Paper 64: Scale space theory

The basic idea of the scale space method is to introduce a parameter which is regarded as the scale in the visual information processing model, obtain the visual processing information at different scales through the continuous change scale parameter, then synthesize the information to dig into the essence characteristic of the image. The scale space method incorporates the traditional single-scale visual information processing technology into the dynamic analysis framework, so it is more easy to obtain the essential features of the image. The goal of scale space is to simulate the multi-scale feature of image data. The Gaussian convolution kernel is the only linear nucleus that realizes the scale transformation.

The motive of the scale space theory:

• Real-world objects are made up of structures of different scales;
• In the human visual, the object observes the scale to be different, the object presents the way also is different;
• For computer vision, it is meaningful to not predict the structure of a certain scale, so it is necessary to express the structure of all the scales.
• From the point of view of measurement, the measurement data of an object is necessarily dependent on a certain scale, such as the acquisition of temperature curve, can not be infinite, but in a certain temperature range of quantitative acquisition. The temperature range is the scale of the selection;
• The theory of scale space is used to model objects, and the concept of scale is fused into physical models.

Axiom of scale space:

• Linear
• Translational invariance
• Semigroup characteristics:g(x,y,t1) * g(x,y,t2) = g (x,y,t1 + T2)
• Rotational invariance
• Scale invariance
• Positive qualitative
• Formal (integral is 1)
• No new poles will be introduced.
• Does not enhance the Poles
• There are infinitely small operators (micro)

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

Thermal diffusion equation:
According to the theory of differential equations, the above-mentioned nuclear function family can be expressed as the following solutions of the thermal diffusion equation:

initial conditions are L( x, y; 0) = F( x, y)

Multi-scale edge detection and blob detection:

• Gradient operator for edge detection
• Over 0 points detection: two differential invariance equations

  Three differential invariance inequalities are satisfied:

• Blob detection: Laplascause equation or determinant of Hessian matrix

Automatic scale selection and scale invariant feature selection:

• The local scale may be selected in the actual problem, and then further analysis
• The invariant feature is a feature that satisfies the invariant nature of the scale, which is detected at one scale and can easily be mapped to the corresponding position of another scale.

Other multi-scale representation methods:

• Pyramid representation
• Nonlinear scale space
• Affine Gaussian scale space
• Wavelet theory

Paper 64: Scale space theory