WO2011057318A1  Representing a surface of an object appearing in an electronic image  Google Patents
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 WO2011057318A1 WO2011057318A1 PCT/AU2009/001370 AU2009001370W WO2011057318A1 WO 2011057318 A1 WO2011057318 A1 WO 2011057318A1 AU 2009001370 W AU2009001370 W AU 2009001370W WO 2011057318 A1 WO2011057318 A1 WO 2011057318A1
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 G06—COMPUTING; CALCULATING; COUNTING
 G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
 G06T17/00—Three dimensional [3D] modelling, e.g. data description of 3D objects
 G06T17/20—Finite element generation, e.g. wireframe surface description, tesselation
Abstract
Description
REPRESENTING A SURFACE OF AN OBJECT APPEARING IN AN
ELECTRONIC IMAGE
Technical Field
The invention concerns representing threedimensional surface information of an object appearing in an electronic image. For instance, but not limited to, the invention is used to represent the threedimensional surface information of an organ for medical diagnostic purposes. Aspects of the invention includes a method, software and,, apparatus.
Background Art
In medical applications, the ability to assess the shape of an organ captured in an image is important. The better the representation of the object in the image, the better the ability to assess the shape for clinical and diagnostic reasons. This may include identifying that a particular shape is indicative of certain medical conditions, such as diseases and organ failure. This may involve comparing the organ captured i an image over time to identify changes or by comparison to a test organ shape.
In the following. description a "shape" is a surface of genus zero; for example, a. sphere has genus zero, while a torus has genus one (the hole). The surface may be closed (the sphere) or open (the two halves of a sphere cut in two). A "shape descriptor" is a vector describing a shape. "Curvature" is used to describe the shape of a surface at a given point. It is also a mathematical term where the shape of a curve at a point is measured by the curvature, and the shape of a surface at a point is . measured by two principal curvatures. . "Shape index" and "curvedness" are two different measures of shape deriving from the principal curvatures. The shape index describes the nature of the shape (say convex or concave), the curvedness describes its "size". If the shape is say convex, is it strongly so or is it rather planar? "Scalespace" concerns a description of a shape at different scales. W
2
"Partitioning" is where the object under consideration is partitioned into several parts according to some methodology.
A "part" is any one of the regions of the object resulting from partitioning this object.
5
Summary of the Invention
In a first aspect the invention provides a computer implemented method for representing threedimensional surface information of an object appearing in an electronic image, the method comprising the steps of:
10 (a) partitioning the object into multiple threedimensional parts; and
(b) for each part, determining a representation of the threedimensional surface of that part using a scale space analysis of the shape index and curvedness of the surface of that part.
15 The representation offers a complementary understanding of the nature of the shape of the object using a shape index and curvedness for surface description, a scale space analysis of the surface and a localisation of that analysis through a partitioning of the surface.
20 It is an advantage of the invention that the representation of the threedimensional surface of the object is specific to particular regions (i.e. parts) of the shape. In this way the shape descriptor is not a single global descriptor of the shape of the object. By being specific to particular regions the representation of the surface is better able to capture subtle differences in the shape classes (i.e. roughness vs smoothness).
25
The object appearing in the image is a data set defining points on the surface of the object, such as a mesh. Examples are genus zero voxelbased meshes or triangular . meshes,
30 Partitioning of step (a) may be based on identifying the smallest threedimensional box that will contain the object, and dividing the object based on axes of the box.
Step (b) may comprise determining for points on the surface of the object a shape index that describes the nature of the shape at that point. The shape index of a point may be 35 based on a determination of the principal curvatures at that point. Where the object appearing in the image is represented as a voxelbased mesh, the method may comprise approximating the principal curvatures at a point using a covariance approach and assigning a sign to the shape index of that point based on an assessment of the sighs of the minimum and maximum onedimensional curvature at that point. The shape index may represent one of the following classes of shape nature:
concave ellipsoid
concave cylinder
hyperboloid .
convex cylinder
convex ellipsoid.
Step (b) may comprise determining the curvedness (i.e. degree) of the points on the surface of the object The curvedness may be based on a determination of the principal curvatures at that point. The method may further comprise determining the mean curvedness of points over consecutive scales.
