I have a set measurements f ={ fk|k = 1,··· ,K} that define lung structure and function.
Each measurement fk (Table3.1) provides information that is sensitive to COPD and lung
structure. I have K= 5 image features: f1= Jacobian determinant, f2= eigenvalue trace,
f3= eigenvalue variance, f4= inspiration voxel intensity and f5= expiration voxel inten-
sity.
I am interested in going beyond global averages that are tradtionally used as features in lung CT image classification. In this section, I propose a new methodology to sample each feature fkthroughout the lung to capture local feature distributions (Figure3.3). These local
distributions are parameterised to build a rich feature set that models local aspects of lung pathophysiology. The collection of local measurements stemming from each fk represents
the feature set that can be exploited as training data to build a classifier.
3.2.3.1 Feature distributions
I propose to sample the local variation of features ( fk) to quantify their distribution across
the lung. This is performed by considering histograms (hi( fk; xj, φi)) of the local distribu-
tions of fk. Each local feature distribution is centered at a voxel xj ( j= 1···J) within a
neighbourhood ω governed by the scale φi, where i= 1···n indexes the scale of the neigh-
bourhood ω and j is the jthsampled neighbourhood. Thus, distributions at increasing scales of analysis (φi) can be computed (Figure3.3). The histograms are modelled by the first 4
statistical moments and the median. The feature fk within ω centered at xjis defined by:
Hj fk(xj)
={µ(h1) ν(h1) σ (h1) γ1(h1) γ2(h1) ···
µ(hn) ν(hn) σ (hn) γ1(hn) γ2(hn)}
(3.5)
where µ is the mean, ν the median, σ the variance, γ1 the skewness and γ2 is the
kurtosis. The vector Hj fk(xj)
thus contains the mean, median, variance, skewness and kurtosis of the local distribution of fk centered at xjand measured at the scales φ = φ1,···,
φn.
the lung. This enables to creation of a patient-specific matrix (Hp) such that Hp= H1 f1(xj) ··· H1 f k(xj) %LAA1ins/exp(xj)∀φ .. . ... ... HJ f1(xJ) ··· HJ f k(xJ) %LAAJins/exp(xJ)∀φ . (3.6)
Each row of Hprepresents the set of measurements for the feature set f at the sampled
region centered at xj. Each column of Hprepresents the collection of a particular histogram
measurement across the lung at sampled regions x= xj,··· ,xJ.
The Jacobian determinant (det(F)), the trace (∑ λ ) and variance (Var (λ )) of the strain eigenvalues and the voxel densities in Iins and Iexp are modelled locally across the lung (k= 5). I also incorporate the %LAA− 950HU and %LAA − 856HU for all φi, leading
to 27n features per xj. The number of sampled regions is determined by the sampling
frequency of xjat the finest scale (φ1). For example, if there were J= 1000 sampled regions
and n= 3 scales of analysis, Hpwould be∈ R1000x27·3. It is the matrix Hpthat is used as a
feature within this chapter.
1
Hj(fk(x0)) ={µ(h1) ⌫(h1) (h1) 1(h1) 2(h1) · · ·
µ(hn) ⌫(hn) (hn) 1(hn) 2(hn)}
2
Figure 3.3: Illustration of the multi-scale sampling for a feature fk. 1) A feature fk (e.g. ∑ λ ) at
xj is sampled at n= 3 scales, leading to 3 local histograms hi( fk; xj, φi). 2) Statisti-
cal moments and the median of hi( fk; xj, φi) are calculated for all φi, leading to the set
Hj fk(xj). © 2014 Springer Nature. Reprinted, with permission from F. Bragman et
al., Multi-scale Analysis of Imaging Features and its Use in the Study of COPD Ex- acerbation Susceptible Phenotype, Medical Image Computing and Computer Assisted Interventions, 2014.
3.2.3.2 Statistical feature analysis
A. Hypothesis testing using Hp
Analysis of the distribution of values contained within each Hp allows hypotheses of
changes in the global nature of local features to be made. For instance, consider the distribution of the variance of det(F) at all xj. Each value demonstrates the local vari-
ation in volume change. The distribution of this measure across the lung will illustrate how the local variation is expressed, which may vary across subtypes. This facilitates a direct comparison of patient-specific distributions across phenotypes. To evaluate the distribution of various measures, one must analyse Hpcolumn-wise.
B. Principal Component Analysis of X
I am interested in modelling the distribution of parameters across the studied population. To perform this, the feature matrix Hp for each patient P is concatenated such that
X=
H>1 ··· H>P
where X∈ RNJx 27nand N
J is total amount of regions sampled
across all subjects P (Figure 3.4). I apply PCA on X. This seeks a low-dimensional projection (d<< 27n) of X into Y (Figure 3.5), where the variance of the projected features is maximised.
Figure 3.4: Illustration of the population matrix X. Each patient matrix Hpis concatenated to pro-
duce X. The columns of X represent the different features obtained by analysis of the local distributions. The rows of X represent the local neighbourhood of a patient p. Each patient matrix Hpcan be extracted from X by considering the row index i of Xi, j. For
the first patient in X, Hp=1can be extracted by considering the sub-matrix Xi=(1,···,J),∀ j.
The entries of X are representative of the local histogram features measured at multiple scales. PCA of X allows me to compute the component scores within each neighbour- hood defined by xj (Figure 3.5). Thus, the computed scores can be projected to the
image space to assess their distribution across the lung for each patient. Since the com- ponent scores are linear projections of the features measured at various scales, they will
Figure 3.5: Dimensionality reduction of the population matrix X into Y. The column-wise dimen- sionality of X is reduced from 27n to d by seeking a linear projection of the histogram features. The projection of each patient-specific matrix Hpwithin X into the new space
of Y is known since the row indices of Y are equivalent to those in X. Thus, the projec- tion of Hp=1within X to Y can be found by Yi=(1,···,J),∀ j. This allows one to project the
principal component scores back into the image space since each yijbelongs to a region of the lung from a patient p. One can also calculate new features by considering the column-wise mean and variance of the linear projections.
capture potential fractal properties in line with the nature of the lung anatomy. The dis- tribution of the principal component scores can be analysed to model patient-specific distributions by computing their respective mean and variance. This is done by calcu- lating the column-wise mean and variance in Y for each patient (Figure 3.5). Thus, phenotype-specific distributions can be parameterised to produce a clinically meaning- ful classifier. Importantly, classification in the PCA subspace prevents overfitting as PCA removes colinearity in the features.