I've been doing layout analysis recently , The text line detection aspect , Many articles deal with Anigauss Anisotropic Gaussian filtering .

seeing the name of a thing one thinks of its function , The simple understanding is two-dimensional Gaussian filtering with different parameters .

In the article Fast Anisotropic Gauss Filtering It is clarified in the paper Anisotropic Gauss Filter It can be broken down into thinking ,y One dimensional Gaussian filtering with two different directions , among x,y It doesn't have to be vertical .

Now let's look at the decomposition process .

A simple isotropic two-dimensional Gaussian convolution filter is as follows

When x,y When the direction scaling is different, we get the anisotropic two-dimensional Gaussian filter, as shown in the figure

Of course, we can also do filtering in other directions , Let's set the rotation angle to θ Pictured , Then the coordinates after rotation are the same as x,y The coordinate relationship is

So we can get a general expression

among u Shaft for θ Direction ,v The axis is perpendicular to θ Direction .

According to the Fourier transform, we know that , Spatial convolution is equal to frequency multiplication

So if the Fourier transform of a linear filter can be written as two respectively about Wx,Wy The form of multiplication of equations , So in space, he can write the convolution of two subterms

So we can   The Fourier transform of is written as follows :

(7)

That is to say

However, we are only interested in x,y Interested in , So we will Decompose into

 (10)

Then we can get a11,a12,a13 The expression of

Once again, our goal is to decompose the anisotropic Gaussian filter into a x Direction and an extension One dimensional filtering of direction .

Easy to decompose , We (10) Write all and Wy Nothing to do with Wy About the additive form

And put in (10) And then we get

According to the nature of the index, it can also be written as

Then we do the inverse Fourier transform and we get

The first item means x Direction filtering

The second means wave filtering

among ,

So far, the decomposition is complete :

In this paper, the author applies recursive filter Go and do it , Specific need to refer to the source code anigauss.c

The specific effect is as shown in the figure

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