Scale space analysis of step (b) comprises tracking a point across multiple scales of the three dimensional surface. Scale space analysis of step (b) may comprise obtaining multiple scales of the shape by smoothing the surface and determining the shape index and curvedness of points for each scale. A point may be tracked in the scale space provided it meets a predetermined criteria based on shape index. The tracked point may have a shape that changes from convex to concave (i.e. fold). The method may comprise determining whether the point is described by a predetermined shape index for consecutive scales, and if so for how many scales did the shape index of that point remain substantially the same (i.e depth).
Step (b) may comprise determining a representation of the three dimensional surface of . that part by:
determining a measure that distinguishes between surfaces based on number of tracked points and the number of consecutive scales the shape index remained substantially the same for each of these tracked points; and
. weighting that measure based on the curvedness of the tracked points.
The method may further comprise forming a descriptor . to represent the three dimensional surface information of the object comprised of the representations of each part. The descriptor may be a vector. The method may comprise using the threedimensional representation to classify and compare the shape of the object according to given criteria. For .example, compare the surface information of the same objects over time or over a population.
The object may be an internal organ of a human or other animal, and the comparison may be for diagnostic purposes or for making clinical measures. For example, identifying neurodegenerative diseases (such as Parkinson's) and potentially organ failure.
The method may be repeated on a different image in which the same type of object appears, and the method may further comprise using the threedimensional representation to compare the surface of the object. This may comprise aligning and comparing each part
Steps (a) and (b) may be automatic.
In another aspect the invention is software (i.e. instructions recorded on a computer readable storage medium storing executable instructions for implementing the method described above).
In yet a further aspect the invention comprises a computer apparatus adapted determine a representation of a threedimensional surface information of an object appearing in an electronic image, the apparatus comprising:
a partitioning module to partition the object into multiple threedimensional parts; and
a representation module, that for each part, is operable to determining a . representation of the threedimensional surface of that part using a scale space analysis of the shape index and curvedness of the surface of that part.
The apparatus may be further adapted to perform the method described above. Brief Description of the Drivings
An example of the invention will now be described with reference to the accompanying drawings, in which:
Fig. 1 is a schematic diagram of an example apparatus of the invention; Fig. 2 is a flow chart of this example of the invention;
' Fig. 3 shows the mapping of shapes to the index [1 , +1] ;.
Fig. 4(a) and Fig. 4(b) are tables showing the shape classification using the shape index of this example;
Fig. 5 shows a sample genus zero voxelbased mesh representing a hippocampus;
Figs. 6(a), (b) and (c) show the partitioning of the hippocampus of Fig. 5;
Figs, 7(a) and (b) schematically shows the determining of a shape index of. the hippocampus of Fig. 5, shown again as Fig. 5(a), and determining the shape index is repeated again for the smoothed version of the shape as shown in Fig, 7(b);
Fig, 8 shows the location of transition points on the object as a black line;
Figs. 9(a), (b) and (c) shows the tracking of transition points over different scales; and
Fig. 10 is a table showing the results of one example of the invention.
·
Best Modes of the Invention
This example is performed on an apparatus, such as a computer system shown in Fig. I, The computer system 10 comprises an input port 12 to receive information of the shape. In this example the input shape information is genus zero voxelbased meshes that describe the . object that was originally captured in an electronic image 22. As an alternative example triangular meshes could be used. The computer system also comprises memory 16 to store the voxelbased meshes and has installed software that causes its processor 1 to retrieve the voxelbased meshes from memory 16. The processor 14 comprised of a partitioning module 14a, a determination module 14b and a comparative module 14c which in this example are a combination of hardware and software. Using the retrieved voxelbased meshes the processor 14 performs the method shown in Fig, 2 to produce a representation of threedimensional surface information. Alternatively, the processor 14 can also operate to calculate the voxel based meshes from the electronic images 22 themselves that are received at the input port 12.
