# Noise reduction

Noise reduction

## Nature of noise

In order to deal the image including a lot of noise, knowing well the nature of the noise is important. Show as following some factors about noise related to electron microscope.

- Quantum noise by lack of electron dose: White Noise(Whole field)
- Noise which exists many in the low-resolution side by energy lack of electron or chromatic aberration: Colored Noise(Whole field)
- Noise by radiation or dust on CCD, fluorescent screen, or film of camera: Local Noise

and so on.

Quantum Noise is one kind of White noise, and it is known that this is often noise distribution depending on Poisson Process. This noise is spread onto a sphere of high frequency.

In the low-resolution side There is noise by Inelastic scattered electrons or chromatic aberration, with blur. Therefor, this noise is Colored.

Noise by radiation or dust on CCD, fluorescent screen, or film of camera is one example of Local Noise. Factor by cosmic rays or radiation by collapse of the fluorescent screen gives pixels which have considerable high contrast. Because of MTF(e.g. by CCD), it is not one point, but occurs blur. However, it does not depend on CTF is occurred by imaging of electron microscopy. Thus, it has some mysterious pattern on a ring, after it performs CTF Correction.

Basically, Smoothing technology supports it, but this purpose is reduction of noise with keeping information as edge.

## Remove abnormal value

On Eos, mrcImageAbnormalValueRemove can remove value which has obviously high or low intensity.

### 6-sigma remove

It calculates the standard deviation among intensity value of image, and it can remove the point which has markedly different value. It can remove pixels which has high intensity according to sudden dust(dark) or radiation on CCD camera. mrcImageAbnormalValueRemove performs this role.

## Linear Filter

On Eos, mrcImageLowPassFilter and mrcImageSmoothing perform this role.

### Low-Pass Filter

It extracts information in the region of low spatial frequency. It uses properties that signal has high intensity in low frequency region, and noise has same intensity in each frequency region. mrcImageLowPassFilter supports to several kinds kernel. Here, describe about Low-Pass Filter that is supported by Eos.

#### ideal

Cut off data above a spatial frequency by using Box Function. This is sometimes called Brick-wall Filter. It is ideal Low-Pass Filter, but because its convolution function on Real Space is Sinc Function, it occurs sometimes serious ripple(vibration). This phenomenon is called Gibbs phenomenon.

#### cos

Cut off High-frequency components by Cosine Windowing Function. Cut edge becomes rounded. Therefore, the influence of convolution function decreases on Real Space.

#### exp

Multiply Exponential Function, on Fourier Space. This is similar to blur by CCD. The Convolution Function on Real Space is Lorentz Distribution. Therefore, it does not occur a ripple. Compared with Gaussian Function, it is good at keeping high frequency elements, but not good at keeping low frequency elements.

#### Gaussian

This filter is often used, and it multiplies Gaussian Function, on Fourier Space. Convolution Function on Real Space is Gaussian Function. Therefore, it does not occur a ripple.

#### Lorenz

Multiply Lorenz Function, on Fourier Space. Convolution Function on Real Space is Exponential Function. This is similar to blur by Film.

### Real Space Filter

It smooths by convoluting at a suitable kernel. On Eos, mrcImageSmoothing supports about filter on Real Space. Here, Here, show filters that is implemented or can be implemented on Eos.

#### Averaging Filter

Arithmetic mean of around value is treated as the representative value. Convolution by Box Function is equivalent to it. On Eos, mrcImageSmoothing supports to it.

#### B-spline Filter

Perform multiple times(k+1 times) the above convolution by Box Function, it becomes basis spline which is called kth-order Cardinal B Spline. If it is repeated infinite number of times, it converges to Gaussian Function Basis. Especially, 3rd-order spline function is often used. Currently, Eos can perform by using mrcImageSmoothing with Averaging Filter of multiple times (Option: -times).

## Nonlinear Filter

On Eos, It is implemented as Real Space Filter, and mrcImageSmoothing performs its role.

### Median Filter

The medium value in specified region is treated as the representative value. It is strong at Poisson Noise or Salt-and-Pepper Noise, and it keeps edge.

### Quadratic Surface Fitting

It performs fitting and determination of the intensity by Minimum Square Methods on Quadratic Surface.

### Lee-Sigma Filter (σ-Filter）

Assume the center point as a average value(avg), and calculate the standard deviationσ(=sqrt(sigma(xi-avg)*(xi-avg)/n) among specified around region, finally determine average value about set of point whose value exists in range(avg+-thres*σ) of constant multiple of this σ, as representative value. It can perform smoothing, keeping edge.

### Bilateral Filter

Prepare a kernel(Normally, Gaussian Function) which is suitable about spatial-direction and intensity-direction, then according to use this multiply as the weight, you can perform smoothing, keeping edge.

### NonLinear Diffusion Equation Filter

### dge-Preserving Smoothing Filter