Level Set Method: Advancing Computer Vision, Exploring the Level Set Method

By Fouad Sabry

What is Level Set Method

The Level-set method (LSM) is a conceptual framework for using level sets as a tool for numerical analysis of surfaces and shapes. LSM can perform numerical computations involving curves and surfaces on a fixed Cartesian grid without having to parameterize these objects. LSM makes it easier to perform computations on shapes with sharp corners and shapes that change topology. These characteristics make LSM effective for modeling objects that vary in time, such as an airbag inflating or a drop of oil floating in water.

How you will benefit

(I) Insights, and validations about the following topics:

Chapter 1: Level-set method

Chapter 2: Navier-Stokes equations

Chapter 3: Green's function

Chapter 4: Hemorheology

Chapter 5: Autoregressive model

Chapter 6: Blasius boundary layer

Chapter 7: Total variation diminishing

Chapter 8: Godunov's theorem

Chapter 9: Vortex lattice method

Chapter 10: Phase-field model

(II) Answering the public top questions about level set method.

(III) Real world examples for the usage of level set method in many fields.

Who this book is for

Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of Level Set Method.

• Language: English
• Publisher: One Billion Knowledgeable
• Release date: May 11, 2024

Book preview

Chapter 1: Level-set method

When it comes to numerical analysis of surfaces and forms, level-set methods (LSM) provide a conceptual foundation for employing level sets. The level-set model is useful because it allows one to execute numerical computations involving curves and surfaces on a fixed Cartesian grid without needing to parameterize these objects (this is called the Eulerian approach). The level-set technique also facilitates effortless tracking of structures that undergo topological changes, such as those that undergo bifurcation, occlusion, or the inverse of these processes. These features make the level-set approach useful for simulating phenomena that change over time, such as an airbag inflating or an oil drop floating in water.

The right-hand diagram elaborates on numerous key points of the level-set approach.

One may make out a form in the top left corner; that is, a contained space with an orderly border.

Below it, the red surface is the graph of a level set function \varphi determining this shape, in which the blue area stands for the xy plane.

The boundary of the shape is then the zero-level set of \varphi , while the shape itself is the set of points in the plane for which \varphi is positive (interior of the shape) or zero (at the boundary).

In the first column, the shape undergoes a topological transformation by bifurcating. Parameterizing the shape's boundaries and tracking its evolution to provide a numerical description of this transition would be challenging. To create parameterizations for the two resulting curves, one needs an algorithm that can identify the point at which the shape divides in half. In contrast, the level set function only translated downward in the bottom row. In cases like these, where dealing with the shape itself would require thinking about and accounting for all the different deformations the shape might suffer, it can be considerably more convenient to work with the shape via its level-set function.

Thus, within a single plane, the level-set method amounts to representing a closed curve \Gamma (such as the shape boundary in our example) using an auxiliary function \varphi , level-setting function, for short.

\Gamma is represented as the zero-level set of \varphi by

{\displaystyle \Gamma =\{(x,y)\mid \varphi (x,y)=0\},}

and the level-set method manipulates \Gamma implicitly, through the function \varphi .

This function \varphi is assumed to take positive values inside the region delimited by the curve \Gamma and negative values outside.

If the curve \Gamma moves in the normal direction with a speed v , then the level-set function \varphi satisfies the level-set equation

\frac{\partial\varphi}{\partial t} = v|\nabla \varphi|.

Here, |\cdot | is the Euclidean norm (denoted customarily by single bars in PDEs), and t is time.

A partial differential equation looks like this:, an equation of the Hamilton-Jacobi kind, and numerical solutions exist, for example, using a Cartesian grid and finite differences.

Consider a unit circle in {\textstyle \mathbb {R} ^{2}} , Continually contracting in on itself, i.e.

At a constant rate, all points on the rim of the circle advance along the direction of the normal that points inward.

The sphere will diminish until it becomes a point.

When a distance field is created from the outset (i.e.

sign-based Euclidean distance to the boundary function, positive interior, outside (negative on the first circle), The circle normal is the normalized gradient of this field.

If a constant value is subtracted from the field over time, the resulting fields will have a circular zero level (the original boundary) that eventually collapses to a point. Since this is the same as the time-integration of the Eikonal equation for a constant front velocity, we may conclude that.

The G equation is used to characterize the instantaneous flame surface in combustion.

In 1979, Alain Dervieux introduced the level-set approach, which was later popularized by Stanley Osher and James Sethian. It has found widespread use in fields as diverse as computational biology, computational geometry, and computational fluid dynamics.

Several level-set data structures have been created to make the level-set technique more accessible for usage in computational settings.

Computational fluid dynamics

Combustion

Trajectory planning

Optimization

Image processing

Computational biophysics

Parameter space and dynamic space visualized in discrete complicated dynamics

In order to run a mathematical model at the interface of two fluids, we must dampen the interactions between the fluids. This calls for the use of a very particular function: Method for Condensing Level Sets.

As a spin off, The LSM's companion, the CompactLSM, that aids in the resolution of LSM equations.

It has applications in numerical flow simulation, for example, If the water-air interaction is to be discretized, then, 6th-order compacts, guarantees a quick and precise solution to the equations at the interface (Monteiro 2018).

The LSM employs a distance function to identify specific liquids. The smallest value of a distance function between the point of analysis and the interface is 1. Positive values of this distance function indicate the presence of one fluid, negative values indicate the presence of the other, and zero indicates the location of the interface, as shown by the corresponding isolines (2D) or isosurfaces (3D).

When using the Compact Level Set Approach, how does one incorporate the Heaviside function?

Due to the discontinuity in specific mass and viscosity at the contact, Inadequate management of the fluid close to the interface will lead to both excess diffusion problem (interface widening) and numerical oscillations.

Reduce the impact of these issues, The Level Set approach takes a gradual, cell-related Heaviside function that explicitly defines the interface position ( \varphi =0 ).

The interface remains fluid throughout, nevertheless, with a thickness on the same scale as a cell, so that disruptions on the same scale as the mesh are not introduced, due to the fact that the interface implies a discontinuous transition from one cell to the next (Unverdi and Tryggvason), 1992).

Reconstructing the flow's material characteristics, such as density and shear stress, yet another function of markers, {\displaystyle I(\varphi )} , type Heaviside is employed here:

where \delta is an empirical coefficient, the norm is 1; 5 and \Delta is the characteristic discretization of the problem, that vary with the phenomenon being modeled.

The value of \delta represents an interface with a thickness of three cells, and thus {\displaystyle \delta \Delta } represents half the thickness of the interface.

Keep in mind that this technique, Virtual thickness exists at the interface, since it is modeled after a continuous function.

Physical properties, include kinematic viscosity and specific mass, are figured out as:

where \rho _{1} , \rho _{2} , v_{1} and v_{2} are the specific mass and kinematic viscosity of fluids 1 and 2.

Similar analogies hold true when applying Equation 2 to the other fluid characteristics.

{End Chapter 1}

Chapter 2: Navier–Stokes equations

The Navier–Stokes equations (/nævˈjeɪ stoʊks/ nav-YAY STOHKS) are partial differential equations which describe the motion of viscous fluid substances, Claude-Louis Navier and George Gabriel Stokes were the namesakes. They were both French engineers and physicists.

They