Phương pháp chẩn đoán hình ảnh medical image analysis methods (phần 7)

Phương pháp chẩn đoán hình ảnh medical image analysis methods (phần 7)

2089_book.fm Page 271 Tuesday, May 10, 2005 3:38 PM Three-Dimensional Multiscale Watershed Segmentation of MR Images Ioannis Pratikakis, Hichem Sahli, and Jan Cornelis CONTENTS 7.1 7.2 7.3 7.4 Introduction Watershed Analysis 7.2.1 The Watershed Transformation 7.2.1.1 The Continuous Case 7.2.1.2 The Discrete Case 7.2.1.3 The 3-D Case 7.2.1.4 Algorithms about Watersheds 7.2.2 The Gradient Watersheds 7.2.3 Oversegmentation: A Pitfall to Solve in Watershed Analysis Scale-Space and Segmentation 7.3.1 The Notion of Scale 7.3.2 Linear (Gaussian) Scale-Space 7.3.3 Scale-Space Sampling 7.3.4 Multiscale Image-Segmentation Schemes 7.3.4.1 Design Issues 7.3.4.2 The State of the Art The Hierarchical Segmentation Scheme 7.4.1 Gradient Magnitude Evolution 7.4.2 Watershed Lines during Gradient Magnitude Evolution 7.4.3 Linking across Scales 7.4.4 Gradient Watersheds and Hierarchical Segmentation in Scale-Space 7.4.5 The Salient-Measure Module 7.4.5.1 Watershed Valuation in the Superficial Structure-Dynamics of Contours 7.4.5.2 Dynamics of Gradient Watersheds in Scale-Space 7.4.6 The Stopping-Criterion Stage 7.4.7 The Intelligent Interactive Tool Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 272 Tuesday, May 10, 2005 3:38 PM 272 Medical Image Analysis 7.5 Experimental Results 7.5.1 Artificial Images 7.5.2 Medical Images 7.6 Conclusions References 7.1 INTRODUCTION The goal of image segmentation is to produce primitive regions that exhibit homogeneity and then to impose a hierarchy on those regions so that they can be grouped into larger-scale objects The first requirement concerning homogeneity can be very well fulfilled by using the principles of watershed analysis [1] Specifically, our primitive regions are selected by applying the watershed transform on the modulus of the gradient image We argue that facing an absence of contextual knowledge, the only alternative that can enrich our knowledge concerning the significance of our segmented pixel groups is the creation of a hierarchy, guided by the knowledge that emerges from the superficial and deep image structure The current trends about the creation of hierarchies among primitive regions that have been created by the watershed transformation consider either the superficial structure [1–4] or the deep image structure [5, 6] alone In this chapter, we present the novel concept of dynamics of contours in scale-space, which integrates the dual-image structure type into a single one Along with the incorporation of a stopping criterion, the proposed integration embodies three different features, namely homogeneity, contrast, and scale Application will be demonstrated in a medical-image analysis framework The output of the proposed algorithm can simplify scenarios used in an interactive environment for the precise definition of nontrivial anatomical objects Specifically, we present an objective and quantitative comparison of the quality of the proposed scheme compared with schemes that construct hierarchies using information either from the superficial structure or the deep image structure alone Results are demonstrated for a neuroanatomical structure (white matter of the brain) for which manual segmentation is a tedious task Our evaluation considers both phantom and real images 7.2 WATERSHED ANALYSIS 7.2.1 THE WATERSHED TRANSFORMATION In the field of image processing, and more particularly in mathematical morphology, gray-scale images are considered as topographic reliefs, where the numerical value of a pixel stands for the elevation at this point Taking this representation into account we can provide an intuitive description of the watershed transformation as in geography, where watersheds are defined in terms of the drainage patterns of rainfall If a raindrop falls on a certain point of the topographic surface, it flows down the surface, following a line of steepest descent toward some local surface minima The set of all points that have been attracted to a particular minimum defines the catchment basin for that minimum Adjacent catchment basins are separated by divide lines or watershed lines A watershed line is a ridge, a raised line where two sloping Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 273 Tuesday, May 10, 2005 3:38 PM Three-Dimensional Multiscale Watershed Segmentation of MR Images raindrops 273 Watershed Line Regional Minima FIGURE 7.1 (Color figure follows p 274.) Watershed construction during flooding in two dimensions (2-D) surfaces meet Raindrops falling on opposite sides of a divide line flow into different catchment basins (Figure 7.1) Another definition describes the watershed line as the