application of radus of curvature for desgning road in hill

8/8/2019 Application of Radus of Curvature for Desgning Road in Hill

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A

TERM PAPER OF

MATHS

TOPIC: application of radius curvature for

designing road in hills

Submitted To: Submitted By:M/s.

Lovely Institute of

Technology

Jalandhar-Delhi G.T. Road (NH-1), Phagwara, Punjab (INDIA) - 144402. TEL:

+91-1824-404404 Toll Free: 1800 102 4431 [email protected]


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ACKNOWLEDGEMENT

With regards I would like to thanks my Lect. M/s. who helped me in

completing my Term Paper on the topic CYCLOTRON, SYNCHROCYCLOTRON,BETATRON AND OTHER PRACTICAL ACCELERATORS USED AROUND THE

WORLD , AVANCEMENT IN PARTICAL ACCELATOR TECHNOLOGY, BRIFING

OF LARGE HADRON COLLIDER, FUTURE PROSPACTIVE AND APPLICATIONS.

. Of subject PHY. Due to his proper guidance and under the shower of his

eternal knowledge I was able to complete my Term Paper comfortably which

might not be possible without his efforts.

I must say thanks to my friend who helped me in the completion of my Term

paper. I must say sorry for the errors if I have committed in my Term Paper.

Date: -11-2010


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INDEX


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ABSTRACT

Model-based segmentation approaches, such as those employing Active Shape Models (ASMs),have proved to be useful for maths image segmentation and understanding. To build the model,however, we need an annotated training set of shapes wherein corresponding landmarks are

identified in every shape. Manual positioning of landmarks is a tedious, time consuming, anderror prone task, and almost impossible in the 3D space. In an attempt to overcome some of these

drawbacks, we have devised several automatic methods under two approaches: c-scale based andshape variance based. The c-scale based methods use the concept of local curvature to find

landmarks on the mean shape of the training set. These landmarks are then propagated to all theshapes of the training set to establish correspondence in a local-to-global manner. The variance-

based method is guided by the strategy of equalization of the shape variance contained in thetraining set for selecting landmarks. The main premise here is that this strategy itself takes care

of the correspondence issue and at the same time deploys landmarks very frugally and optimallyconsidering shape variations. The desired landmarks are positioned around each contour so as to

equally distribute the total variance existing in the training set in a global-to-local manner. Themethods are evaluated on 40 MRI foot data sets and compared in terms of compactness. The

results show that, for the same number of landmarks, the proposed methods are more compactthan manual and equally spaced methods of annotation, and the variance equalization method

tops the list.Keywords: curvature, shape description, variance equalization, shape models, landmarks,

segmentation.

1. INTRODUCTION

Segmentation and modeling of organs using model-based approaches such as Active Shape

Models (ASMs) 1requires a priori information often provided by manual annotation of a training

set of shapes. Manual positioning of landmarks is a tedious, subjective, time-consuming, and

error prone task, and almost impossible in the 3D space. To overcome the drawbacks of

manually creating a training set, it is necessary to perform automatic landmark tagging, which

could avoid the errors and the drudgery associated with manual annotation.Automatic landmark

tagging for model building consists of two main tasks: accurate landmark positioning in one

reference shape or in all shapes of the training set (depending on the method used), and

establishing landmark correspondences among shapes of the training set. These two steps are

interdependent, with many possible approaches for each of them. Some methods handle bothaspects in a tightly coupled manner, however, most of the methods focus on one of the two tasks.

Many papers just treat the correspondence part, taking one reference image on which landmarks

are positioned somehow, say manually. The order in which these two tasks are accomplished

leads to a classification of the existing methods into two groups: local-to-global and globalto-

local. In local-to-global approaches, local operations first define landmarks, among which

correspondence is subsequently established by global operations. In global-to-local methods,


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global operations, first carried out, lead to localization of landmarks which simultaneously

maintain and establish correspondence. Most of themethods found in the literature are local-to-

global approaches. Some of the local-to-global methods existing in the literature distinguish

between those focusing on the landmark positioning task and those centered on the landmark

correspondence task.

