Scale Space Theory

The basic idea of the scale space method is:Introduce a parameter to be regarded as a scale in the visual information processing model, and obtain the visual processing information at different scales through continuous scale parameters, then, the information is integrated to deeply explore the essential features of the image. The scale space method incorporates traditional single-scale visual information processing technology into a dynamically changing scale analysis framework, making it easier to obtain essential features of images. The purpose of generating a scale space is to simulate multi-scale features of image data. Gaussian convolution kernel is the only linear kernel that implements scale transformation.

Motivation of the Scale Space Theory:

• Objects in the real world are composed of objects of different scales;
• In human vision, objects are displayed in different ways when they are observed on different scales;
• For computer vision, it is meaningful to predict the structure of objects of a certain scale. Therefore, it is necessary to express the structures of all scales;
• From the measurement point of view, the measurement data of an object must be dependent on a certain scale. For example, the collection of temperature curves cannot be infinite, but can be quantified within a certain temperature range. The temperature range is the selected scale;
• We use the dimensional space theory to model objects and integrate the concepts of scale into the physical model.

Criterion of scale space:

• Linear
• Translation immutability
• Semi-group features:G(X,Y,T1 )*G(X,Y,T2) =G(X,Y,T1 +T2)
• Rotation immutability
• Scale immutability
• Positive Definite
• Normalization (point: 1)
• Will not introduce new poles
• Not extreme Enhancement
• Operator with an infinitely small number)

According to the preceding conditions, the only possible scale space kernel function is the Gaussian Kernel function.

Heat diffusion equation:
According to the theory of differential equations, the above kernel function family can be expressed as the solutions of the following heat diffusion equations:

The initial condition is L( X, Y; 0)
= F( X, Y)

Multi-scale edge detection and blob Detection:

• Gradient operator for Edge Detection
• Zero Point crossing detection: Quadratic Differential immutability Equation

  Three times of differential immutability inequality:

• Blob Detection: the determining factor of Laplace Gaussian equation or Hessian Matrix

Automatic scale selection and constant scale feature selection:

• In actual problems, you may need to select a local scale for further analysis.
• The scale-unchanged feature is a feature that satisfies the scale-unchanged nature. This feature can be easily mapped to the corresponding position of another scale under one scale.

Other multi-scale Representation Methods:

• Pyramid Representation
• Nonlinear Scale Space
• Affine Gaussian Scale Space
• Wavelet Theory