connected points that lie along the singularities (i.e., creases or curvature discontinuities) in the distance transform It can also be considered as the crest line, which consequently can be interpreted by two descriptions: firstly, as the line that consists of the local maxima of the modulus of the gradient, and secondly, as the line that consists of the zeros of the Laplacian These intuitive descriptions for the watershed-line construction have been formalized in both the continuous and discrete domain 7.2.1.1 The Continuous Case In the continuous domain, formal definitions of the watershed have been worked out by Najman [7] and Meyer [8] The former definition is based on a partial ordering relation among the critical points that are above several minima Definition 1: A critical point b is above a if there exists a maximal descending line of the gradient linking b to a Definition 2: A path γ: ] −∞, +∞ [ → R2 is called a maximal line of the gradient if ∀s ∈] − ∞, +∞[, γ (s ) = ±∇f [ γ (s )] ≠ and lim γ (s) = lim γ (s) = s→−∞ s→−∞ Definition 3: A maximal line is descending if ∀s ∈] − ∞, +∞[, γ (s ) = −∇f [ γ (s )] Definition 4: Let P(ƒ) be the subset of the critical points a of ƒ that are above several minima of ƒ Then the watershed of ƒ is the set of the maximal lines of the gradient linking two points of P(ƒ) This definition of Meyer [8] is based on a distance function that is called topographical distance Let us consider a function ƒ: Rn→R and let supp(ƒ) be its support The Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 274 Tuesday, May 10, 2005 3:38 PM 274 Medical Image Analysis topographical distance between two points p and q can be easily defined by considering the set Γ(p,q) of all paths between p and q that belong to supp(ƒ) Definition 5: If p and q belong to a line of steepest slope between p and q (ƒ(q) > ƒ(p)), then the topographical distance is equal to TD(p,q) = ƒ(q) − ƒ(p) Definition 6: We define a catchment basin of a regional minimum mi, CB(mi), as the set of points x ∈ supp(ƒ) that are closer to mi than to any other regional minimum with respect to the topographical distance j ≠ i ⇒ TD(x, mi) < TD(x, mj) Definition 7: The watershed line of a function ƒ is the set of points of the support of ƒ that not belong to any catchment basin  Wsh ( f ) = supp ( f ) ∩   ∪ i  [CB(mi )]  7.2.1.2 The Discrete Case Meyer’s definition [8] can also be applied for the discrete case if we replace the continuous topographical distance TF by its discrete counterpart Another definition is given by Beucher [1] and Vincent [9] The basic idea of the watershed construction is to create an influence zone to each of the regional minima of the image In that respect, we attribute a one-to-one mapping between the regional minima and the catchment basin Definition 8: The geodesic influence zone IZA(Bi) of a connected component Bi of B in A is the set of points of A for which the geodesic distance to Bi is smaller than the geodesic distance to any other component of B IZA(Bi) = {p ∈ A, ∀j ∈ [1,k]\{i}, dA(p,Bi) < dA(p,Bj)} Definition 9: The skeleton by influence zones of B in A, denoted as SKIZAB, is the set of points of A that not belong to any IZA(Bi) SKIZA(B) = A/IZA (B) with IZA(B) = ∪i∈[1,k] IZA(Bi) Definition 10: The set of catchment basins of the gray-scale image I is equal to the set X hmax obtained after the following recursion (Figure 7.2) X hmin = Thmin ( I ) ∀h ∈[ hmin , hmax − 1], X h +1 = h +1 ∪ IZ Th +1 ( I ) ( X h ) where hmin, hmax are the minimum and maximum gray level of the image, respectively Th(I) is the threshold of the image I at height h minh is the set of the regional minima at height h Definition 11: The set of points of an image that not belong to any catchment basin correspond to the watersheds Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 275 Tuesday, May 10, 2005 3:38 PM Three-Dimensional Multiscale Watershed Segmentation of MR Images Level Level 275 Level X1 X2 X3 Watershed lines FIGURE 7.2 (Color figure follows p 274.) Illustration of the recursive immersion process 7.2.1.3 The 3-D Case A brief but explicit discussion about watersheds in three-dimensional (3-D) space was initiated by Koenderink [10], who considered watersheds as a subset of the density ridges According to his definition, “the density ridges are the surfaces generated by the singular integral curves of the gradient, that is, those integral curves that separate the families of curves going to distinct extrema.” In cases where we consider only families of curves that go to distinct minima, then the produced density ridges are the watersheds For a formal definition of the watersheds in 3-D, the reader can straightforwardly extend the definitions in Sections 7.2.1.1 and 7.2.1.2 For the definition of Najman [7] in