Landmarkpositioning

The selection of landmarks is usually achieved manually or automatically using methods to findcharacteristic features on shapes, which may be anatomical or mathematical2 . Themain premise

is to represent each shape of the training set with a set of landmarks or key points, to generate acompact Point Distribution Model (PDM)3 that will best capture the shape variation among the

training shapes considered. Therefore, each landmark must be located in the same manner in allthe examples of the training set. However, manual selection of landmarks in certain anatomical

or biological structures is not always trivial, and the same is true in finding correspondences

among landmarks in a non-homogeneous class of shapes, as in the case of disease states.Baumerg and Hogg4 proposed a method to select landmarks automatically on contours of humanfigures in walking pedestrian images. For each contour of the training set, a reference point is

defined at the lowest position of the point of intersection of the principal axis with the contour.This point is then used as the starting point of a cubic length-wise uniformly spaced B-spline,

where the control points are the selected landmarks. Walker et al.5 aimed at automaticallytraining appearance models of human face sequences. They use first and second order

normalized Gaussian partial derivatives to locate feature vectors on the first image of a sequenceand construct saliency images for different scales at coarse and fine levels. The correspondence

is tracked across frames distributing the features evenly across each object under certainconstraints of scale and distance. This work demonstrates that coarse scale features are more

reliable than fine scale features to obtain landmarks. One of the problems of this method comesto fore when features move or the shape of the object in different frames is not the same; then thecorrespondence fails. Also, features that are salient in one frame are not necessarily salient in

another frame. Another approach was introduced by Rohr6, 7 for landmark-based registration.He described differential operators for detecting landmarks using first order partial derivatives on

images to avoid instabilities of higher order partial derivatives. Souza and Udupa8 use polygonalapproximation of contours to find landmarks in the mean shape of the training set. The

landmarks are propagated to all the examples of the training set using a closest point strategy toestablish correspondence. This method obtains better results than manual and equally spaced

annotations, but the location of the landmarks does not precisely correspond to the dominantpoints on the shape, since the location of the points is dependant on the location of the initial

points.

Landmark correspondence

Other methods of automatic landmark tagging focus more on the correspondence task, and in

most of these cases, the initial selection of landmarks is achieved manually. The correspondencecan be established using closest point propagation, registration, parameterization, or optimization

approaches such as Minimum Description Length (MDL) strategies. Bookstein studied


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extensively the use of landmarks for the statistical study of biological shapes. In his work, initialanatomical landmarks are located manually on the shapes. Thin-plate splines and Procrustes

analysis are used to establish correspondence between curves among shapes of the training set.Landmarks are allowed to move along the contours in order to minimize the corresponding

bending energy of the splines. Other approaches, such as the one studied by Frangi et al., use

registration as a way of establishing correspondence among shapes. In this work, a 3D atlas isbuilt automatically using non-rigid registration in a training set of segmented images. A meanbinary volume is computed from the training set of binary images to create a 3D binary atlas

representative of the class of shapes considered. The landmarks are found on the shaperepresented by this binary volume by using the marching cubes algorithm to obtain dense

triangulation of the boundary surface. Decimation is applied to retain only the minimum numberof nodes (landmarks) necessary to represent the surface. Then, by using 3D elastic registration,

based on maximizing the mutual information, the landmarks are propagated to all the shapes ofthe training set. Hill et al. and Brett and Taylor describe a pairwise non-rigid correspondence

method for 2D and 3D respectively, using a binary tree of merged shapes. They achievecorrespondence by matching sparse polygonal representations without using curvature

estimations. Landmarks are placed on the mean shape at the root of the tree and propagated tothe original training set, corresponding to the leaves of the tree. The charge for registration is the

use of the reference shape to establish correspondences. Another popular method is the MDLapproach described by Davies et al. and Thodberg among others. This method treats landmark

correspondence as an optimization problem. In this type of work, the initial landmarks have to bedefined on a shape (in an equally spaced manner or with a priori knowledge of where they

should be), and then, the correspondence is optimized by minimizing a certain objectivefunction. This method performs better than the manual method and finds correspondences by

reparameterizing each shape of the training set. However, the method is complex,computationally slow, and the results obtained depend on the different steps used to implement

it. Furthermore, the algorithm does not make explicit use of known properties of the shapes andsometimes places landmarks in locations that do not seem appropriate for the human notion of

landmark correspondence. It has been shown by Thodberg that incorporating urvatureinformation into an MDL description gives better results than using MDL alone. However, this

paper also observed that this will be true only for shapes with low noise, due to the curvaturecalculations that were used.