The predetermined presentation is then provided to an output port 18 where they can be used in further analysis, such as diagnostic analysis. Alternatively, this further analysis is performed by the processor 16 and the outcome of the analysis is provided to the output port 18. In this example, the further analysis is performed by the processor 14 in the comparative module 14c that uses the determined representation and comparative data that is also stored in memory 16. In this example, the output port 18 is connected to a display device 20, such as a monitor, to present to the user in a graphical user interface the result of the further analysis. The method performed by the processor 14 will now be described with reference to Fig. 2. First, the mesh data of the images is received at the input port 12 the partitioning module 14a uses a partitioning scheme to automatically identify multiple three dimensional parts of the object 200. Then for each part, the determination module 14b determines a representation of the threedimensional surface of that part using a scale space analysis of shape index and curvedness of the surface of that part 202. These steps will be described in fivrther detail below, where in this example the object is a hippocampus. . .
Firstly, step 202 will be explained in further detail,
Shape Index
Firstly, the shape index will be explained. Following oenderink [10] we define the shape index at a point ^ as
S =— arctaa f¾ > ¾
where ^{K}'i and ^^{2} are the principal curvatures at ^ , The shape index effectively maps the surface at into ^ ^ ^{"}f^{"} ψϋ¾ the convention that &— +/— 1 f_{or} umbilic points (when ¾ ^ we have a mapping of all the shapes, except the plane, into[0, +1], This is pictured in Fig. 3 (from Koenderik [10]).
In this example use a simplified mapping that provides an easy shape classification according to the shape index. This is given in Fig, 4(a), Curvedness
The shape index is a measure of the nature of the shape of a neighbourhood at a point. To get a measure of "size", that is, to be able to determine if a convex region is rather planar or rather very "curved", we introduce the curvedness at a point [10], which is defined as
■
Surface Representation and Alignment
In this example the 3D shapes are hippocampal shapes and are represented as genus zero voxelbased meshes ([1]). They are aligned using the ellipsoid resulting from computing the coefficients of the 1st order spherical harmonic basis function ([7]). Voxelbased meshes are formed of the vertices, edges and faces of the genus zero surface voxels. We say that two points are neighbours ,if they are joined by an edge. Principal Curvatures Computation
A Discrete Method:
To compute the curvature at each point, we adopt a discrete approach introduced by Caelli et al. [8] which approximates the principal curvatures using a covariance method. This is particularly adapted to our data representation as a voxelbased mesh. Using traditional parametric methods to compute the curvature on this mesh would introduce singularities.
We briefly describe the covariance method here. It essentially consists of two steps. Assume we want to compute the principal curvatures at some point . Let ^ be the set of points whose distance from ^ is no more than some fixed ^{w}. For all practical purposes, what is meant here by distance between two points is the length of the shortest path between them: for example, a point and its neighbour are at distance one from each other. First, we compute the covariance matrix of all the points in & . Using the eigenvalues · of the covariance matrix we compute the tangent plane ^~ at ^ as the plane that maximizes the variance of the point projections on it. In the second step, we compute the covariance matrix of the projections on the tangent plane 1^{"} of the normals of the points in . The two eigenvalues of this matrix are then the approximation to^{'} the principal curvatures ^{Λ}ι and ^{Ki} at P . We set >¾ and ^{K}» so that ≥ ¾ .The Problem of Signed Principal Curvatures
As is apparent from the description above the principal curvatures obtained are always positive; that is, we have lost the information that describes the shape as being convex, concave, or hyperbolic (anticlastic).
Following an idea in [11] we may render the curvatures as a signed quantity as follows. Let ^{p} and Q be two points on the surface and let ^{n}P and ^{n}Q be their respective normals. Then the sign of the onedimensional curvature at ^ in the direction of Q is set as being positive if ll^  QII≤ I ** + ^) ^{~} and negative otherwise.
To set the signs for ^{K}i and ^{Λ}¾ at ^{P} , we proceed as follows ( ® is defined as above):
(i) By choosing points Q in ® that are reasonably "far" away from ^\ we compute the minimum "^™ and the maximum ^{maa>} onedimensional curvature at P in the direction of Q,
(ii) Tfboth ^{KmiV?} and ^{K}™^{atc} are of the same sign, we set ^{K}i and to this sign.