the 3-D case, we have to consider that the points in P(ƒ) are the maxima and that the two types of hypersaddles are connected to two distinct minima These points have, in one of the three principal curvature directions, slope lines descending to the distinct minima; the two slope lines run in opposite directions along the principal curvature direction These points make the anchor points for a watershed surface defined by these points and the slope lines that connect them 7.2.1.4 Algorithms about Watersheds The implementation of the watershed transformation has been done by using the following methods: iterative, sequential, arrowing, flow-line oriented, and flooding The iterative methods were initiated by Beucher and Lantuéjoul [11], who suggested an algorithm based on the immersion paradigm The method Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 276 Tuesday, May 10, 2005 3:38 PM 276 Medical Image Analysis expands the influence zones around local minima within the gray-scale levels via binary thickenings until idempotence is achieved The sequential methods rely on scanning the pixels in a predefined order, in which the new value of each pixel is taken into account in the processing of subsequent pixels Friedlander and Meyer [12] have proposed a fast sequential algorithm based on horizontal scans The arrowing method was presented by Beucher [1] and involves description of the image with a directed graph Each pixel is a node of the graph, and the node is connected to those neighbors with lower gray value The word “arrowing” comes from the directed connections of the pixels The flow-line oriented methods are those that make an explicit use of the flow lines in the image to partition it by watersheds [5] The flooding methods are based on immersion simulations In this category, there are two main algorithms The algorithm of Vincent and Soille [9] and the algorithm of Meyer [13] For an extensive analysis and comparisons of the algorithms that are based on flooding, the interested reader can refer to the literature [14, 15] 7.2.2 THE GRADIENT WATERSHEDS Whenever the watershed transformation is used for segmentation, it is best to apply it only on the gradient magnitude of an image, because then the gradient-magnitude information will guide the watershed lines to follow the crest lines, and the real boundaries of the objects will emerge It has no meaning to apply it on the original image Therefore, from now on, we will refer to gradient watersheds, thus explicitly implying that we have retrieved the watershed lines from the modulus of the gradient image Examples of gradient watersheds in two dimensions (2-D) and 3-D can be seen in Figure 7.3 and Figure 7.4–7.5, respectively The singularities of the gradient squared in 2-D occur in the critical points of the image and in the points where the second-order structure vanishes in one direction This can be formulated as: L2x + L2y = 0, ( L x = ∧ Ly = ) L2x + L2y ≠ ∧ Lww = ∧ Lwv = FIGURE 7.3 Gradient watersheds in 2-D Copyright 2005 by Taylor & Francis Group, LLC (7.1) (7.2) 2089_book.fm Page 277 Tuesday, May 10, 2005 3:38 PM Three-Dimensional Multiscale Watershed Segmentation of MR Images 277 (a) (b) FIGURE 7.4 (a) The cross-sections of the 3-D object and (b) their 3-D gradient watersheds where x, y denote Cartesian coordinates and w, v denote gauge coordinates [16] The gradient can be estimated in different ways It can be computed as (a) the absolute maximum difference in a neighborhood, (b) a pixelwise difference between a unit-size morphological dilation of the original image and a unit-size morphological erosion of the original image, and (c) a computation of horizontal and vertical differences of local sums guided by operators such as the Roberts, Prewitt, Sobel, or isotropic operators The application of gradient operators as in case c reduces the effect of noise in the data [17] In the current study, the computation of the gradient magnitude is done by applying the Sobel operator Accordingly, in the case of 3-D, the singularities of the gradient squared occur due to the following conditions L2x + L2y + L2z = 0, ( L x = ∧ Ly = ∧ Lz = ) L2x + L2y + L2z ≠ ∧ Lww = ∧ Lwv = ∧ Lwu = (7.3) (7.4) where x, y, z denote Cartesian coordinates and w, v, u denote gauge coordinates with w in the gradient direction and (u, v) in the perpendicular plane to w (the tangent plane to the isophote) Similar to the 2-D case, the gradient magnitude in 3-D can be estimated in different ways All of the existing approaches issue from a generalization of 2-D Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 278 Tuesday, May 10, 2005 3:38 PM 278 Medical Image Analysis FIGURE 7.5 A rendered view of the 3-D gradient watershed surface and the orthogonal sections edge detectors