Divious transformation – Wikipedia, free encyclopedia

An image in gray scale can be seen as a topographic relief, where the gray level of a pixel is interpreted as its height in the relief.
A drop of water that falls on a topographic relief flows along a road to finally reach a local minimum. Intuitively, the Divisoria of a relief corresponds to the limits of the adjacent hydrographic basins.

In image processing, various dividing lines can be calculated. In graphs some can be defined on nodes, edges or hybrid lines on nodes and edges. Divisories can also be defined in continuous domain. ^{[ first ]} There are also many different algorithms to calculate the divisories.

For a purpose of image segmentation, the length of the gradient is interpreted as lifting information.

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  Gradient divisions (relief)

Classification [ To edit ]

Flood dividing [ To edit ]

Dividing transformation according to the interpretation of the flood. The terrain is indicated by a continuous black line, the dividing lines are marked by black dot lines, the arrows indicate the local minimums (where floods begin), the blue slopes lines indicate the water levels, and the line Wide and black mark the dam.

The idea was introduced in 1979 by S. Beucher and C. Lantuéjoul in. ^{[ 2 ]} It consists of placing a water source in each regional minimum to flood the relief from the sources and build barriers where the different sources are joined. The resulting set of barriers constitutes a flood divide.

DIVISORY BY TOPOGRAPHIC DISTANCE [ To edit ]

Dividing transformation according to the interpretation of the rain. The terrain is indicated by a continuous black line, the dividing lines are marked by discontinuous black lines, the circles indicate the local minimums and the arrows indicate the water direction.

Intuitively, a drop of water that falls on a topographic relief flows more quickly towards a minimum. This condition is not verified in the previous class.

Division between-pixel [ To edit ]

S. Beucher and F. Meyer introduced into ^{[ 3 ]} An algorithmic definition of inter-pixel dividing, giving the following procedure:

1. Estuve each minimum with a different label. Initiate a set S with the nodes labeled.

2. Remove a node x of minimum altitude f, that is, f (x) = min {f (y) | and in s}.
Assign the label of x to each node and not labeled adjacent to x, and insert and in S.

3. Repeat step 2 until S is empty.

Topological dividing [ To edit ]

The above notions focus on the basins, but not on the separation line. The topological dividing was introduced by M. Couprie and G. Bertrand in 1997. ^{[ 4 ]} It benefits from the following fundamental property: A W function is a divide of a function F (F is a function that assigns pesos in the edges of the graph associated with the image) If and only if W ≤ F and W retains the contrast between the regional minimums of F, where the contrast between two regional minimums m _first y M ₂ It is defined as the minimum height to which you have to go to go _first a M ₂ . ^{[ 5 ]} ​

Various approaches can be used to use the principle of dividing for image segmentation:

• The local minimums of the image gradient can be chosen as markers, in this case there is excessive segmentation and a second step implies the fusion of regions.
• The dividing transformation based on markers makes use of positions of specific markers that have been explicitly defined by the user or automatically determined with morphological operators or other ways.

Meyer flood algorithm [ To edit ]

One of the most common dividing algorithms was presented by F. Meyer in the 90s.

The algorithm works on a gray -scale image. During the successive floods of the relief with gray values, the divisories with adjacent basins are built. This flood process is carried out in the image of the gradient, that is, the basins must arise along the edges. Normally, this will lead to an excess of image segmentation, especially for images with noise, for example, a tac. Or the image must be preprocessed or the regions must be fused later on the basis of a similarity criterion.

Steps:

1. A set of markers, the pixels where the flood will begin, are chosen. Each receives a different label.
2. The neighboring pixels of each marked area are inserted into a priority tail with a priority level corresponding to the gray level of the pixels.
3. The pixel with the highest priority level is extracted from the priority tail. If all the neighbors of the extracted pixel who have already been labeled have the same label, then the pixel is marked with its label. All unmarked neighbors who are not yet in the priority tail are placed in the priority tail.
4. Remove step 3 until the priority tail is empty.

The unique pixels are the dividing lines.

Optimal expansion forest algorithms (Watershed cuts) [ To edit ]

Divisories as an optimal expansion forest have been introduced by Jean Custy et al. ^{[ 6 ]} They established the compatibility of these divisors: they can be equivalently defined by their “capture basins” (through the fastest descent property) or by the “dividing line” that separates the basins (through the principle of the fall of the water). Then, they prove, through an equivalence theorem, their optimality in terms of minimum expansion forests. Then, they introduce a linear time algorithm to calculate them. It is worth noting that similar properties are not verified in other frames and that the proposed algorithm is the most efficient algorithm, both in theory and in practice.

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  An image with two markers (green) and a minimum expansion forest calculated on the image gradient.

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  Result of segmentation by minimum expansion forest.

Links with other computer vision algorithms [ To edit ]

Graph Cuts [ To edit ]

In 2007, C. Allène et al. ^{[ 7 ]} They established links that relate graph cuts to optimal expansion forests. More precisely, they show that when the power of the pesos of the graph is above a certain number, the cut that minimizes the energy of the Graph Cuts is a cut by maximum expansion forests.

Shorter road forests [ To edit ]

Falcao et al. ^{[ 8 ]} It is a procedure to calculate shorter road forests. J. Cousy et al. ^{[ 9 ]} They demonstrated that when the IFT markers correspond to the ends of the weight function, the cut -induced cut is a watershed cut.

Random Walker [ To edit ]

The Random Walker algorithm is an algorithm of segmentation that solves the Dirichlet problem, adapted to segmentation of images by L. Gray in 2006. ^{[ ten ]} In 2009, C. Couprie et al. They showed that when the power of the graph weights converge towards infinity, the cut that minimizes the energy of the Random Walker is a cut by maximum expansion forest. ^{[ 11 ]} ​

Hierarchies [ To edit ]

A hierarchical transformation of divisoriums transforms the result into a graphic deployment (that is, the neighborhood relations of the segmented regions) are determined and then applies dividing transformations recursively.

References [ To edit ]

• Fernand Meyer. An optimal algorithm for the water sharing line. In 8 ^me Congress of recognition of forms and artificial intelligence , Vol. 2 (1991), pages 847-857, Lyon, France.

• Luc Vincent and Pierre Soille. Watersheds in digital spaces: an efficient algorithm based on immersion simulations. In IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 13, Num. 6 (1991), pages 583-598.

• L. Najman and M. Schmitt. Geodesic saliency of watershed contours and hierarchical segmentation. In IEEE Transactions on Pattern Analysis and Machine Intelligence Vol. 18, Art. 12 (1996), pages 1163-1173.

• J.B.T.M. Roerdink and A. Meijster. The watershed transform: definitions, algorithms, and parallelization strategies. In Informatic basis 41 (2000), pp. 187–228.

• Laurent Najman, Michel Couprie and Gilles Bertrand. Watersheds, mosaics, and the emergence paradigm. In Discrete Applied Mathematics Vol. 147, Num. 2-3 (2005), pages 301-324.

external links [ To edit ]