(iii) If ^{K}'^{min} and ^{κ}**^{αιχ)} are not of tlie same sign we set ^{K}l and ^{A}'^{2} to opposite signs. We can see that some ambiguity still remains from (iii) since we cannot distinguish between a saddle ridge or a saddle valley. Moreover, since we have — l^{Ka}', the range of the shape index function is actually [0, 1]. That is, even if our method allows us to correctly compute the mean ^{ft}'^{2}^ and the Gaussian * ^{K<}^ curvatures at each point, that is, we can distinguish between convex, concave and hyperbolic shapes, the shape index function in this case only allows us to distinguish between hyperbolic and nonhyperbolic shapes. The points with a shape index above 3/8 denote a convex or convex shape, while the points with a shape index between 0 and 3/8 denote hyperbolic or saddle points (see Fig. 4(b)). Surface Characterization
Tracking Shape Change and Stability Analysis;
For each part . we determine a representation that reflects both the. density of the wrinkles or folds on a given hippocampal surface as well as the depth height of these folds. Computation of this representation is in two steps: (i) finding transitions points (i.e. computing the shape index of points at different scales and identifying the transition points), and
(ii) tracking those across the successively smoothed surfaces (i.e. at different scale spaces).
The fact that in our case the shape index has range [0, 1] is not a hindrance since we are only interested in how the shape changes along the surface. Essentially we are interested in the changes between el iiptic. (convex or concave) and hyperbolic shapes, By setting £\= Ό for planar shapes (for which S is actually undefined), we are able to track shape change by tracking across consecutive scales the points for which S— 3/S _{as seen} i_{n} jg_{,} (¾)_{,}
We perform a shape/stability analysis based on ideas in [12] and [6]. Let's call the original surface ·'^° and the surface obtained from ^^{0} and smoothing (i.e. scaling) ^{•} steps "^*. As before ^^{ΰ} is a voxelbased mesh; we smooth while preserving its neighbourhood topology: the numbers of mesh points and the neighbourhood of points is preserved in all "^* meshes. Smoothin is achieved via a 3D Gaussian convolution G(ar, z; σ) « e*p ·^{■}¾ +· _ _{ff} p _{¾ ¾} ^ _{fa M}^ ^ _{p<} ^ ^ obtained by convolving * times and which thus belongs to "^*. Let be the total number of convolution/smoothing steps:
(i) For each mesh "^* we find the points whose shape index has a value of 3/8. We call these points crossings (i.e. fold or transition points). That means the shape has changed from concave to convex., or vice versa, and must involve a saddle point. We identify edges in the voxelbased mesh for which the values of the shape index of the endpoint vertices indicate such a transition. They are the edges with an endppoint having a shape index value smaller than 3/8 while the other end point vertex has a shape index value larger than 3/8. Such an edge indicates a transition from a convex or concave patch to saddle and we make the midpoint as a transition point. .
(ii) We then track each transition point across the resulting successively smoothed surfaces. For each point ^^{€} we form a characteristic vector ^{Vjp'} of length^{€} + ^{1} so that »P[*J = 1 if Pi is a crossing in ^{Mi}, and ^{V}P W = ^{0} otherwise (°≤ ^{i}≤ ^{c}). ■ Call a subsequence of ^{Vp} consisting of consecutive ones ^{Sp}. There may be more than one such subsequence in ^{Vp}, we denote them by ^{Si} f' ^— $— ^{Up}, where ^{N}P is the number of subsequences of ^{Vp} consisting of consecutive ones. Also, let ^^{j}p be the length of ^{s}3P, which we call the depth of ^{8}3 , SO for the point P, the number of consecutive smoothings (i.e. scales) for which P is a transition point is its depth. It may be the case that^{*} a point is a transition point over several successive sequences of consecutive smoothings, in which case .p denotes the depth of the/th such transition sequence. The number of points ? for which there exists subsequences and their respective depth will inform us of the nature of the surface. The challenge is to best capture this information to describe the surface in a meaningful way. This is explored below.
A Measure for Surface Characterization
There are many different ways to use the information given by the vectors ^{Vp} to derive a measure characterizing the surface. As already noted, ^{Vp} provides us with two pieces of information:
(i) the number ^{N}P of subsequences ^{&3} in ^{V p},
(ii) and their respective length *^>^{p}.