In contrast with the methods presented previously, a global-to-local approach was introduced by

Rueda et al. recently for 2D shapes. The method consists of equalizing the variance contained ina training set for selecting landmarks. The strategy itself takes care of the correspondence issue

and at the same time deploys landmarks very frugally and optimally considering shapevariations. The desired landmarks are positioned around each contour in such a manner as to

equally distribute the total variance existing in the training set. In this method, landmarks maynot correspond to dominant points on the shapes, however, the method produces a compact

model with good representation of the variability existing in the training set.

In this paper, we compare two distinct and novel landmark tagging methods. The first method isa localto-global strategy that establishes landmark correspondence using the mean shape in the

training set and its curvature, whereas the second strategy is a global-to-local approach. Thelocal-to-global approach uses a robust shape descriptor based on curvature, called c-scale to


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define automatically mathematical landmarks with different levels of detail and in digitalboundaries. Landmarks are detected at different scales to vary the level of detail depending on

the application. Previous curvature estimations were not accurate in extracting object features,such as high curvature or inflection points, in digital boundaries, and considering different scales.

Because of the existing differences among shapes in the training set, we cannot assume that

corresponding points will lie on regions that have same curvature. Therefore, in order to use theconcept of curvature for landmark tagging, one way of establishing correspondence is throughthe mean shape. The mean shape represents the average variations that occur among shapes of

the training set. When we locate landmarks in the mean shape using c-scale, we are capturing themain features in terms of curvature from the training set. After detecting these landmarks, we can

propagate them into all the shapes of the training set, and thus, establish landmarkcorrespondence.

In this paper, which focuses on 2D shapes, we present the c-scale local-to-global landmark

tagging approach based on curvature in Section 2. Section 3 introduces the global-to-localstrategy based on variance. In Section 4, we compare the methods and show results in a maths

application in terms of compactness. Our conclusions and future directions are stated in Section5.

2. CURVATUREBASED LANDMARK TAGGING (LOCAL-TO-

GLOBAL)

2.1 c-Scale

Let b1, , bc be the points or boundary elements (bels) defining a boundary B. We define localcurvature scale segment or c-scale segment at any point b on a boundary B as the largest

connected set of points of B symmetrically situated with respect to b such that the distance d ofany of these points from a line connecting the two end points of the connected set is within a

fixed value t. We will associate with each point b = bi its c-scale segment C(b). This set is anindirect indicator of the curvature at b, as shown by Rueda et al.19 . To determine C(bi), we

progressively examine the neighbors, first the set of points bi2, bi, bi+2, then the set bi3, bi2,bi, bi+2, bi+3, and so on (Fig. 1). At each examination, we calculate the distance of the points in

the set from the straight line connecting the two end points of the set. If the maximum distance ofthese points from the line is greater than a threshold t, we define the c-scale segment C(b) of b as

the last set of connected points found for which the distance was still within the threshold. The c-scale value we assign to bi, denoted Ch(b) is the chord length corresponding to C(b), which is the

length of the straight-line segment between the end points bb and bf of C(b). If Ch(b) is large, it

indicates small curvature at b, and if it is small, it indicates high curvature. c-scale values arevery helpful in estimating actual segments


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and their curvature, independent of digital effects. The actual arc length A(b) at b corresponds to

the c-scale segment C(b) which can be derived from a knowledge of Ch(b) and by assuming thatC(b) locally represents a circular arc :

where r is the radius of the osculating circle at b, and s is the distance between the chord Ch(b) ofthe osculating circle and the boundary segment C(b) at b.

