AD).
Data acquisition
For each subject, at least three T1-weighted magnetization-prepared rapid gradient
echo (MP-RAGE) images were obtained according to the following protocol: 128
sagittal slices, matrix = 256 × 256, TR = 9.7 ms, TE = 4 ms, flip angle = 10°,
resolution = 1 mm × 1 mm × 1.25 mm. The images were gain-field-corrected and
averaged in order to improve the ratio of signal to noise. Then, images from all
subjects were aligned within the Talairach reference frame (voxel size = 1× 1 × 1
mm3) via the affine transform T, and their skulls were masked out18. After abnormal
intensities such as highlight intensities from residual skull were adjusted to normal
levels via a histogram analysis, the histogram equalization was applied to normalize
the intensity ranges to [0, 1] in all images.
Classification method
The proposed volumetric MRI classification method predicts the group (e.g.,
control/patient) to which an unlabeled volumetric MR image belongs according to a
training set. It involves four steps: linear registration, scale-invariant feature transform,
probabilistic modeling, and Bayes classification.
Linear registration
This step removes uncorrelated environment information including body-location
differences in image acquisition and the uninterested affine parameters. Thus we
could focus on the remaining appearance and geometric variability of local
anatomical patterns. In addition, due to the approximate alignment of same underlying
patterns, correspondences between subjects could be first constrained by their
locations.
Scale-invariant feature transform
Given that the anatomical patterns of human brain are naturally characterized by
different scales, e.g., width of ventricles or thickness of cortices, local features are
desired to be adaptive to scales, other than at arbitrary global or voxel-level11,19,20. The
efficient scale-invariant feature transform (SIFT) based on the scale-space theory21-23
is applied to extract such features24-26.
For each volumetric image, SIFT first builds a Gaussian pyramid with
incrementally blurred versions of the original image. Then, the Difference of
Gaussian (DoG) space is constructed by subtracting two nearby images in Gaussian
pyramid, and its local extrema are selected as candidate feature points. In order to
improve their stability for further corresponding, these points are refined at sub-voxel
level by interpolation in the DoG space27. These points could not be reliably localized
in all spatial directions, and could be identified via an analysis of the 3  3 Hessian
matrix24. Fig. 1 illustrates examples of scale-invariant features extracted in a
volumetric image.
Probabilistic modeling
After the SIFT procedure, each volumetric image is modeled as a collage of local
features. The probabilistic modeling aims at robustly quantifying the anatomical
variability and informativeness of local features based on such statistical regularity in
a training set.
In the present study, the unlabeled image that needs classification is modeled as a
set of local features as:
F   fi , i  1,...N ,
(1)
where N is the total number of local features extract from the unlabeled image, and
each individual feature is represented as fi  ai , gi  . ai is the appearance descriptor
representing the measurements of a local image appearance pattern of f i , and
gi  xi ,  i  represents the geometry of f i in terms of image location xi and
scale  i .
In order to quantify the informativeness of f i , we use the following likelihood
ratio12:
dlr ( fi ) 
p( f i | C , T )
,
p( f i | C , T )
(2)
where p( fi | C , T ) represents the conditional probability density of a feature f i
given (C, T). C is a discrete random variable of groups from which the unlabeled
image may sample, and T represents the linear registration transform that
approximately aligns images into normalized space. In order to ensure that the ratio in
Eq. (2) is well defined even when the denominator is zero, the Dirichlet regularization
which adds small artificial counts to both the numerator and the denominator of the
ratio is applied28.
To estimate the conditional probability density p( fi | C , T ) for likelihood ratios,
several issues are taken into considerations here. Firstly, local features that do occur
with f i in the training set are the basis of the density estimation. Secondly, the
underlying distribution of a local feature is complex and unknown as a prior due to
the various anatomical patterns. Finally, the inter-subject anatomical variability may
introduce much noise into the distribution and thereby disturbing the statistic
regularity of the occurrences.

