Application 2 Detect Filarial Worms Source BTT Remove Noises Threshold The first 12 points of the skeleton branches, counting from their extremities, are eliminated. The structures that were not eliminated will be the markers for extracting the two worms. Skeleton Eliminate short structures Reconstruction Final result

Ultimate Erosion Ultimate Erosion (UE) is based on Recursive Erosion operation. “Keep aside each connected components just before it is removed throughout the recursive erosion process”.

Geodesic Influence Geodesic Influence (GI) is based on Recursive Dilation operation with mask which also called conditional dilation. Reconstruct the seeds by the restriction of the mask, and distribute the pixels on the interface by means of “first come first serve”.

UE and GI UE: split a connected region (have to be convex) gradually and record the iteration number. GI: Reconstruct the split regions and get the segments.

Application Segment connected organs: RE: region shrinking to generate all the candidate seeds GI: region reconstruction to recover separated organs

(c) structuring element(d) initial point inside the boundary; Figure 4.22: Region filling: (a) boundary of an object; (b) complement of the boundary; (c) structuring element(d) initial point inside the boundary; (e)-(h) various steps of the algorithm; (i) final result, obtained by forming the set union of (a) and (h).

Watershed transform A grey-level image may be seen as a topographic relief, where the grey level of a pixel is interpreted as its altitude in the relief. A drop of water falling on a topographic relief flows along a path to finally reach a local minimum. Intuitively, the watershed of a relief correspond to the limits of the adjacent catchment basins of the drops of water. Cardiac MRI image Gradient image Relief of the gradient Watershed of the gradient Watershed of the gradient (relief)

GENERAL DEFINITION A drainage basin or watershed is an extent or an area of land where surface water from rain melting snow or ice converges to a single point at a lower elevation, usually the exit of the basin, where the waters join another waterbody, such as a river, lake, wetland, sea, or ocean

INTRODUCTION The Watershed transformation is a powerful tool for image segmentation, it uses the region-based approach and searches for pixel and region similarities. The watershed concept was first applied by Beucher and Lantuejoul at 1979, they used it to segment images of bubbles and SEM metallographic pictures

𝑓(𝑥) is the gray value of the image at point 𝑥 IMAGE REPRESENTATION We will represent a gray-tone image by a function: 𝑓: ℤ 2 →ℤ 𝑓(𝑥) is the gray value of the image at point 𝑥 A section of 𝑓 at level 𝑖 is a set 𝑋 𝑖 (𝑓) defined as: 𝑋 𝑖 𝑓 = 𝑥∈ ℤ 2 :𝑓 𝑥 ≥𝑖 And in the same way we define 𝑍 𝑖 (𝑓) as: 𝑍 𝑖 𝑓 = 𝑥∈ ℤ 2 :𝑓 𝑥 ≤𝑖 ⟺ 𝑋 𝑖 𝑓 = 𝑍 𝑐 𝑖+1 𝑓

REMINDER-IMAGE GRADIENT An image gradient is a directional change in the intensity or color in an image. Image gradients may be used to extract information from images.

IMAGE GRADIENT an intensity image  a gradient image in the x direction measuring horizontal change in intensity a gradient image in the y direction measuring vertical change in intensity

IMAGE GRADIENT The morphological gradient of a picture is defined as 𝑔 𝑓 = 𝑓⨁𝐵 − 𝑓⊝𝐵 Where 𝑓⨁𝐵 is the dilation of 𝑓 and 𝑓⊝𝐵 is its erosion. But because 𝑓 is continuously differentiable, 𝑔 𝑥 is nothing more than the modulus of the gradient of 𝑓: 𝑔 𝑓 = 𝑔𝑟𝑎𝑑 𝑓 𝑥 = 𝜕𝑓 𝜕𝑥 2 + 𝜕𝑓 𝜕𝑦 2 1/2

GEODESIC DISTANCE For two points 𝑥,𝑦 𝜖𝑋 when 𝑋⊂ ℤ 2 we define the geodesic distance 𝑑 𝑋 (𝑥,𝑦) as the length of the shortest path (if any) included in 𝑋 and linking 𝑥 and 𝑦. Let 𝑌 be any set included in 𝑋, then: 𝑅 𝑋 𝑌 ={𝑥∈𝑋:∃𝑦∈𝑌, 𝑑 𝑋 𝑥,𝑦 𝑓𝑖𝑛𝑖𝑡𝑒} is the set of all points of 𝑋 that are at a finite geodesic distance from 𝑌.

GEODESIC ZONE OF INFLUENCE The geodesic zone of influence of 𝑌 𝑖 (when 𝑌 is composed of 𝑛 connected components 𝑌 𝑖 ) is the set of points in 𝑋 whose finite distance is closest to 𝑌 𝑖 (among all 𝑌 components) 𝑔𝑧 𝑋 𝑌 𝑖 ={𝑥∈𝑋: 𝑑 𝑋 𝑥, 𝑌 𝑖 finite and ∀ 𝑗≠𝑖, 𝑑 𝑋 𝑥, 𝑌 𝑖 < 𝑑 𝑋 𝑥, 𝑌 𝑗 }

GEODESIC SKELETON BY ZONES OF INFLUENCE The boundaries between the various zones of influence give the geodesic skeleton by Zones of influence of 𝑌 in 𝑋. 𝐼𝑍 𝑋 𝑌 = 𝑖 𝑔𝑧 𝑋 ( 𝑌 𝑖 ) 𝑆𝐾𝐼𝑍 𝑋 𝑌 =𝑋/ 𝐼𝑍 𝑋 (𝑌)

