A Brief Introduction to the Snake Model Based on the Active Contour Model of image cutting (5)

A Brief Introduction to the Snake Model Based on the Active Contour Model of image cutting (5)

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In "image cutting (I) Overview", we have briefly understood the mainstream image cutting methods. Below we will mainly learn the energy functional-based cutting method. Here we will learn a simple knowledge of the Snake model. The level set model will be described in the following blog post.

 

Energy Functional-based cutting method:

This method mainly refers to the Active Contour Model and the algorithms developed based on it. Its basic idea is to use continuous curves to express the target edge, and define an energy functional to make its independent variable contain the edge curve. Therefore, the cutting process is transformed into the process of solving the minimum value of the energy functional. the curve where the energy reaches the hour is the target contour.

The Active Contour Model is a top-down image feature locating mechanism. The user or other active processes place an initial contour line near the target of interest in advance, and the internal energy (Internal Force) and external energy (External Force) deformation of external energy to attract the active contour moving toward the edge of the object, while the internal energy to maintain the smooth and topology of the active contour, when the energy reached the minimum, the active contour converges to the edge of the object to be checked.

 

I. curve evolution theory

The curve evolution theory is applied horizontally, but I feel that this knowledge is common in the cutting method of the Active Contour Model, so here we will briefly understand it.

Curves can be divided into several simple types:

There is curvature in the curve, and there is a positive and negative curvature. Therefore, under the force of the normal curvature, the movement direction of the curve is different: some parts expand outward, while some parts move inward. Such a situation is shown in the following example. In the figure, the curvature at the Blue Arrow is negative, and the curvature at the Green Arrow is positive.

The evolution of a simple curve driven by the curvature force (that is, the quadratic derivative of the curve) has a special mathematical nature: All simple curves, no matter how serious they are distorted, just a simple curve, the curvature force will finally degrade into a circle, and then disappear (as you can imagine, the curvature force of all points of the circle is toward the center of the circle, so it will gradually narrow down and eventually disappear ).

Two important measures used to describe the geometric features of a curve are the unit vector and curvature. The unit vector describes the direction of the narrative curve, and the curvature indicates the degree of curve bending. The theory of curve evolution is to study the deformation of a curve over time by using the unit method vector and geometric Gini numbers such as curvature of the curve. The evolution process of the curve can be thought to indicate that the curve is driven by the force F, and N is directed to the normal direction at the velocity V evolution. The velocity is divided into positive and negative values. Therefore, assume that the velocity V is negative, indicating that the evolution process of the active contour is in the external direction. If the velocity V is positive, it indicates the evolution towards the internal direction, the Activity Curve evolves in a single direction and cannot evolve in two directions at the same time.

Therefore, the curve evolution process is the process of different forces acting on the curve, and the force can also be expressed as energy. All things in the world tend to have the least energy. Because it is the most balanced at this time, it consumes the least (do not understand it ?). In image cutting, the goal is to find the target contour. In the target contour, the energy of the entire contour is the smallest, so the curve is in no matter where the image is located, because the force evolves towards the contour of the smallest energy, when the target contour is evolved, the force balance will not change because the energy is the smallest and the speed is 0, at this time, the goal was cut out by us.

Now the key lies in: 1) How do we express this Contour; 2) How do we construct these forces and what forces can we construct to minimize the energy of the target contour?

Many cutting methods based on the Active Contour Model are derived from the description and solution of these two problems. The answer to the first question forms two major schools: assuming that the contour is represented by the number of workers, it is the parametric Active Contour Model ), the typical model is the snake model. Assume that the contour is geometric representation, that is, the Geometric Active Contour Model, that is, the level set method ), it embeds a two-dimensional contour into a zero-Plane Surface of a three-dimensional surface (it can be understood as a contour line of a mountain, and a contour line switches the mountain, the horizontal shape of this high mountain is coming out, that is, the contour), so the low-dimensional evolution curve or surface, indirect expression of a zero-Level Set expressed as a high-dimensional functional surface (this Contour change allows us to intuitively adjust the shape of a mountain or adjust the height of the climbing line ).

The second problem is a problem that both schools have encountered and is the most critical issue they have to solve. What capabilities can be used to achieve the goal of cutting? This will be discussed later.