ExampleThe above principle is illustrated by using an example shape in Fig.2. This shape is constructed

from theoretical functions. It includes different parts such as a rotated rectangle, circular arcs ofdifferent radii (convex and concave), and a sine wave. The starting point of the boundary b1 is

represented in the figure by a cross. The boundary is oriented and follows the direction of thearrow, leaving always the inside of the object to the left. The order of the bels is defined using

this direction. To this shape, we apply the c-scale representation method


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with t = 0.02. This parameter controls the level of detail or the global scale. For digitalboundaries, we usually set t ! 3, to preserve appropriate boundary details and at the same time

ward against digitization noise. The values of the arc length (i.e., c-scale) for the bels along theboundary are represented in Fig.3.

The peaks in A(b) correspond to middle points of flat segments or to inflection points. The

valleys in A(b) represent high curvature points or corners. By detecting these peaks and valleys,we can detect the characteristic points of the shape for use in the automatic landmark tagging

process. We can use peaks, or valleys, or both.

2.2Shape Description via c-scale

In this section we present the method of boundary shape description based on c-scale concepts.

Given a (digital) boundary B and a scale parameter t, our goal is to obtain a partition PB of Binto segments and a set sL of landmarks (or characteristic dominant points).

First, A(b) is estimated for B. Then, A(b) is smoothed with a median filter of width 2w+1

centered at every element b, where w is the half width of the window. We repeat this process mtimes to get a smoothed version of A(b), denoted Af (b). This is necessary only for digital

boundaries. Then, we detect automatically the peaks and the valleys of Af (b) by usingmathematical morphology. The peaks correspond to the middle point of straight line segments in

the boundary and the valleys to the middle point of curved segments. To find the valleys, we

apply to Af (b) a bottom-hat filtering operation, which is the difference between Af (b) and itsclosing. Similarly, to find the peaks, we use a top-hat filtering operation, which is the differencebetween Af (b) and its opening. By selecting a different size of the structuring elements used in

these operations, we can vary the num ber of valleysand peaks detected. We can select mostprominent peaks and valleys by keeping only those that are greater than a certain value. This

allows us to fully control the number of dominant points we want for a given application.


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For the example in Fig.2, the peaks and valleys in Af (b) are shown in Fig.5. Thecorrespondingmiddle points of straight segments (or inflection points) and the corners ( or high

curvature points) detected on the shape in Fig.2 are shown in Fig.4.

Figure 4. The dominant points detected for the shape in Fig.2 using c-scale.

Figure 5. Peaks and valleys in Af (b) detected for the shape in Fig.2.

2.3 Landmarkselection via c-scale

Given a training set of M segmented images, the given training images are first aligned by usingaffine registration. Because we want to capture shape dominant points characterizing the training

set and not particular dominant points corresponding to a particular instance of the training set,we calculate the mean shape by using a signed distance transformoperation on the training set.

The boundary of thismean shape is then extracted on which the set of landmarks sL is selected by using the c-scale shape descriptor (Fig.6). The selected landmarks are subsequently

propagated from the mean shape to all shapes of the training set18 . With all landmarks located


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in all shapes of the training set, we can create the PDM needed to build the ASM. The mainaspects of

this process are the selection of landmarks on the mean shape, and the propagation of landmarks

to the shapes of the training set.

Two strategies were studied for landmark selection: hierarchical and non-hierarchical.

In hierarchical selection, we fix the parameters for a certain application to get the maximumnumber of

landmarks necessary to create a model. Then, to select the desired number of landmarks, the peaks are arranged in a descending order and the valleys are arranged in an ascending order

using their corresponding Af (b) values. In this manner, we can select the peaks and valleys

hierarchically from the most prominent to the least. This allows us to select precisely the desirednumber n of landmarks for the application by just considering only the n most prominent peaksand/or valleys. Higher peaks in Af (b) correspond to longer straight regions, whereas lower

valleys correspond to regions of higher curvature.

As opposed to the above hierarchical approach, we can also select n landmarks by adjusting the parameters each time independently. We call this method non hierarchical. This process of

adjusting the parameters needs to be repeated each time n changes, whereas in the hierarchicalapproach, the calculation is performed only once, at the beginning, and the number of landmarks

can be varied easily and without further processing.

Once we have detected the peaks and valleys, we locate the landmarks on the boundary of theshape by considering only valleys, only peaks, or both. This gives us overall six possibilities for

selecting landmarks via the c-scale method.

Two different approaches were devised for landmark propagation: closest point and parametric.


