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... algorithm. Published in the IEEE Transactions on Image Processing, Vol. 2, No. 2, pp. 176{201, April 1993. 1s1 Introduction Reconstruction isavery useful operator provided by mathematical morphology =-=[18, 19]-=-. Although it can easily be de ned in itself, it is often presented as part as a set of operators known as geodesic ones [7]. The reconstruction transformation is relatively well-known in the binary c...

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...tion of setY , de ned as the intersection ofX and the standard dilation ofY . Note that some authors use a di erent terminology and utilize the word \conditional" for what this paper calls \geodesic" =-=[5]-=-. (a) 4-connectivity (b) 8-connectivity Figure 6: Boundaries of the successive geodesic dilations of a set (in black) within a mask. When performing successive elementary geodesic dilations of a setY ...

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...he minimal number of edges of the grid to cross to go from pixelpto pixelq. In 4-connectivity, this distance is often called city-block distance whereas in 8-connectivity, itisthe chessboard distance =-=[2]-=-. The elementary ball in distancedG is denotedBG, or simplyB. We denote byNG(p) the set of the neighbors of pixelp for gridG. In the following, we often consider two disjoined subsets ofNG(p), denoted...

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... too many of these crest lines are due to noise in the original data. Therefore, the watersheds of Fig. 15.b yield the over-segmented result of Fig. 15.c. As explained in numerous recent publications =-=[28, 20,30,13]-=-, the correct way to use watersheds for grayscale image segmentation consists in rst detecting markers of the objects to be extracted. The design of robust marker detection techniques involves the use...

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...one orientation for which the vessels are not completely removed by opening. After taking the supremum of these di erent openings, one gets Fig. 10.b, which is still an algebraic opening of Fig. 10.a =-=[19]-=-. It is used as marker to reconstruct the blood vessels 9sentirely. Fig. 10.c is the result of the grayscale reconstruction of Fig. 10.a from Fig. 10.b. Since the aneurisms are disconnected from the b...

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... mark the co ee beans. By marker of an object, we mean a connected component of pixels located inside the object to be extracted. Once correct markers have been obtained, the watershed transformation =-=[1]-=- allows us to achieve the desired segmentation automatically. It is therefore crucial to design robust marking procedures. In the case where the objects to be separated are roughly convex, the ultimat...

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... on the notion of regional maxima and uses breadth- rst image scannings enabled by a queue of pixels [25]. The second one is a combination of this scanning technique with the classical sequential one =-=[14]-=-, and it turns out to be the fastest algorithm in almost all practical cases. We shall exclusively be concerned here with the discrete case. The algorithms are described in 2D, but their extension to ...

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...ted are roughly convex, the ultimate erosion usually provides a satisfactory marking [28]. This transformation is obtained as the regional maxima of the distance function of the original binary image =-=[2, 15]-=-|Recall that the distance function dist(I) of binary imageI assigns with every pixelp its distance to the background, i.e.. to the closest pixel with value 0. However, in many cases, due to the fact t...

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... of this image yields Fig. 13.d, which is an accurate set of cell markers. Additional examples of application of theh-dome transformation can be found in [23, 26, 3], and more details can be found in =-=[4]-=-. Note that the results of this section can easily be \inverted" to extract minima andh-basins in grayscale images. 3.4 Grayscale reconstruction and binary segmentation The examples reviewed above ill...

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...is a local maximum, but the plateau may have neighboring pixels of higher altitude and thus not be a regional maximum. An alternative de nition can also be proposed for the notion of regional maximum =-=[23]-=-: De nition 3.2 Aregional maximum at altitudehof grayscale imageI is a connected component C ofTh(I) such thatC\Th+1(I) =;. (Recall from eq. (4) thatTh(I) is threshold ofI at levelh.) Determining the ...

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...nstruction isavery useful operator provided by mathematical morphology [18, 19]. Although it can easily be de ned in itself, it is often presented as part as a set of operators known as geodesic ones =-=[7]-=-. The reconstruction transformation is relatively well-known in the binary case, where it simply extracts the connected components of an image which are \marked" by another image (see x 2). However, r...

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...tegically located pixels [23, 26, 22]. These two class of methods can be used to e ciently implement such complex morphological operations as propagation functions [9, 16], watersheds [30], skeletons =-=[24]-=- and many others [17]. Here, we shall be concerned with the second class of algorithms. The breadth- rst scannings involved are implemented by using a queue of pixels, i.e., a First-In-First-Out (FIFO...

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...highly dependent on the type of connecticity which is used. This notion is illustrated by Fig. 5. Geodesic distance was introduced in the framework of image analysis in 1981 by Lantuejoul and Beucher =-=[6]-=- and is at the basis of several morphological operators [7]. In particular, one can de ne geodesic dilations (and similarly erosions) as follows: De nition 2.2 LetX Z 2 be a discrete set of Z 2 andYX....

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...given mask [16], whereas the algorithms of 19sthe second category regard the images under study as graphs and realize breadth- rst scannings of these graphs starting from strategically located pixels =-=[23, 26, 22]-=-. These two class of methods can be used to e ciently implement such complex morphological operations as propagation functions [9, 16], watersheds [30], skeletons [24] and many others [17]. Here, we s...

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...esI taking their values inf0; 1;:::;N, 1g, it su ces to consider the successive thresholdsTk(I) ofI, fork= 0toN, 1: Tk(I)=fp2DIjI(p)kg: (4) They are said to constitute the threshold decomposition ofI =-=[10, 11]-=-. As illustrated by Fig. 7, these sets obviously satisfy the following inclusion relationship: 8k2[1;N, 1];Tk(I)Tk,1(I): When applying the increasing operation to each of these sets, their inclusion r...

