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.

1. Quantum noise by lack of electron dose: White Noise(Whole field)
2. Noise which exists many in the low-resolution side by energy lack of electron or chromatic aberration: Colored Noise(Whole field)
3. 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.

実空間フィルタ

　実空間に、適切なカーネルを畳み込むことで平滑化します。Eosでは、実空間でのフィルタに関しては、mrcImageSmoothingにより対応しています。ここでは、Eosで実装、あるいは、実現可能なフィルタを示します。

平均化フィルタ

　周辺の値の算術平均を代表値とします。Box関数による畳み込みに対応しています。Eosでは、mrcImageSmoothingが対応しています。

B-splineフィルタ

　上記のBox関数による畳み込みを複数回(k+1回)実施すると、k次のCardnal B Splineと呼ばれるスプライン基底になります。無限会繰り返すと、Gauss関数基底に収束します。また、３次のスプライン関数はよく利用されます。Eosでは、今のところ、mrcImageSmoothingの平均化フィルタを複数回適用(-timesにより指定できる)することで実現できます。

非線形フィルタ

Eosでは、実空間フィルタとして実装されており、mrcImageSmoothing等がその役割を担っています。

メジアンフィルタ

　指定した領域の中央値を代表値として採用します。ポアッソンノイズやごま塩型のノイズに強く、エッジを保存します。

２次曲面フィッティング

　強度を２次曲面に最小自乗法でフィットし、推定します。

Lee-Sigmaフィルタ(σフィルタ）

　中心点を平均値(avg)と仮定し、その周辺の指定した領域の標準偏差σ(=sqrt(sigma(xi-avg)*(xi-avg)/n)を計算し、そのσの一定倍数内(avg+-thres*σ)の値をもつ点の集合の平均値を代表値として採用します。エッジを保存しながら、平滑化が可能です。

Bilateralフィルタ

　空間方向及び強度方向に適切なカーネル（通常は、ガウス関数）を用意し、その積をウェイトとして用いる事で、エッジを保存しながら平滑化することが可能です。