[dimension reduction strike] Hilbert curve

How to reduce dimensions ？

《 Trisomy 》 Dimension reduction attack mentioned in , Humans in the three-dimensional world look at the four-dimensional space as fish in the pond look at the world outside the pond , Four hit three-dimensional , Just like human crushing ants . So how can we reduce the dimension ？

• such as ： How to reduce three-dimensional to two-dimensional ？ Imagine how we turn the earth into two dimensions ？
  • Directly flatten ？
  • Unfold into orange petals ？ It just unfolds the surface , And the earth is solid .
    

1. Explore the connection between dimensions

2 Dimension and 1 Dimensional relationship , Our commonly used raster scanning sequence is one 2 Peace-keeping 1 Dimensional relationship . Classic order

• Morton curve
• z Font curve
• Hilbert curve

1.1 Hilbert curve

1.1.1 Two dimensional Hilbert curve

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When the order tends to infinity , You can think that the Hilbert curve fills the whole plane . Is this mathematically rigorous ？ This involves （ discrete ——> continuity ） The problem of , The plane of theory is actually continuous , The position of each point is infinitely fine ,n Is an integer , Is discrete , When n As we go to infinity , Can you think of 1/n Is the value of continuous ？

• differential
• limit
• discrete , continuity
  

1.1.2 Three dimensional Hilbert curve

1.2 Characteristics of Hilbert curve

• The local adjacent points in the high dimension are 1 Dimension is still close , The spatial continuity is well maintained

1.3 Application of Hilbert curve

What modern computers deal with is discrete The data of , We don't have to worry about whether mathematics is rigorous .

1.3.1 Two dimensional to one dimensional

• Use Hilbert curve to transform two-dimensional digital pictures into 1 Dimension vector .
  The picture is actually composed of many pixel blocks , Use the pseudo Hilbert curve to pass through each pixel block , Then extract the pixel blocks in the order of Hilbert curve traversal , It realizes two-dimensional to 1 Dimensionality reduction
  
  

1.3.2 Three dimensional to two dimensional

Reference material

A visualization scheme of dimensionality reduction attack