MINIMA AND MAXIMA The set of points in the function 𝑓 can be seen as topographic surface 𝑆, The lighter the gray value of the function at the point 𝑥 the higher the altitude of the corresponding point on the surface

MINIMA AND MAXIMA An ascending path is a sequence {𝑠 1 , 𝑠 2 ,..} On the surface such that: ∀ 𝑠 𝑖 𝑥 𝑖 ,𝑓 𝑥 𝑖 , 𝑠 𝑗 𝑥 𝑗 ,𝑓 𝑥 𝑗 𝑖≥𝑗 ⟺𝑓 𝑥 𝑖 ≥𝑓( 𝑥 𝑗 ) A point 𝑠 belongs to a minimum if there is a no ascending path starting from 𝑠. It can be considered as a sink of the topographic surface (see next slide). The set 𝑀 of all the minima of 𝑓 is made of various connected components 𝑀 𝑖 (𝑓)

ASCENDING PATH

NON-ASCENDING PATH

THE WATERSHED TRANSFORMATION If we look at the image 𝑓 as a topographic surface, imagine that we pierce each 𝑀 𝑖 (𝑓) of the topographic surface 𝑆 and then we plunge this surface into a lake, the water entering through the holes floods the surface and if two or more floods coming from different minima attempt to merge, we avoid this event by building a dam on the points of the surface where the floods would merge. At the end of the process only these dams will emerge and this is what define the watershed of the function 𝑓

THE WATERSHED TRANSFORMATION

THE WATERSHED TRANSFORMATION http://cmm.ensmp.fr/~beucher/lpe1.gif

BUILDING THE WATERSHED Suppose the flood of the surface has reached the section 𝑍 𝑖 (𝑓), when it continue and reach 𝑍 𝑖+1 𝑓 the flooding is performed in the zones of influence 𝐼𝑍 𝑍 𝑖+1 𝑓 𝑋 𝑖 𝑓 . The components of 𝑍 𝑖+1 𝑓 which are not reached by the flood are the minima at this level and must be added to the flooded area

BUILDING THE WATERSHED

BUILDING THE WATERSHED If we define 𝑊 𝑖 𝑓 as the catchment basins of 𝑓 at level 𝑖 and 𝑀 𝑖 (𝑓) as the minima of 𝑓 at height 𝑖+1 then: 𝑊 𝑖+1 𝑓 = 𝐼𝑍 𝑍 𝑖+1 𝑓 𝑋 𝑖 𝑓 ∪ 𝑀 𝑖+1 𝑓 𝑀 𝑖+1 = 𝑍 𝑖+1 (𝑓)/ 𝑅 𝑍 𝑖+1 𝑓 ( 𝑍 𝑖 𝑓 ) The initiation of this iterative algorithm is 𝑊 −1 𝑓 =⊘ In the end the watershed line is 𝐷𝐿 𝑓 = 𝑊 𝑐 𝑁 𝑓 when 𝑁=max⁡(𝑓) Visual illustration

Skeletonization

Skeleton by distance transforms Maxima of distance transform

Distance Transform

Skeleton Reconstruction: the original object can be reconstructed by given knowledge of the skeleton subsets Si(F), the SE K, and i: Examples of skeleton: Shift-invariant operator

The Distance Transform on Curved Space (DTOCS)

Distance Transform on Curved Space (DTOCS) Calculates minimal distances between 2 points along a curved surface Calculates minimal distances between areas and areas/points on curved surface Uses a 3x3 calculation kernel with different metrics: Chessboard City block Is a gray-level extension to the Rosenfeldt-Pfaltz-Lay algorithm (which calculates a distance transform for binary images) Presented by Toivanen and Vepsäläinen in 1991 and 1993. Applications: Texture feature extraction and classification (e.g. paper roughness) (Kuparinen and Toivanen 2006, 2007) Shortest distance calculations (Ikonen and Toivanen 2006) Image compression

Weighted Distance Transform on Curved Space (WDTOCS) Calculates minimal distances between 2 points along a curved surface Calculates minimal distances between areas and areas/points on curved surface Uses a 3x3 calculation kernel with different metrics: Chessboard City block Measures the differences between adjacent pixels by their Euclidean distance + (1 or 1,4 for the xy-surface displacement) Presented by Toivanen and Vepsäläinen in 1991 and 1993. Applications: Texture feature extraction and classification (e.g. paper roughness) (Kuparinen and Toivanen 2006, 2007) Shortest distance calculations (Ikonen and Toivanen 2006) Image compression

Definition of the Distance Transform On Curved Space (DTOCS)

The 3x3 kernel used in DTOCS algorithm pne pn pw pc pe psw ps pse

The 3x3 kernel used in DTOCS algorithm pne pn pw pc pe psw ps pse

The Distance Transform on Curved Space (DTOCS)

Original image Distance image after forward pass

Distance image after backward pass Distance image after 2nd iteration (= forward+backward pass second time)

Curves in which DTOCS distance > binary distance Control points chosen along the curves Original Lena image 521 x 521 x 8 bits

(a) LWC (b) SC (c) Cardboard (d) LWC (e) SC (f) Cardboard

Shortest route calculation with Route DTOCS.

Original image a b

a b Shortest route between a and b

Fig. 2. a) Original image, b) distance from source point, c) distance from destination point, d) sum of distance images, e) route by DTOCS, f) route by WDTOCS.

Original labyrinth Shortest routes by DTOCS

Shortest routes by DTOCS Original labyrinth

The End