Let F   f j , j  1,...M
 represent a training set of M local features that are
extracted from the training images. The local feature f j is identified as an occurrence
of f i if they are similar in terms of geometry and appearance. Thus, we could
express the occurrence cluster Si as:
Si  Gi
Ai ,
(3)
where Gi represents a geometry set whose elements are similar with f i in terms of
geometry, and Ai represents a appearance set whose elements are similar with f i in
terms of appearance.
The geometry set Gi is determined by a binary measure of geometrical similarity
where feature f j is said to be similar with f i if the distances of their locations and
scales are less than certain thresholds:





Gi   f j : xi  x j   x  ln i     ,
 j




(4)
where xi and x j represent the location of f i and f j respectively, and their distance is
measured by Euclidean norm;  i and  j represent the scale of f i and f j
respectively23.  x and   represent the location threshold and the scale threshold, and
reflect the maximum acceptable deviations of the location and the scale of a local
feature occurring in different images respectively.
It is also noted that the location threshold  x is generally multiplied by the
scale  i in previous study, so as to make the binary measure for geometry
scale-independent13. This part is determined by the resampling rate  i of the octave
where f i locates. Therefore, we set the location threshold as:
 x   x  2  i .
(5)
Such location threshold makes the measure for geometry similarity adaptive to
octaves with different resampling rates.
Next, the appearance set Ai is determined by a binary measure of appearance
similarity where local feature f j is said to be similar with f i if their Euclidean distance
of appearance descriptors is less than the threshold  ai :


Ai ( ai )  f j : ai  a j   ai ,
(6)
where ai and a j represent the location of f i and f j respectively. The threshold  ai
represents the maximum acceptable deviation of the appearance descriptor of a local
feature occurring in different images.
A key issue of clustering is to determine the three thresholds   x ,  ,  ai  , as
improper thresholds may increase the false-positive or false-negative occurrences.
The optimal value for  ai is automatically determined as:


 a  sup  a [0, ) : Ai ( a ) Gi  Ai ( a ) Gi .
i
(7)
The optimal values for thresholds  x and   are selected via cross-validation on a
training set.
After the identification of the cluster of occurrences for each local feature, the
next step is to estimate the probability density of f i based on the spatial distribution
of the occurrences. KDE is appropriate for this situation, as it can estimate the
probability density function without any assumptions of the underlying
distribution16,17,29. Given the cluster of occurrences Si , p( f i C , T ) can be estimated
as:
2

i  ai 
a


1
1
pˆ ( fi | C , T ) 
exp  

,
d
2
2
2
NC fiSi C  2 h 2 
2

h
a
i




where ai and ai are the appearance descriptors of f i and f i ; d is the dimension of
(8)
appearance descriptors; N C is the count of training images in group C, and h is the
bandwidth of the kernel function. In the Eq. (8), the appearance threshold  ai is used to
normalize the distances of feature descriptors, which makes the estimation more
adaptive to patterns with different anatomical variabilities. The bandwidth h is a
free parameter to control the smoothness of the probability density function17, and its
optimal value can be obtained via the cross-validation approach.
Bayes classification
The classification of an unlabeled image is based on the informativeness of each
local feature quantified in probabilistic modeling in terms of likelihood ratio30,31.
Given the knowledge of (T, C), it is assumed that local features are conditional
independent13. Then, the Bayes classifier makes a decision using the maximum a
posteriori (MAP) rule as:
 p C,T | F  


C  arg max 

C
 p C , T | F 
*


 p C,T  N p  f | C,T  


i
 arg max 
,

C
 p C , T i 1 p fi | C , T 



(9)