Lui [18] has proposed to generalize the Roberts operator in 3-D by using a symmetric gradient operator Zucker and Hummel [19] have extended to 3D the Hueckel operator [20] They propose an optimal operator that turns out to be a generalization of the 2-D Sobel operator The morphological gradient in 2-D has been extended to 3-D by Gratin [21] Finally, Monga [22] extends to 3-D the optimal 2-D Deriche edge detector [23] For the implementation of the gradient watersheds in 3-D, the current study has adopted the 3-D Zucker operator for the 3-D gradientmagnitude computation 7.2.3 OVERSEGMENTATION: A PITFALL TO SOLVE IN WATERSHED ANALYSIS The use of the watershed transformation for segmentation purposes is advantageous due to the fact that • • • Watersheds form closed curves, providing a full partitioning of the image domain; thus, it is a pure region-based segmentation that does not require any closing or connection of the edges Gradient watersheds can play the role of a multiple-point detector, thus treating any case of multiple-region coincidence [7] There is a one-to-one relationship between the minima and the catchment basins Therefore, we can represent a whole region by its minima Those advantages can be useful provided that oversegmentation, which is inherent to the watershed transformation, can be eliminated An example of oversegmentation is shown in Figure 7.6 This problem can be treated by following two different strategies The first strategy considers the selection of markers on the image and their introduction to the watershed transformation, and the second considers the construction of hierarchies among the regions that will guide a merging process The next sections of this chapter are dedicated to the study of methods following the second strategy Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 279 Tuesday, May 10, 2005 3:38 PM Three-Dimensional Multiscale Watershed Segmentation of MR Images 279 FIGURE 7.6 Example of an oversegmented image 7.3 SCALE-SPACE AND SEGMENTATION 7.3.1 THE NOTION OF SCALE As Koenderink mentions [24], in every imaging situation one has to face the problem of scale The extent of any real-world object is determined by two scales: the inner and the outer scale The outer scale of an object corresponds to the minimum size of a window that completely contains the object and is consequently limited by the field of view The inner scale corresponds to the resolution that expresses the pixel size and is determined by the resolution of the sampling device If no a priori knowledge for the image being measured is available, then we cannot decide about the right scale In this case, it makes sense to interpret the image at different scales simultaneously The same principle has been followed by the human visual front-end system Our retina typically has 108 rods and cones, and a weighted sum of local groups of them make up a receptive field (RF) The profile of such an RF takes care of the perception of the details in an image by scaling up to a larger inner scale in a very specific way Numerous physiological and psychophysical results support the theory that the cortical RF profiles can be modeled by Gaussian filters (or their derivatives) of various widths [25] 7.3.2 LINEAR (GAUSSIAN) SCALE-SPACE Several authors [24, 26–35] have postulated that the blurring process must essentially satisfy a set of hypotheses like linearity and translation invariance, regularity, locality, causality, symmetry, homogeneity and isotropy, separability, and scale invariance These postulates lead to the family of Gaussian functions as the unique filter for scale-space blurring It has been shown that the normalized Gaussian Gσ(x) is the only filter kernel that satisfies the conditions listed above: Gσ (x ) = Copyright 2005 by Taylor & Francis Group, LLC  x⋅x exp    2σ  (2 πσ ) d / (7.5) 2089_book.fm Page 280 Tuesday, May 10, 2005 3:38 PM 280 Medical Image Analysis (a) (b) (c) (d) FIGURE 7.7 An MR brain image blurred at different scales (a) σ = 1, (b) σ = 4, (c) σ = 8, (d) σ = 16 Here x⋅x is the scalar product of two vectors, and d denotes the dimension of the domain The extent of blurring or spatial averaging is defined by the standard deviation σ of the Gaussian, which represents the scale parameter An example of this spatial blurring can be seen in Figure 7.7 From this example, it can clearly be seen how the level of detail in the image decreases as the level of blurring increases and how the major structures are retained The scale-space representation of an image is denoted by the family of derived images L(x,σ) and can be obtained as follows: let L(x) be an image acquired by some acquisition method Because this image has a fixed resolution determined by the acquisition method, it is convenient