Over a specific surface patch «M, we interpret ∑P€M ^{1ip} as the number of crossings over M.. We have chosen to count the presence of each subsequence to denote one crossing. On one hand, we want to distinguish between surfaces with many crossings and those with few crossings; but we also need to account for the fact that some crossings have a larger depth than others. For example, a surface with heavy wrihkliness will have many crossings of small depth, while a smooth surface will have fewer crossings but which are likely to be of significant depth (for example picture a closed surface that is smoothed over and over). To distinguish between those surfaces we need a measure 202c that privileges the number of crossings against their depth:
(1)
P<SM 3=1 In (1) no account is taken of the curvedness of the shapes. For a given subsequence ^{s}$p let ^{C}3 P be the mean curvedness of the points ^¾ across multiple scales which are the crossings making up ^{&}$P. That is, ¾ P is the mean of the curvedness of the points which denote a crossing in i.e. a transition point at scale i.
We modify (1) to be weighted 202d by ^{c}ap thus:
M_{M}  ∑ ¾P p) (2)
PGM 3=1 In (2) the points with large curvedness are privileged; indeed for a point to be a crossing we intuitively expect it to be reasonably curved. The aim here is to capture the shape index and curvedness information collected across multiple scales. In this example we use (2) as the representation characterizing a surface. Partitioning the Hippocampus
Step 200 will now be described.
Since our measure (2) is based on the shape index it is more a local measure than a global measure. That is, it is more meaningful to compute (2) over a defined surface patch than over the whole surface. The idea is then partitioning the shape under consideration in a way that is meaningful and helpful in interpreting the value of (2) for the resulting patches. Below we describe one approach in partitioning the hippocampus; it is clear that the paititioning itself will be dependent of the particular shapes/surfaces.
Automatic Partitioning
Partitioning the hippocampus could be performed following the functional anatomy as outlined by Duvernoy in [3], but this would require performing manual segmentations on the data which is one way of performing this step. Clearly, an automated approach is preferable and multiple automatic methods could be used. We describe there the preferred method. Considering the smallest 3 rectangular box containing the hippocampus it is natural to distinguish three axes: the longest or longitudinal axes, the shortest axis, and the middle axis. In this example partitioning the hippocampus as follows:
(i) Along the longitudinal axis the hippocampus is subdivided into a head, a body and a tail following three different methodologies which we describe below.
(ii) Each of the head, body and tail are further subdivided into two lateral parts along the shortest axis, and into a ventral and dorsal part along the middle axis.
In total, the hippocarapal surface is thus subdivided into 12 parts. See Fig. 6 for an illustration.
Anterior, Central, and Posterior parts
We adopt three different methodologies for partitioning the hippocampus into anterior, central and posterior parts. They are due to J. Mailer [2], First we note that the hippocampal surfaces do not contain the posterior section of the tail (here the posterior part) as they have been handtraced using the Watson et al, protocol [9]. This posterior section has been estimated at being 11 ,32% of the overall length (largest axis) of the hippocampus [4]. We have named the three methodologies ISO, ISOTAIL and HACKERT respectively. Partitioning using ISO into anterior, central and posterior sections is 1/3, 1/3, 1/3 respectively of the longest axis of the hippocampus without the posterior section, of the tail. Partitioning using ISOTAIL into anterior, central and posterior sections is 1/3, 1/3, 1/3 respectively of the longest axis of the hippocampus considered together with the estimated (11.32%) posterior section of the tail. Finally, partitioning for HACKERT into anterior, central and posterior sections is 35%, 45%, 20% respectively of the hippocampus considered together with the posterior section of the tail. The later subdivision is due to Hackert et al. [5]. Medial and Lateral
To define the medial and the lateral parts of the anterior, central and posterior sections, consider the smallest 3D rectangular boxes containing each of them. Subdivide these further into three boxes along the middle axis to obtain the medial box, the lateral box, and two remaining boxes. We call these boxes the superior and inferior section respectively. For example, taking the 3D box containing the anterior part of the hippocampus we obtain the medial box and the lateral box for the anterior part by delirninating an equal percentage of length along the middle axis at both extremities. See Fig.6 for a sample partitioning.