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In the first approach, for each landmark P in the mean shape, the point identified on a trainingshape is that point on its boundary that is closest to P. This identification is accomplished via a

distance transform applied to each P. We will call this method closest point propagation.

The second method considers a reparameterization of the contours of the training set and the

mean shape to have the same number of boundary elements and equal distance betweensuccessive points in each contour. After finding the appropriate landmarks on the mean shape,we directly propagate them to the contours of the training set by placing them at their

corresponding parametric positions.We will call this method parametric propagation.

We note that, from the three types of landmarks (peaks only, valleys only, peaks and valleys)identified hierarchically or non hierarchically, and from the two approaches of propagation, we

have described 12 possible methods of tagging landmarks altogether via c-scale.

3. VARIANCEBASED LANDMARK TAGGING (GLOBAL-TO-

LOCAL)

Given a training set ofM segmented images, we align them by using affine registration andextract the boundaries {Bj; j = 1, ...,M} of the segmented structures. After identifying the

corresponding starting point in all shapes Bj, we parameterize them so they have the samenumber N of equally spaced (roughly equal to the pixel spacing) points along each boundary. We

define N to be the minimum number of points among all Bj, because we want to retain as muchinformation in these boundaries as possible. This way we obtain a parametric description of the

boundaries; that is, {Bj(i); i = 1, ...,N; j = 1, ...,M}. A good model should pick more points wherethere is more variation so the variation can be captured as best as possible in the PDM. These

regions are where the landmarks are less correlated. In other regions, the landmarks will behighly correlated. From the M parameterized boundaries, we can determine the variance in the

location of any ith point, i = 1, ...,N, and express it as a function V ar(i). The total variance ateach ith point over all M shapes can be estimated by adding the eigenvalues of the covariance

matrix of the x and y coordinates at the ith point over all M shapes. Our goal is to placelandmarks by making use of this information in V ar(i). We wish to select more landmarks in

regions where V ar(i) is higher, and less where it is small. We will achieve this by equalizing thevariance V ar(i). Therefore, we will use the variance to guide the selection of points so that the

variance is more or less equally distributed with point density on the contours.

Once we specify the number of landmarks n we want, we distribute them around each contour ina way that the area under the variance curve between each successive pair of landmarks is

roughly the same. The selection of points will be done separately for each contour and they willget their point arrangement depending on their shape and its relationship to the variance graph.

Let xi, i = 1, ...,N, denote the ith point generically in any Bj,

let V ar(xi) be the total variance at xi, and let del be the spacing between points in Bj. Then,given V ar(xi), n, and del, we wish to find the points p1, ..., pn on Bj such that

V ar(p1) |p2 p1| + . . . + V ar(pn) |p1 pn| = V ar(x1)del + V ar(x2)del + . . . + V ar(xN)del.


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The right side of Equation 2 corresponds to the total variance, denoted TAj, in Bj over all itspoints, which is also the area under the V ar(xi) curve. | | denotes the distance along the contour

Bj between successive points. Therefore, |p2 p1| = (q + 1) del, if there are q points in Bjbetween p1 and p2 excluding p1 and p2. The algorithm may be summarized as follows:

Algorithm VE

Input:B j(i) : j = 1, ...,M, i = 1, ...,N; n; del.

Output: Landmarks pj 1, ..., pj n, j = 1, ...,M.

Step 1: Compute V ar(xi) for i = 1, ...,N.Step2: For j = 1, ...,M, compute the area per point for Bj as Ak = TAj n , where TAj is the total

variance (or area under the V ar(xi) curve) listed before, and k = 1, . . . , n.Step 3: Select the first landmark on each contour as the point on Bj at which V ar(xi) is

maximum. Call this point pj 1.

Step 4: From pj 1, go forward on Bj and skip points until we reach pj 2 such that V ar(pj 1) |pj 2

pj 1| " Ak. Once pj 2 is found, go forward from pj 2 and repeat this process until we come back

to pj 1.Step 5: Output pj 1, ..., pj n. 4

4. RESULTSWe tested the proposed methods on a set of 40 MRI foot images, the object of interest being the

talus bone of the foot (Fig.7). These objects were segmented by using an operator-steered LiveWire technique to create the

Figure 7. The talus bone of the foot in an MR image slice.


