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Level set evolution

level set equation elegantly combines spatial and temporal changes of the zero level set. Speed term F = edge term (uses the image content to guide the level set evolution to track image edges), advection term (same principle of the balloon term), curvature term encodes properties of the resulting contour) Edge term can be calculated starting with a edge detection filter. Curvature term can be calculated by using the curvature of the level set surface (calculated in normal direction9

Snake equation:

Equilibrium between force terms, modeling the image content(fitting & properties) and constant inflation (ensure snake evolves constant to in or outwards Components are: Damping term: Optional à lowers oscillations around optimum point à adding robustness Tension force term: Curvature or bending force term: Balloon term: expands or contracts the contour (+ expansion, - contraction) Image interaction: aims to guiding the fitting of snake to image content (through edge information) Equation ins solved for each node individually!

Narrow banding is used in level sets on order to…. Improve accuracy Speed up calculations Avoid minima solutions Incorporate neighboring information’s

- Speed up calculations

Important aspects when implementing deformable models:

Local optimization: optimization problem à initialization, optimization step, stopping criteria Refining step: fitting or matching our flexible and deformable shape template to image content Tracking: only for snakes àtracking of points over the temporal evolution of the snake representation Smoothing: enforcing properties of the resulting snake model through internal forces

What are snakes?

A set of points with nearest neighbor interactions governed by a force or energy Basic representation are meshes or polyenes in 2D à hard to extend to 3D! Governed by physical connected mass equations à each mesh or node is given a virtual mass on which forces, and an energy equation can be associate to Connective function can be of any degree (e.g. zero order à straight line Topology of the snake does not change! à each point is always connected to its neighbors (also over time) à allows tracking of points over time

Name an implicit deformable model:

Level-sets

Disadvantages of level set:

No explicit correspondence à no fixed number of points on the contour surface; difficult for temporal tracking of shape applications Larger memory requirements N3 Higher computational requirements O(N3) Need to keep track of entire grid à redundant method of object representation

Which of the following is (are) true statements(s) about level set based segmentation? Can handle n-dimensional problems Uses an explicit representation Can handle topological changes Can handle point tracking

Can handle n-dimensional problems Can handle topological changes

Snake energy equation:

Total energy as a combination of three terms E_int: internal energy of snake and model forces created by tension E_image: how image infos is used to derive the fitting of the snake to the image content E_con: models any other constraint imposed on the snake (not typical used)

Speeding up level sets:

- Narrow banding: calculation of the level set function on the Cartesian grid is only performed around the current zero level set tradeoff between correctly capturing the zero level set (large enough band) and improving the computational efficiency of the level set evolution (narrower band)

Pros and cons of segmentation modeled by conturs:

Robust option to edge extraction Effective way to represent biological organs à possible to encode properties of the resulting contour as a prior information (Useful when in the presence of large anatomical variability) Affected by the quality of the edge extractor filter that is used to construct them Susceptibility (anfällig) to image artefacts (Missing infos from low resolution or noise)

The image term of sakes and level sets is based on: Image derivatives to track borders Texture information to drive blob detection Voxel intensity statistics via a random forest classifier

Image derivatives to track borders

Name an explicit deformable model:

Snakes

Advantages of level set:

No bookkeeping of vertices required – elegant solution Topologically adaptable Grid allows straightforward multiresolution implementation Extendible to nD with no change in parameterization Grid nature allows straightforward parallelization implementation of evaluation equation

Concept of snake evolution:

Each node has a virtual mass m Relationship between the position, speed and acceleration of the mass particles (Newtons 2nd law) Solving of equation system à finding equilibrium of the system and derivation of update equations

Which of the following is (are) not true about snakes? Each point (snakes) moves individually Snakes uses an implicit modelling Snake based deformation is computationally expensive Snakes enables point tracking

Snakes uses an implicit modelling Snake based deformation is computationally expensive

Advantages and disadvantages of snakes:

Ability to track nodes over time à makes it possible to track points over time (useful to track physiological changes such as heartbeat, respiratory motion etc. Self-intersecting contours: Occur when snake evolution calculates position updates independently for each node in the snake à does not reflect biological processes Handling stretching of the contour à single or portion of nodes moves in direction away from snake: Tensile forces become too large à help: change topology of snake and add nodes through interpolation

Level set:

uses an eulerian reference frame and can be used for high-dimensionality problems level set a is a numerical technique used to track interfaces and shapes no parameterization is needed Intersection is called zero-level (f(x)=0); Inside and outside area of cutting plane are either negative or positive Image plane is moved upwards or downwards a different cutting plane is generated as well as a different zero level Adding a temporal dimension à level set evolution level set equation elegantly combines spatial and temporal changes of the zero level set.

Basic Idea of deformable models? What we aim for?

Basically, contrary to the previous techniques we visited, where each voxel is classified, in deformable models we aim at solving the segmentation problem by adapting a given template to the image content Template is very simple (circle/sphere) and envolves

Snake based segmentation is…. Voxel based segmentation Contour deformation Contour regression Voxel- wise front propagation

- Contour deformation

Function of snakes:

Initial circle is initialized in center à after n iterations, the snake evolves to fit the image

Info cartes d'apprentissage

MIA_deformable Models