 

Ii. Snake Model

Since Kass proposed the Snake Model in 1987, various image cutting comprehension and Recognition Methods Based on Active Contour have sprung up. The basic idea of the snail Kes model is very easy. It uses some control points that make up a certain shape as the template (contour), through the elastic deformation of the template itself, and matches with the local features of the image to reconcile, that is, an energy function is minimized to cut the image. Then, we can further analyze the template to understand and recognize the image.

To put it simply, the snake model is a deformation curve of the Gini number and its corresponding energy function. It aims to minimize the energy target function and control the deformation of the Gini number curve, A closed curve with minimum energy is the target contour.

The purpose of constructing the snake model is to reconcile the contradiction between upper-layer knowledge and bottom-layer image features. All image features are local, regardless of the brightness, gradient, angle, texture, or optical flow. The so-called locality means that the features of a certain point in an image only depend on the neighborhood of the point, and are irrelevant to the shape of the object. However, people's understanding of objects mainly comes from their contour. How to effectively integrate the two is just the benefit of the snkes model. The contour line of the Snake model carries the upper layer knowledge, while the matching between the contour line and the image integrates the underlying features. These two items represent the internal force and image force of the energy function in the snkes model, respectively.

The deformation of a model is controlled by many different forces acting on the model at the same time. Each force generates part of the energy, this part of energy represents an independent energy item of the energy function of the Active Contour Model.

The snake model first needs to provide an initial curve near the region of interest. Then, the energy function is minimized to make the curve deform in the image and keep approaching the target contour.

The original snake model proposed by Kass is composed of a set of control points: V (S) = [x (s), y (s)] s ε [0, 1, these vertices start and end with a straight line. X (s) and Y (s) represent the coordinates of each control point in the image. S describes the independent variables of the border in the form of Fourier transformation. Define an energy function (reflecting the relationship between energy and the contour) at the control point of the snake ):

Among them, the first derivative model of V is elastic energy, the second derivative model of V is bending energy, and the second derivative model of V is external energy (external force ), in the basic model, only the local features of the image where the control point or link is located, such as gradients, are taken:

Also called image power. (When contour C is close to the edge of the target image, the gray gradient of contour C will increase, and the above formula will have the minimum energy, and the curve evolution formula will know that the speed of this point will change to 0, that is, stop the exercise. In this way, C stops at the edge of the image, and the cut is completed. The premise is that the target edge in the image is more obvious than the margin, otherwise it will be very easy to cross the edge .)

Elastic energy and Bending energy are collectively referred to as internal energy (Internal Force), which is used to control the elastic deformation of the contour and maintain the continuity and smoothness of the contour. The third item represents the external energy, also known as the image energy, indicating that the deformation curve is consistent with the local features of the image. The internal energy is only related to the shape of the snake, but not the image data. External energy is only related to image data. The α and β values at a certain point determine the degree to which the curve can stretch and bend.

Finally, image cutting is transformed to solving the energy function etotal (v) minimization (minimizing the energy of the contour ). In the process of minimization of energy functions, elastic energy quickly compresses the contour line into a smooth circle, and the curved energy drives the contour line into a smooth curve or a straight line, the image force draws the contour line closer to the position of the image's high gradient. The basic snake model works under the combination of the three forces.

Since the points on the image are discrete, the algorithms used to optimize the energy functions must be defined in the discrete domain. Therefore, solving the energy function etotal (v) minimization is a typical Variational Problem (in a differential operation, the independent variable is usually a coordinate variable, and the dependent variable is a function; in a variational operation, the independent variable is a function, the dependent variable is a function of a function, which is a mathematical function. The variational method is used to calculate the Extreme Value of a functional ).

Under the discretization condition (digital image), the Euler's equation shows that the answer to the final question is equivalent to solving a set of difference equations: (Euler's equation is a differential expression of the extreme functional condition, after solving the Euler's equation of the functional, we can obtain a resident function that obtains the extreme values of the functional and converts the variational problem into a differential problem .)

Record external force F = ?? P, Kass, etc. After the above formula is discretization, the linear equations of x (s) and Y (s) are constructed respectively for the two five diagonal arrays, which are solved through iterative calculation. In practical application, you can manually point out the control point around the object as the starting position of the Snake model, and then iteratively solve the energy function.

 

The above is just a simple understanding of Snake. If you want to learn more, please refer to many other professional documents. The level is limited, and errors are inevitable.

 

Reference:

Li tianqing and others, Snake Model summary, Computer Project, 2005, vol. 31st, issue 1

A Brief Introduction to the Snake Model Based on the Active Contour Model of image cutting (5)