A New Shape Descriptor
Combining (2) with the parts of the hippocampus ; results in a 12elements shape descriptor, with one value representing each part, as follows:
{M_{AI},M_{AL} ,M_{A}„,M_{AL},M_{C>} ,M_{CI} ,M_{CU} ,M_{CI} ,M_{PT} ,M_{PI},M_{PU},M_{H} ] . Applications of the invention include analysis of threedimensional digital data or . neuroanatomical structures like the hippocampus with the .aim of correlating shape changes over time with the onset of Alzheimer's disease. If changes in the hippocampus that are indicative of this disease are. very subtle, then the invention can be used to detect the subtle differences that other previously used methods such as volumetric measures are not able to detect.
Also, the shape descriptor will help in classifying and discriminating shapes for various factors like sex, age, clinical diagnosis. The shape descriptor of the invention may be used in any situation requiring understanding of the nature of a shape or surface. Other applications include object recognition and object classification in computer vision problems.
For example; the invention is able to detect surface change of the hippocampus which may be an early predictor of dementia onset in nondemented elderly subjects. This example of the invention is able to describe the shape of the hippocampus at a more localised level which can be more sensitive as it may reflect neuro degeneration in smaller clusters of cells which may not affect previously used global shape descriptors.
This application will now be described in further detail. Fig. 5 shows an input genus zero voxelbased mesh representing a hippocampus that is provided as input 198. Then, this mesh is partitioned 200 as shown in Figs. 6(a), (b) and (c). Partitioning is performed by partitioning hippocampus into anterior, central, and posterior sections, these three parts being further divided into superior, inferior, medial and lateral parts. In Fig, 6 the anterior, central, and posterior sections are indicated, by the physical partition of the shape along the longitudinal axis; the superior and inferior parts are shown in the area of 300 and 302 respectively; the medial and lateral parts are shown at 304 and 306 respectively.
Next the representation of the threedimensional surface of each part is determined 202. Computing the shape index at different scales is schematically shown in Figs. 7(a) and (b) with shading on the surface representing the shape index for that point. Computation of the shape index of points on a non smoothed surface is shown in Fig. 7(a) and on the same surface after smoothing as shown in Fig. 7(b). The transition points are then identified, that is points at a particular scale that show a transition between saddle or plane regions to convex or concave regions. The solid lines drawn on the surface of the object in Fig. 8 shows the location of these transition points. Then these transition points are tracked over several scales. The result of tracking points over three scales is shown schematically at Figs. 9(a), (b) and (c).
The resulting representation of each part is then computed using Equation (2). Computing the surface index of each of the twelve parts we obtain a vector of length 12 describing the shape of the hippocampus under consideration.
Assuming one is looking at a certain population of hippocampus shapes, and assuming these shapes have been aligned (say, using a standard method such as alignment to the . first order ellipsoid), one may then compare the values of each of the parts. over that population by doing standard statistical analysis.
In this example, the population consists of 169 right hippocampal shapes from the . publicly available OASIS database (www.oasis~brains.org) which have been hand traced by an expert neurologist We define the following:
A J[ to be the surface index value for the inferior anterior part;
A_S superior anterior part;
C_I inferior central part;
A whole anterior part;
W whole shape/hippocampus. ^{"}We do a logistic regression with clinical dementia rating (CDR) as the dependent variable, The data set is divided into two classes: the class of hippocampi with CDR = 0, that is without a diagnosis for Alzheimer's disease (AD); and the class of • hippocampi with CDR = 1 , that is with a diagnosis for AD.

The population we are considering comes with other attributes, among them: sex, age, years of education, and nWBV, the normalized whole brain volume. For ease of exposition we denote by shape the surface index measure for any of the subregion of the hippocampus.
For the logistic regression, we consider four models in turn:
m_l) with covariates sex, age, educ, and shape;
mJ2) with covariates sex, age, educ, nWBV, and shape;
m_3) with covariates sex, age, educ, volume, and shape;
m_4) with covariates sex, age, educ, nWBV, volume, and shape.
In addition to logistic regression we. also perform a best fit analysis. We compare each model ra__l , .. m_4, to its reduced counterpart without the shape covariate.
That is, given the fours models
m'_l) with covariates sex, age, educ;
nV_2) with covariates sex, age, educ, nWBV;
m'_3) with covariates sex, age, educ, volume;
m'_4) with covariates sex, age, educ, nWBV, volume;
we compare m_l with m'_l, m_2 with m'_2. m_ 3 with m'_3, and m_4 with m^{1}_4.
