Introduction to Gaussian filtering of image filtering

1 Gaussian Filter Introduction

　　Before we understand Gaussian filtering, we first get familiar with Gaussian noise. Gaussian noise refers to a class of noises whose probability density function obeys Gaussian distribution (i.e. normal distribution ). If a noise, its amplitude distribution obeys the Gaussian distribution, and its power spectral density is evenly distributed, it is called Gaussian white noise. The second moment of Gaussian white noise is not correlated, and the first-order moment is a constant , which refers to the correlation of the signal in time, and the Gaussian white noise includes the thermal noise and the bulk-particle noise .

Gaussian filter is a kind of linear smoothing filter which chooses weights according to the shape of Gaussian function. Gaussian smoothing filters are very effective in suppressing noise that obeys a normal distribution. The one-dimensional 0 mean Gaussian function is:

g (x) =exp (-x^2/(2 sigma^2)

The Gaussian distribution parameter Sigma determines the width of the Gaussian function. For image processing, the two-dimensional 0 mean discrete Gaussian function is used as smoothing filter, the graph of Gaussian function:

　　　　　　　　　　　　　　　　　　

2 Gaussian filter function

For images, the Gaussian filter is a 2 -dimensional convolution operator using a Gaussian kernel for image blurring (removal of detail and noise).

1) Gaussian distribution

One Gaussian distribution:

Ivigos Distribution:

2) Gaussian core

Theoretically, Gaussian distributions have non-negative values on all domain definitions, which requires an infinitely large convolution core. In fact, you only need to take the value of 3 times the standard deviation around the mean value, and remove it directly from the external part. As a standard deviation of 1.0 , the integer value of the Gaussian nucleus.

3 Gaussian Filter Properties

The Gaussian function has five important properties which make it particularly useful in early image processing. These properties show that the Gaussian smoothing filter is a very effective low-pass filter both in the spatial domain and in the frequency domain, and is effectively used by engineering personnel in the actual image processing. The Gaussian function has five very important properties, which are:

(1) The Ivigos function has rotational symmetry, that is, the smoothing degree of the filter in all directions is the same. In general, the edge direction of an image is unknown in advance, so it is not possible to determine in one direction more smoothing than the other on the other side before filtering. Rotational symmetry means that the Gaussian smoothing filter will not be biased in either direction in subsequent edge detection.

(2) The Gaussian function is a single-valued function. This shows that the Gaussian filter uses the weighted mean of the pixel neighborhood to replace the pixel value of the point, and the pixel weights of each neighborhood are monotonically increasing with the distance between the point and the center point. This property is important because the edge is an image local feature, and if the smoothing operation still has a significant effect on pixels far from the center of the operator, the smoothing operation can distort the image.

The

(3 ) the Fourier transform spectrum of the Gaussian function is single-lobe. As shown below, this property is a direct corollary of the fact that the Gaussian function pays the Fourier transform equal to the Gaussian function itself. Images are often contaminated with unwanted high-frequency signals ( Noise and fine textures

(4) The width of the Gaussian filter ( which determines the degree of smoothness ) is characterized by the parameter σ, and the relationship between σ and smoothness is very simple. The larger the σ, the wider the frequency band of the Gaussian filter, and the better the smoothing level. By adjusting the smoothing degree parameter σ, the excessive amount of unwanted mutations caused by noise and fine textures in image features over-Blur ( over-smoothing ) and smoothed images ( Under-Smoothing ) to achieve a compromise .

(5) due to the separation of Gaussian function, the large size Gaussian filter can be effectively implemented. Ivigos function convolution can be carried out in two steps, first the image and a Gaussian function convolution, and then the convolution result and direction perpendicular to the same Gaussian function convolution. Therefore, the computational amount of Ivigos filtering increases linearly with the width of the filter template rather than the square.

4 Gaussian filter applications

The degree to which the image is smoothed after Gaussian filtering depends on the standard deviation. Its output is a weighted average of the domain pixels, and the higher the pixel weight is, the closer the center is. As a result, it has a softer smoothing effect relative to the mean filter (mean filter), and the edges retain better.

The essential reason for Gaussian filtering being used as a smoothing filter is that it is a low-pass filter, see. Moreover, most of the convolution smoothing filters are low-pass filters.

Figure . Gaussian filters (standard deviation =3 the frequency response of the pixel). the spatial frequency axis is marked

In cycles per pixel, and hence no value above 0.5 have a real meaning.

5 Gaussian filtering steps

( 1 moves the central element of the associated nucleus so that it is directly above the input image pending pixel

( 2 ) takes the pixel value of the input image as a weight, multiplied by the relevant kernel

( 3 Add the results from the above steps as output

Introduction to Gaussian filtering of image filtering