to fix the inner scale as zero The linear scale-space L(x,σ) of the image is defined as L(x,σ) = L(x) ⊗ Gσ(x) (7.6) where ⊗ denotes spatial convolution Note that the family of derived images L(x,σ) depends only on the original image and the scale parameter σ Lindeberg [29] has pointed out that the scale-space properties of the Gaussian kernel hold only for continuous signals For discrete signals, it is necessary to blur with a modified Bessel function, which, for an infinitesimal pixel size, approaches the Gaussian function Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 300 Tuesday, May 10, 2005 3:38 PM 300 Medical Image Analysis FIGURE 7.20 (Color figure follows p 274.) Segmentation of the fingers in the HAND100 volume (Provided by Koen Vincken from the Image Sciences Institute, Utrecht, Netherlands.) Volume rendering of the segmented “whole hand” in artificial volume HAND100 is compared with the volume rendering of the thresholded versions of the original data (Figure 7.23) We observe that in the case that we use only thresholding, parts of the whole hand are either obscured by noise or disappear due to a high threshold However, the segmented whole hand preserves the features of the original data 7.5.2 MEDICAL IMAGES The dynamics of contours in scale-space algorithm in 3-D has been tested for the segmentation of the cerebellum of the brain The produced hyperstack consisted of four volumetric levels, which are shown in Figure 7.24, Figure 7.25, Figure 7.26, and Figure 7.27 In these figures, the different levels are demonstrated and compared by showing the same orthogonal views for each level For the segmentation of the Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 301 Tuesday, May 10, 2005 3:38 PM Three-Dimensional Multiscale Watershed Segmentation of MR Images 301 FIGURE 7.21 (Color figure follows p 274.) Segmentation of the palm in the HAND100 volume (Provided by Koen Vincken from the Image Sciences Institute, Utrecht, Netherlands.) cerebellum, the optimum coarse partitioning is obtained by selecting level four After the level selection, we clicked once with the mouse inside the area of the cerebellum The result can be seen in Figure 7.28 The red line indicates the delineation of the assigned object This coarse segmentation step produces a volume that is rendered in the same figure While the orthogonal views (which can be seen in Figure 7.28) indicate that the coarse segmentation is very close to the real object, the rendered view indicates that structures that not belong to the cerebellum have been assigned as such For example, the elongated part at the bottom of the rendered view does not belong to the cerebellum Thus, we have to browse through the selected hierarchical level for the coarse segmentation and indicate the parts of the 3-D object that have to be corrected Figure 7.29 and Figure 7.30 are examples of the coarse segmentation from other orthogonal views In these figures we can see that inclusion and exclusion of areas that not belong to the cerebellum are needed Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 302 Tuesday, May 10, 2005 3:38 PM 302 Medical Image Analysis FIGURE 7.22 (Color figure follows p 274.) Segmentation of the whole hand in the HAND100 volume (Provided by Koen Vincken from the Image Sciences Institute, Utrecht, Netherlands.) To refine our segmentation, we superimpose the coarse segmented object onto a level of the hyperstack that can provide the segments needed to refine the segmentation This superposition is shown in Figure 7.31, Figure 7.32, and Figure 7.33, which correspond to the coarse segmentation of Figure 7.28, Figure 7.29, and Figure 7.30, respectively Then the final segmentation can be achieved by including/excluding regions using mouse clicks or dragging over the regions The application of the refinement step has resulted in a final segmentation, which can be seen in Figure 7.34, Figure 7.35, and Figure 7.36 An improvement after the refinement step can also be seen in Figure 7.37, where a volume rendering from different views is given for the cerebellum in the case of (a) the coarse segmentation and (b) the segmentation after refinement with manual correction Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 303 Tuesday, May 10, 2005 3:38 PM Three-Dimensional Multiscale Watershed Segmentation of MR Images (a) 303 (b) FIGURE 7.23 (Color figure follows p 274.) Volume rendering of HAND100 in the case of (a) thresholding and (b) segmentation (Provided by Koen Vincken from the Image Sciences Institute, Utrecht, Netherlands.) 