The table of Fig. 10 shows the results. Each column shows results for a specific model. Each entry shows a) the subregion with a statistically significant (p < 0,05) correlation with the CDR variable in a given model; and b) a pair of pvalues. The first pvalue corresponds to the correlation of the shape variable with CDR, and the second pvalue corresponds to the best fit of the given model with its reduced counterpart.
For example, the entry in the first row and first column from the table of Fig. 10 indicates that the subregion A_I is statistically correlated to CDR with p = 0.014562 and that m_l is reliably different from m'_l with p = 0.001442, Importantly, note that all entries in the table show statistically significant correlation and best fit. In other words, we only show subregions with a statistically significant relation to CD , and we omit results for the other subregions. For the right hippocampus, we see that^{'}A_I shows more discriminative power overall. A_I is significantly related to CDR after accounting for demographics (demo = age + sex + education) and volume (models m_l and m_3).
The results for model m_2 reveal a mixed picture, where A_l is significant in the 6089 and 6099 age bracket, and again in the 8099 age band. The three age bands starting at age 70 show that A_S is significant. Finally, we can see that other regions (C_I, A, W) show a significant relation to CDR after accounting for demographics . and volume (model m_3). In summary we may say that A_l is significantl correlated to CDR after accounting for demographics and volume.
It will be appreciated by persons skilled in the art that numerous variations and/or modifications may be made to the invention as shown in the specific embodiments without departing from the scope of the invention as broadly described. The present embodiments are, therefore, to be considered in all respects as illustrative and not restrictive.
References
[1] Lieby, P., Barnes, N., McKay, B.D.: Topological Repair on VoxelBased Quadrangular Meshes. In: Proceedings of the Workshop on Mathematical Foundations of Computational Anatomy MICCAI 2006 (2006) 146155
. [2] Mailer, J.: Personal communication (2006)
[3] Duvernoy, H.M.: The Human Hippocampus. 3 edn. Springer Verlag (2005)
[4] Mailer, J.J., R'egladeMeslin, C, Anstey, K.J., Sachdev, P.: Sex and Symmetry Differences in Hippocampal Volumetrics: Before and Beyond the Opening of the Crus of the Fornix. Hippocampus 16(1) (2005) 8090 [5] Hackert, V.H., den Heijer, T., Oudkerk, M., KoudstaH, P.J., Hofman, A., Breteler, MM.: Hippocampai Head Size Associated with Verbal Memory Performance in Nondemented Elderly. Neuroimage 17(3)^{'} (2002) 136572 [6] Mokhtarian F,. Khalili N.. Yuean P.: Curvature Computation on Freeform 3D Meshes at Multiple Scales. Computer Vision and Image Understanding 83 (2001) 118 139
[7] Kelemen, A., Szekely, G., Gerig, G.: Elastic ModelBased Segmentation of 3D Neuroradiological Data Sets. IEEE Trans. Medical Imaging 18(10) (1999) 828839
[8] Caelli, T., Osman, E., West, G,: 3D Shape Matching and Inspection Using Geometric Features and Relational Learning. Computer Vision and Image Understanding 72(3) (1998) 340350
[9] Watson, C, Jr., C.J., Cendes, F.: _{.}Volumetric magnetic resonance imaging. Clinical applications and contributions to the understanding of temporal lobe epilepsy. Archives of Neurology 54 ( 1997) 15211531 [10] Koenderink, J.: Solid Shape. MIT Press (1990)
[11] Flynn, P.J., Jain, A.K.: On Reliable Curvature Estimation. In: Proceedings CVPR '89, IEEE Computer Society Conference on Computer Vision and Pattern Recognition (1989) 110116
[12] Bischof, W.F., Caelli, T.: Parsing ScaleSpace and Spatial Stability Analysis, Computer Vision, Graphics, and Image Processing 42 (1988) 192205
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JP2015504543A (en) *  20111023  20150212  ザ・ボーイング・カンパニーＴｈｅＢｏｅｉｎｇ Ｃｏｍｐａｎｙ  Shape modeling of composite parts, including ply laminate and a resin 
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