 N

 arg max  P0  dlr ( fi ) 
C
 i 1

where p  C, T | F  represents the posterior probability density of (C, T) given
F. p(C , T ) represents a joint prior distribution over (C, T), and p ( F ) represents the
probability density of evidence of feature set F. Accordingly, the classification is
primarily driven by the data likelihood ratio (DLR) of the unlabeled volumetric
image:
N
DLR   dlr ( f i ) .
(10)
i 1
Evaluations of classification performance
The threshold of data likelihood ratio was adjusted to generate the receive operating
characteristic (ROC) curve32. Besides, we also reported other two
threshold-independent metrics from the ROC curve—the equal error classification
rate (EER) and the area under the ROC curve (AUC).
RESULTS
The bandwidth was originally set as h  0.2 . Cross-validation in grid-search
manner was performed to check each combination of thresholds:  x 1, 2, ...,10 ,
 0.6, 0.7, ..., 2.0 . Different values of the bandwidth were then cross-validated in
0.04, 0.06, ...,1.00 . The value that yielded the best classification performance was
selected as the optimal value of the bandwidth. The cross-validation surfaces for
geometry thresholds   x ,    on three divisions were shown in Fig. 2. The maxima
of the surfaces were obtained at the following coordinates: divisions 1)
 x'  7,    1.3 ; divisions 2)  x'  7,    1.3 ; divisions 3)  x'  7,    1.8 .
The cross-validation curves for the bandwidth on three divisions were shown in
Fig. 3. The trends of AUC curves were similar on different divisions. Particularly, the
curves were relatively stationary between the 0.15 and 0.25, and they decreased
drastically when the bandwidth went from the 0.1 to 0, while decreased gradually as
the bandwidth was above 0.4. This coincided with the smoothing mechanism of the
bandwidth. On the one hand, small value resulted in the undersmoothing of
probability density functions, thus the functions became more sensitive to noise from
the inter-subject anatomical variability. On the other hand, large value may result in
the oversmoothing, which makes the function insensitive to fine anatomical patterns.
To avoid such overfitting, the curves derived from different divisions were averaged.
The optimal value for the bandwidth was selected at h  0.23 where the maximum
of the averaged AUC curve was reached.
We evaluated the performance of the proposed method on the three divisions with
different levels of ages and the severity from clinical diagnosis, in comparison with a
recent local-feature based classification method13. Their performances were firstly
tested via the ROC curves (shown in Fig. 4). Overall, the ROC curves of our method
were above that of the Toews’ method. Then, the performances were also compared
using both the EER values and AUC values. Particularly, the EER values of our
method were 0.85, 0.79 and 0.72 respectively, while the corresponding values derived
from the Toews’ method were 0.80, 0.72, 0.70. In addition, AUC values of our
method were 0.92, 0.85 and 0.80 respectively, while these values generated by
Toews’ method were 0.88, 0.81 and 0.73.
Both methods were implemented in C++ programming language and tested on a
computer with a four-core-processor running at 3.46GHz and 8GB RAM. The
average time consumed to classify an unlabeled image was about 2.2 minutes. Such
time can meet the good criterion for a clinical MRI diagnosis. The memory amounts
consumed in the experiments were also listed in Table I. They were generally
acceptable and linearly dependent on the sample size of study cohort.
DISCUSSION
In this paper, we proposed a volumetric brain MRI classification method for
neurodegenerative disease based on statistics of local features using kernel density
estimation. Statistics of local features specifically aimed at making use of the
group-related anatomical patterns whose correspondence between all images were
ambiguous. Nonparametric kernel density estimation technique was adopted to
improve both the accuracy and reliability of the probabilistic densities in statistics,
further enhancing the classification performance. In the experiments on three
divisions of a freely available OASIS dataset18 with different age ranges and different
severity of clinical diagnosis, the proposed method all achieved higher AUC values
than the method currently suggested by Toews13]. The better classification accuracy
indicated that the proposed method may be potentially useful for computer-aided
diagnosis in clinical settings.
The Bayes classifier employed in the volumetric MRI classification method was
built on three steps involving linear registration, scale-invariant feature transform, and
probabilistic modeling. In particular, linear registration was used to achieve
approximate inter-subject alignment of potential corresponding anatomical pattern in
volumetric images. A large number of local features extracted by the 3D
scale-invariant feature transform were used to represent various anatomical patterns of
volumetric images. Based on these local features, the probabilistic modeling step can
