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I am working with wavelet transforms. When we apply a discrete wavelet transform to an image, it decomposes into four components named the approximation component, thevertical component, the horizontal component, and the diagonal component.

These components are indicated as LL, HL, LH, HH respectively. I understand how an image is decomposed by using a wavelet transform, but I don't know why generally in an image high frequency component contain more noise.

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  • $\begingroup$ The most important reason in practice does not seem to have been mentioned in any answer below. Typical images just contain so much less energy in higher frequencies than in the lower ones. See for example www2.compute.dtu.dk/~jerf/code/images/fft_demo.png for a Fourier transform of a photograph. The energy is concentrated in the lower frequencies at the center. That makes high frequencies so much more susceptible for noise, even if that noise does not specifically favor higher frequencies. $\endgroup$ – Jazzmaniac Feb 14 '14 at 9:57
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There are different types of noise sources in imaging process.

Photoelectronic

photon noise: Poisson distribution, whole spectra;

thermal noise: Gaussian distribution, white noise, whole spectra;

Impulse

salt and pepper noise: mainly high frequency noise; (@user7358 made a very good point on the reason).

line drop: part or all of a line in the image lost, can be viewed as a deterministic high frequency noise;

Structured

periodic: low frequency noise;

aperiodic: JPEG noise, low frequency too;

detector striping: calibration differences among individual scanning detectors, mainly low frequency;

If you only observe the evident noise when you zoom in the high frequency band with wavelet, I assume it is mainly the shot noise (salt and pepper). When you are doing some simulations with noise added, try (1) salt and pepper, (2) Gaussian, and (3) Gaussian with a high-pass band applied, and observe the difference after wavelet decomposition.

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In actual photography, this is the case because there is no correlation in the noise of neighboring pixels. They are all independent, other than the contribution of a supposed signal.

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    $\begingroup$ Independence of the noise implies that the spatial autocorrelation is a Kronecker impulse, which makes the noise power spectrum flat. So following your argument all frequency intervals of the same bandwidth should contain the same noise power. So I don't see how this answers the OP's question. $\endgroup$ – Jazzmaniac Feb 13 '14 at 21:13
  • $\begingroup$ If there is noise with a flat power spectrum, the above reasoning will of course not stand. An explanation why the spectrum is not flat would be a little involved. However, it is observable. $\endgroup$ – user7358 Feb 14 '14 at 10:07
  • $\begingroup$ @user7358, I think you misunderstand me. I'm not arguing that flat spectra are observed, I'm saying that your argument necessarily implies a flat spectrum. And therefore the relation of your answer to the question of the OP is not obvious. What are you trying to imply with your answer? $\endgroup$ – Jazzmaniac Feb 14 '14 at 10:11
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    $\begingroup$ @user7358, that reasoning is problematic, because, as I wrote above, the noise is spectrally white. That means each wavelet picks up noise proportional to its bandwidth. Low frequency wavelets have a smaller frequency bandwidth, so they pick up proportionally less noise. But that doesn't really follow from your explanation. Reading your answer you seem to suggest the spatial independence implies a high frequency bias, which is not true. The real reason is the non-uniform bandwidth distribution of the wavelets, not the noise. $\endgroup$ – Jazzmaniac Feb 14 '14 at 10:27
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    $\begingroup$ @user7358, it's just as important, but the math doesn't change if you reason using spatial width. It's just much easier to see what happens if you look at the frequency domain representation. $\endgroup$ – Jazzmaniac Feb 14 '14 at 10:36
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noise is often high frequency in nature. it interrupts the image, you can find them in the high frequencies area in DWT also the same for Discrete Cosine Transform (DCT) matrix, where noises lay in the down-right corner (high frequencies band).

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I don't know why generally in an image high frequency component contain more noise.

This is not always true and depends on the relative envelope of both the signal and noise. Imagine your signal is the image of a bar code while you are in a foggy environment generating cloud-like noise. This will generate a situation where the signal is more white and the noise pink - in that case the opposite would be true: in this image low frequency component contain (relatively) more noise.

This counter-example is to show your are most of the time true: natural images have have an envelope which is mostly pink (in 1/f) due to the fact that they contain images of objects that are self-similar while pixel noise is mostly independent - thus white- noise. Yes, in most images, the high frequency components relatively contain more noise.

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