7.6 CONCLUSIONS In this chapter, we discussed our novel multiscale segmentation scheme, which is based upon principles of the watershed analysis and the Gaussian scale-space In particular, the proposed scheme relies on the concept of the dynamics of contours in scale-space, which incorporates a segment-linking that has been advocated by studying the topological changes of the critical-point configuration An algebraic classification for these topological changes for the gradient-squared evolution in 2D has been studied We have investigated the performance of the algorithm by setting up an objective evaluation method Our conclusion is that it performs better than algorithms using either the superficial or the deep image structure alone There is a very simple explanation for this behavior The proposed approach can integrate three types of information into a single algorithm, namely homogeneity, contrast, and scale, and therefore utilizes far more information to guide the segmentation process This good behavior of the algorithm gave us the hint that its extension to a fully 3D segmentation algorithm would be worthwhile Hence, we implemented this extension to 3-D, and our experimental observations are very optimistic Coupling the production of meaningful 4-D hyperstacks with a user interface adapted to 4-D data manipulation without requiring any training for the user, the 3-D algorithm can lead to robust and reproducible segmentations These conclusions have been drawn after experiments involving the use of both artificial and real medical images Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 304 Tuesday, May 10, 2005 3:38 PM 304 Medical Image Analysis FIGURE 7.24 (Color figure follows p 274.) Hierarchical hyperstack: Level FIGURE 7.25 (Color figure follows p 274.) Hierarchical hyperstack: Level Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 305 Tuesday, May 10, 2005 3:38 PM Three-Dimensional Multiscale Watershed Segmentation of MR Images FIGURE 7.26 (Color figure follows p 274.) Hierarchical hyperstack: Level FIGURE 7.27 (Color figure follows p 274.) Hierarchical hyperstack: Level Copyright 2005 by Taylor & Francis Group, LLC 305 2089_book.fm Page 306 Tuesday, May 10, 2005 3:38 PM 306 Medical Image Analysis FIGURE 7.28 (Color figure follows p 274.) Coarse segmentation FIGURE 7.29 (Color figure follows p 274.) Coarse segmentation Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 307 Tuesday, May 10, 2005 3:38 PM Three-Dimensional Multiscale Watershed Segmentation of MR Images 307 FIGURE 7.30 (Color figure follows p 274.) Coarse segmentation FIGURE 7.31 (Color figure follows p 274.) Coarse segmentation superimposed on a final hierarchical level Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 308 Tuesday, May 10, 2005 3:38 PM 308 Medical Image Analysis FIGURE 7.32 (Color figure follows p 274.) Coarse segmentation superimposed on a final hierarchical level FIGURE 7.33 (Color figure follows p 274.) Coarse segmentation superimposed on a final hierarchical level Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 309 Tuesday, May 10, 2005 3:38 PM Three-Dimensional Multiscale Watershed Segmentation of MR Images FIGURE 7.34 (Color figure follows p 274.) Final segmentation FIGURE 7.35 (Color figure follows p 274.) Final segmentation Copyright 2005 by Taylor & Francis Group, LLC 309 2089_book.fm Page 310 Tuesday, May 10, 2005 3:38 PM 310 Medical Image Analysis FIGURE 7.36 (Color figure follows p 274.) Final segmentation Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 311 Tuesday, May 10, 2005 3:38 PM Three-Dimensional Multiscale Watershed Segmentation of MR Images (a) 311 (b) FIGURE 7.37 Three different views for the segmentation of the cerebellum in the case of (a) coarse segmentation and (b) segmentation after refinement with manual correction Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 312 Tuesday, May 10, 2005 3:38 PM 312 Medical Image Analysis REFERENCES Beucher, S., Segmentation d’Images et Morphologie Mathématique, Ph.D Thesis, École Nationale Supérieure des Mines de Paris, Fontainebleau, 1990 Grimaud, M., A new measure of contrast: the dynamics, SPIE Proc., 1769, 292–305, 1992 Najman, L and Schmitt, M., Geodesic saliency of watershed contours and hierarchical segmentation, IEEE Trans Pattern Analysis and Machine Intelligence, 18, 1163–1173, 1996 Pratikakis, I.E., Sahli, H., and Cornelis, I., Hierarchy determination of the gradient 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Sporring, J et al., Eds., Kluwer Academic, Dordrecht, Netherlands, 1997, pp 147–163 50 Pratikakis, I.E., Sahli, H., and Cornelis, J., Low-level image partitioning guided by the gradient watershed hierarchy, Signal Process., 75, 173–195, 1999 51 Papoulis, A., Probability, Random Variables and Stochastic Processes, McGraw-Hill International, Singapore, 1991 52 Pizer, S.M., Cullip, T.J., and Fredericksen, R.E., Toward interactive object definition in 3-D scalar images, in 3-D Imaging in Medicine, Hohne, K.H et al., Eds., vol F60, NATO ASI Series, Springer-Verlag, Berlin, 1990, pp 83–105 53 Maes, F., Vandermeulen, D., Suetens, P., and Marchal, G., Automatic image partitioning for generic object segmentation in medical images, in Information Processing in Medical Imaging, Bizais, Y et al., Eds., Kluwer Academic, Dordrecht, Netherlands, 1995, pp 215–226 54 Pratikakis, I.E., Deklerck, R., Salomie, A., and Cornelis, J., Improving precise interactive delineation of 3-D structures in medical images, in Computer Assisted Radiology, Lemke, H.U., Ed., Elsevier, Berlin, 1997, pp 215–220 55 Hearn, D and Baker, M.P., Computer Graphics, Prentice-Hall, Englewood Cliffs, NJ, 1994 Copyright 2005 by Taylor & Francis Group, LLC [...]