quantify the anatomical variability and informativeness with respect to groups in
terms of likelihood ratios.
The probabilistic modeling aimed at making use of group-related local features
which did occur with statistical regularity to improve the individual classification
performance. This was achieved by robust feature clustering and probability density
estimation. In the presence of anatomical variability, feature clustering tried to
identify correspondences of the same underlying anatomical patterns based on the
distinctive appearance and geometry of local features. Particularly, the error threshold
for the measurement of appearance similarity was determined in a feature-specific
manner, meaning that clusters with different anatomical variability may have different
error thresholds. In conclusion, feature clustering may provide a more effective
mechanism for statistics of individual anatomical variability, overcoming the
limitation of traditional methods which were based on the fundamental assumption of
one-to-one correspondence between all subjects. After that, KDE was utilized to
estimate the probability densities of local features. Compared with the naive estimator
adopted in Toews’ classification method, the estimator of KDE was evidently
smoother14. In addition, the amount of smoothing controlled by bandwidth of KDE
was cross-validated to select an optimal value, so the estimator could be closer to the
actual distribution of the local feature and more robust to the noise from anatomical
variability16,17.
A limitation of our current study has been the failure to compare different
classifiers. Only the Bayes classifier was tested as it could be naturally applied to the
statistic of local features given the strong independent assumption. Although the
Bayes classifier could obtain good accuracy and robustness in the classification
experiments of Alzheimer’s diseases, other classifiers should be performed to find out
which was more suitable according to the no-free-lunch theorem33. Nevertheless, most
classifier algorithms were applied to an input vector with fixed-length, and the
elements were correspondent between the input vector and the vectors in training
samples. In contrast, the numbers of local features extracted in different image are not
fixed, and the correspondences may not exist in all volumetric images. Therefore,
special feature selection strategies should be introduced to obtain a more suitable
vector for each volumetric image with fixed-length and correspondent elements from
the local features.
In future, we plan to compare various classifiers combined with suitable feature
selection techniques to further improve classification performances for
neurodegenerative diseases. In particular, we will first expand the correspondence of
the local features between subsets of subjects to the whole set via some strategies, and
then generate a new vector for each volumetric images with fixed-length and
correspondent elements. After that, some sophisticated feature selection and reduction
methods will be applied to produce a relatively short vector to avoid the "curse of
dimensionality" phenomenon34. Finally, various tests will be performed on the short
vectors to compare different classifiers (e.g., AdaBoost and support vector machine35)
on the OASIS dataset.
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Figure 1. Scale-invariant features extracted in a three-dimensional structural MR
image. Circles represent the locations and scales of features in the central coronal
slice (left) and the central axial slice (right). The features shown above are located
within 2 mm of the corresponding slice. Note how features reflect the spatial extent of
underlying anatomical structures, e.g., the size of sulci or ventricles.
Figure 2. Cross-validation surfaces for geometry thresholds on three different
divisions: (a) Subjects aged 60–80 years, CDR=1 (66 NC, 20 AD); (b) Subjects aged
60–96 years, CDR=1 (98 NC, 28 AD); (c) Subjects aged 60–80 years, CDR=0.5 and 1
(66 NC, 70 AD). Bandwidth is set as h  0.2 . The surfaces are relatively stable in
the neighborhoods of the maximum points.
Figure 3. Cross-validation curves for bandwidths on three different divisions,
given geometry thresholds respectively: division 1)  x'  7,    1.3 ; division 2)
 x'  7,    1.3 ; division 3)  x'  7,    1.8 . The dotted curve is the average of other
three curves. The optimal value for the bandwidth is selected at point where the
maximum of the averaged AUC curve is reached.
Figure 4. Comparison of ROC curves of two classification methods on three
different divisions: (a) Subjects aged 60–80 years, CDR=1 (66 NC, 20 AD); (b)
Subjects aged 60–96 years, CDR=1 (98 NC, 28 AD); (c) Subjects aged 60–80 years,
CDR=0.5 and 1 (66 NC, 70 AD). The red solid curves are the result of our method,
while the blue dashes are those of Toews.
Table 1. The memory amounts practically consumed to classify a new image on
the three divisions
Division 1)
Division 1)
Division 1)
Sample
size
85
125
135
Memory
(MB)
593
804
855