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training set of talus shapes. The mean shape was estimated from 40 training binary images byusing distance transform operations.

LandmarkSelection

In this part, we show how the selection of landmarks is achieved for the two groups of methods.Both groups are very different and the landmarks resulting from each group do not necessarily

have the same meaning. In the c-scale approach, landmarks are features of the shape, selecteddirectly on the mean shape and based on curvature, whereas, in the VE strategy, landmarks are

distributed around each shape, placing more landmarks in regions where the variance amongshapes is high and less where it is low, taking into account the real variability existing in the

training set.

Figure 8. 18 high curvature landmarks (valleys) detected on the mean shape using c-scale.


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Figure 9. 18 landmarks detected on a training shape via variance equalization.

In Fig. 8, 18 high curvature landmarks are located on the mean shape by using c-scale. In Fig. 9,

the same number of landmarks, obtained with the VE method, is represented in one of the shapesof the training set. The maximum variance in this training set occurred in the bottom part of the

shapes where more landmarks are tagged. Some of the shapes in the set are overlaid in Fig. 10 ingray to show the variability as well as the correspondence among landmarks in different shapes

of the training set. In the upper part of the structures where variance is much lower, the pointstagged are farther apart than in the lower part.

Figure 10. 18 landmarks detected on several training shapes via variance equalization.


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Comparison ofmethods

The methods are compared in terms of compactness of the model. Let xj be the 2n-dimensional

vector of landmarks for the jth training shape, defined as xj = (x1, y1, x2, y2, . . . , xn, yn)T, j =1, . . .,M. Then, we calculate the eigenvalues "i of the covariance matrix of these M vectors,

which correspond to the different modes of variations existing in the shapes of the training set.The manner in which a model captures the variations present in the training shapes is a function

of the number n of landmarks selected and the number l of modes selected. Therefore, weexpressthecompactnessofamodel in termsof both these variablesasa fractionof the variationpermode tototal variation, given by

#n,l indicates how well the variation is captured as a function of both the number of eigenvalues

(modes) and the number of landmarks selected. It is shown as a surface plot in Fig.11 for the VEmethod for the foot data set. To represent the comparison independently of the number of modes,

we calculate the area under the curve

Figure 11. !n,l for n = 1, ..., 28 and l = 1, ..., 56 for the variance equalization method.


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for each of the cases n = 1, ..., L, where L is the maximum number of landmarks considered inthe study. If we integrate #n,l over the l variable, we obtain

The #n values for all the methods compared can be seen in Fig.12. We can see that VE, c-scalefor valleys, and c-scale for valleys and peaks, perform similarly in terms of#n, and better than

equally spaced and manual landmark tagging. If we integrate #n over L landmarks, then we canexpress the overall ability of the method to capture variation by using up to L landmarks:


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Figure 12. !n for n = 1, ..., 18. Comparison between c-scale (peaks, valleys, peaks and valleys),variance equalization, equally spaced, and manual landmark tagging.

The # values are listed in Table 1 for all the methods compared. We notice that VE performs

better than any

Table 1. Comparison of methods in terms of ! value for peaks and valleys.

method, followed closely by the one that considers valleys only. All methods tested performbetter than equally spaced and manual landmark tagging approaches.

5. CONCLUSIONS

In this paper, we have presented and compared several novel methods for automatic landmark

tagging based on the new concepts of c-scale and variance equalization. We evaluated thecompactness of the models considering both the number of landmarks and the modes selected

and compared these methods with manual and equally spaced annotation methods. Our preliminary evaluation indicates that the proposed methods perform better. c-scale approaches

are able to capture dominant points at different scales (levels of detail) creating compact modelsfor different number of landmarks. We also showed that the distribution of landmarks by using

the variance equalization method is a good way of creating more compact models taking intoaccount the variability existing in the training set. These methods are applicable to spaces of any

dimensionality, although we have focused in this paper on 2D shapes. Both local-to-global andglobal-to-local strategies have their merits and demerits. It remains to be seen how these will pan

out especially in the 3D context.

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application of radus of curvature for desgning road in hill

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