Thermodynamics and image processing

Preface
In terms of image processing, we draw a lot of inspiration from nature, such as radiation, gravity, water, and other physics models. One of the most commonly used, it seems, is the thermodynamic model, which describes the process of heat transmission and heat balance in an abstract and concise way. This article will discuss the thermodynamic model in detail, not only exposing the beauty of mathematics, but also the beauty of nature.

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Thermodynamic model heat transfer Law thermodynamic iterative standard thermodynamic system common filter isotropic diffusion and Gaussian fuzzy Gaussian fuzzy equivalence isotropic diffusion anisotropic diffusion and beauty filter algorithm principle Python code for Image segmentation

1. Heat Transfer law of thermodynamic model

∂t (x,t) ∂t=α∂2t (x,t) ∂2x \large \frac{\partial t (x,t)}{\partial T} =\alpha \frac{\partial ^2 t (x,t)}{\partial ^2x}
The thermodynamic model uses the above formula to describe the heat transfer process. Wherein, Α\alpha represents the thermal diffusion rate, the first simple coefficient, determined by the dielectric material; t (x,t) t (x,t) function represents the entire temperature distribution of space X x at T T moment, and X x can be a space of any dimension. This simple formula establishes the direct relationship between temperature distribution in time and space, that is, the trend of temperature variation of a point in space is determined directly by the curvature of temperature distribution in space.

Here is an example of a simple heat transfer. Suppose X x is a one dimensional space, and T (x,t=0) =sin (x) t (x,t=0) =sin (x), i.e. the whole system in the initial case, the temperature is exactly the sine distribution in space. So how does the temperature of the whole system change at the next moment? ∂t (x,t=0) ∂t=α∂2t (x,t=0) ∂2x=α∂2sin (x) ∂2x=−αsin (x) \frac{\partial T (x,t=0)}{\partial T} =\alpha \frac{\partial ^2 t (x,t =0)}{\partial ^2x}=\alpha \frac{\partial ^2 sin (x)}{\partial ^2x}=-\alpha \ sin (x), the direction of change is exactly the opposite of the current distribution. In other words, the current temperature is positive area temperature will drop, the temperature is negative region temperature will rise, and the higher or lower the temperature is the greater the change; This is consistent with our common sense of life. Heat transfer can become extremely complex in some complex spaces and complex distributed systems, but it is inherently simple. Thermal Iteration

What is the point of understanding the law of thermal flow? By mastering the rules, we can deduce the entire thermodynamic system along the timeline. All thermodynamic model applications are based on this.
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