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...ight) blood vessels and mainly located in the dark central area of the image. Obviously, it is impossible to detect these micro-aneurisms via simple thresholdings. Similarly, a top-hat transformation =-=[12]-=- consisting in subtracting from the original image its morphological opening with respect to a small disc would extract all the \white" features, i.e. aneurisms and blood vessels, which is not desirab...

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... theh-domes of Fig. 13.a. An easy thresholding of this image yields Fig. 13.d, which is an accurate set of cell markers. Additional examples of application of theh-dome transformation can be found in =-=[23, 26, 3]-=-, and more details can be found in [4]. Note that the results of this section can easily be \inverted" to extract minima andh-basins in grayscale images. 3.4 Grayscale reconstruction and binary segmen...

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...ct marking of our objects, thereby yielding a correct segmentation, as illustrated by Figs. 14.d{e. For more details on the use of watersheds and grayscale reconstruction for binary segmentation, see =-=[28, 30]-=-. 3.5 Watershed segmentation of grayscale images Similarly, the markers/watersheds methodology applies to grayscale segmentation. This task consists in extracting objects from a gray-level image as pr...

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.... This segmentation methodology is commonly used in morphology and has been successfully applied to various types of images: NMR images [23], digital elevation models [21], corneal endothelial images =-=[29]-=-, succession of images used for motion estimation [3], and many others. 4 Computing reconstruction in digital images In this section, we are concerned with both the binary and the grayscale case, but ...

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... out the image parts which cannot hold the disc. Recall that the opening of a binary image I by a disc is the union of all the possible positions of the disc when it is totally included in the imageI =-=[18, 28]-=-. In some cases, one wishes to lter out all the connected components which cannot contain the disc while preserving the others entirely. The way to do so is to reconstruct the original imageI from its...

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...ted components of the mask image, i.e., each of these components is assigned a unique number. Note that this step can itself be implemented very e ciently by using algorithms based on chain and loops =-=[16]-=- or queues of pixels [23, 26]. 2. Determine the labels of the connected components which contain at least a pixel of the marker image 3. Remove all the connected components whose label is not one of t...

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...el algorithms, the scanning order is essential! This type of algorithm was rst introduced for the computation of distance functions [15] and then extended to a number of morphological transformations =-=[8, 23]-=-. Among others, binary and grayscale reconstruction can be obtained sequentially by using the following algorithm, where information is rst propagated downwards in a raster scanning and then upwards i...

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...s of these graphs starting from strategically located pixels [23, 26, 22]. These two class of methods can be used to e ciently implement such complex morphological operations as propagation functions =-=[9, 16]-=-, watersheds [30], skeletons [24] and many others [17]. Here, we shall be concerned with the second class of algorithms. The breadth- rst scannings involved are implemented by using a queue of pixels,...

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...e on conventional computers. Two algorithms are introduced to bridge this gap. The rst one is based on the notion of regional maxima and uses breadth- rst image scannings enabled by a queue of pixels =-=[25]-=-. The second one is a combination of this scanning technique with the classical sequential one [14], and it turns out to be the fastest algorithm in almost all practical cases. We shall exclusively be...

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... theh-domes of Fig. 13.a. An easy thresholding of this image yields Fig. 13.d, which is an accurate set of cell markers. Additional examples of application of theh-dome transformation can be found in =-=[23, 26, 3]-=-, and more details can be found in [4]. Note that the results of this section can easily be \inverted" to extract minima andh-basins in grayscale images. 3.4 Grayscale reconstruction and binary segmen...

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...esI taking their values inf0; 1;:::;N, 1g, it su ces to consider the successive thresholdsTk(I) ofI, fork= 0toN, 1: Tk(I)=fp2DIjI(p)kg: (4) They are said to constitute the threshold decomposition ofI =-=[10, 11]-=-. As illustrated by Fig. 7, these sets obviously satisfy the following inclusion relationship: 8k2[1;N, 1];Tk(I)Tk,1(I): When applying the increasing operation to each of these sets, their inclusion r...

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...els [23, 26, 22]. These two class of methods can be used to e ciently implement such complex morphological operations as propagation functions [9, 16], watersheds [30], skeletons [24] and many others =-=[17]-=-. Here, we shall be concerned with the second class of algorithms. The breadth- rst scannings involved are implemented by using a queue of pixels, i.e., a First-In-First-Out (FIFO) data structure: the...

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... nition using threshold superposition It has been known for several years that|at least in the discrete case|any increasing transformation de ned for binary images can be extended to grayscale images =-=[18, 19, 31, 20]-=-. By increasing, we mean a transformation such that 8X;Y Z 2 ;YX =) (Y ) (X): (3) In order to extend such a transformation to grayscale imagesI taking their values inf0; 1;:::;N, 1g, it su ces to cons...

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... nition using threshold superposition It has been known for several years that|at least in the discrete case|any increasing transformation de ned for binary images can be extended to grayscale images =-=[18, 19, 31, 20]-=-. By increasing, we mean a transformation such that 8X;Y Z 2 ;YX =) (Y ) (X): (3) In order to extend such a transformation to grayscale imagesI taking their values inf0; 1;:::;N, 1g, it su ces to cons...

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...tion, as illustrated by Fig. 15.f. This segmentation methodology is commonly used in morphology and has been successfully applied to various types of images: NMR images [23], digital elevation models =-=[21]-=-, corneal endothelial images [29], succession of images used for motion estimation [3], and many others. 4 Computing reconstruction in digital images In this section, we are concerned with both the bi...