... 2005 3:38 PM 298 Medical Image Analysis FIGURE 7.17 Cross-sections of the HAND100 volumetric image (Provided by Koen Vincken from the Image Sciences Institute, Utrecht, Netherlands.) FIGURE 7.18 Rendering of the thresholded HAND100 volumetric image (Provided by Koen Vincken from the Image Sciences Institute, Utrecht, Netherlands.) (little finger) From this original image, two different images have been... studies 7.5 EXPERIMENTAL RESULTS The dynamics of contours in scale-space algorithm in 3-D has been tested on artificial test images and on real-world (medical) images 7.5.1 ARTIFICIAL IMAGES For the experiments, we have used the HAND100 artificial 3-D images The original HAND image contains the pixel values zero (background), 500 (thumb), 800 (forefinger), 1000 (palm of the hand), 1250 (middle finger... Obviously, this demonstrates why the placement of watershed Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 286 Tuesday, May 10, 2005 3:38 PM 286 Medical Image Analysis FIGURE 7.8 Successive blurring of the original image analysis into a scale-space framework makes it an attractive merging process Nevertheless, there is a major pitfall In Figure 7.10, it is clearly evident that, during... segmentation methods are known to be unreliable due to the complexity and variability of medical images, and they cannot be applied without supervision by the user On the other hand, manual segmentation is a tedious and time-consuming process, lacking precision and reproducibility Moreover, it is impractical when applied to extensive temporal and spatial sequences of images Therefore, to perform an image- segmentation... user input is minimized by constructing an image description rich in meaningful regions with low cardinality, and an interactive tool ensures accuracy and reproducibility without requiring any special Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 296 Tuesday, May 10, 2005 3:38 PM 296 Medical Image Analysis training by the user A meaningful image description is obtained by following... concepts are used for image- segmentation purposes 7.3.4.1 Design Issues For the implementation of a multiscale image- segmentation scheme, a number of considerations must be kept in mind A general recipe for any multiscale segmentation algorithm consists of the following tasks: Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 282 Tuesday, May 10, 2005 3:38 PM 282 Medical Image Analysis 1 Select... LLC 2089_book.fm Page 284 Tuesday, May 10, 2005 3:38 PM 284 Medical Image Analysis tracking in scale-space In a similar spirit, other authors have produced works in this field: Jackway [46] applied morphological scale-space theory to control the number of extrema in the image, and by subsequent homotopy-linking of the gradient extrema to the image extrema, he obtained a scale-space segmentation via the... robust and reproducible segmentations These conclusions have been drawn after experiments involving the use of both artificial and real medical images Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 304 Tuesday, May 10, 2005 3:38 PM 304 Medical Image Analysis FIGURE 7.24 (Color figure follows p 274.) Hierarchical hyperstack: Level 1 FIGURE 7.25 (Color figure follows p 274.) Hierarchical... that the inner scale σ0 of the initial image is taken to be equal to the linear grid measure ε At coarse scales, the ratio between successive scales will be about constant, while at fine scales the differences between successive scales will be approximately equal 7.3.4 MULTISCALE IMAGE- SEGMENTATION SCHEMES The concept of scale-space has numerous applications in image analysis For a concise overview, the... Watershed Segmentation of MR Images 289 Annihilation Creation (a) (b) (c) FIGURE 7.11 (Color figure follows p 274.) Generic events for gradient-magnitude evolution (a) (b) Scale N + 1 Scale N (c) FIGURE 7.12 Linking in two successive levels: (a) scale N and (b) scale N + 1 Copyright 2005 by Taylor & Francis Group, LLC 2089_book.fm Page 290 Tuesday, May 10, 2005 3:38 PM 290 Medical Image Analysis successive