J V

1a

2a

3a

4a

Figure 18.6 In this figure, the bottom row shows simulated star images with amplitudes of 1,2,3, and 4. The middle shows the same stars but with Gaussian noise (with o = 1), and the top row shows the same data truncated. Reliable noise rejection requires a signal of ~3<X

Figure 18.6 In this figure, the bottom row shows simulated star images with amplitudes of 1,2,3, and 4. The middle shows the same stars but with Gaussian noise (with o = 1), and the top row shows the same data truncated. Reliable noise rejection requires a signal of ~3<X

frequency. Wavelets, as a spatial/frequency hybrid, offer some effective methods of separating and removing random noise from significant signals.

18.2.1 When Is an Image Feature Significant?

Against a noisy sky background, the image of a star consists of a few pixels that have a slightly higher value than their neighbors. If a sky background has a mean value of B and a standard deviation of aB, then 68.3% of sky pixels have a value between B - aB and B + aB, while the remaining 31.7% of pixels lie outside that range—so a pixel that is within 1 aB of the mean sky value is hardly rare enough to qualify as real signal. A pixel that departs 2 oB from the mean is considerably less common—only 4.5% lie outside B ± aB—but still not sufficiently rare to qualify as genuine signal. However, random noise pushes a mere 0.27% of pixels outside the range of values between B -3aB and B + 3aB —and for most practical purposes, a so-called "three-sigma detection" constitutes reasonable evidence that a pixel departing 3 aB from the mean is indeed a significant signal.

Suppose that we try to use this criterion to locate stars in an image, and suppose that the image consists of nothing but an average background sky value of B plus a scattering of stars. (See Figure 18.6.) To find the stars, we measure the standard deviation of the sky background, aB, in star-free region and then set any pixel with a value less than B + 3aB equal to B. Any pixel under this threshold is considered a sky pixel, and any pixel greater than the threshold belongs to a star

Original o = 1000

Figure 18.7 Random noise with a Gaussian deviate follows a predictable pattern. If the standard deviation, a, of the original image is 1.000, then we expect a value of O = 0.800 in Scale 1, 0.274 in Scale 2, 0.120 in Scale 3, and so on. This characteristic allows us to separate real features from random noise.

Original o = 1000

Figure 18.7 Random noise with a Gaussian deviate follows a predictable pattern. If the standard deviation, a, of the original image is 1.000, then we expect a value of O = 0.800 in Scale 1, 0.274 in Scale 2, 0.120 in Scale 3, and so on. This characteristic allows us to separate real features from random noise.

image. Such a scheme can, in fact, locate star images reliably, especially if we set a higher threshold—5 aB—and eliminate all single-pixel detections.

However, there are many things besides stars in astronomical images. The value of a single pixel in an image of a galaxy may never satisfy a B + 3aB crite rion, but a galaxy image is made from hundreds or even thousands of pixels. This is where wavelets enter the scene, because wavelet analysis can tell us whether a cluster of hundreds or thousands of pixels is statistically significant.

Wavelet noise-filtering methods are based on the characteristic behavior of random noise. Consider an image consisting entirely of random noise, such as the example shown in Figure 18.7. For a given standard deviation of the original image, S0, in successive wavelet scales Wj the standard deviation, a7, drops in a predictable way. As each wavelet level is computed, and noise at higher frequencies is stripped away, we're left with smoother and less noisy wavelet scales.

Suppose that we have an image of a faint galaxy that is 32 pixels across. The image will be most evident at wavelet scale 5, and for Gaussian noise, the amplitude of random noise will have dropped by a factor of -40 from aB. If the galaxy's wavelet coefficients at scale 5 exceed B + (3/40)o5, then that cluster of pixels constitute a significant signal, and the galaxy has been detected!

Determining which features in an image are statistically significant and which are due to noise is the basis for wavelet noise filtering. However, we must overcome three obstacles before we can subject an image to a wavelet noise filter and reject the noise. These are:

• Our scheme relies on the properties of Gaussian noise, but in astronomical images, most noise is Poisson noise with quite different characteristics.

• In images that are full of features, we must measure the noise amplitude that would exist if there were no features.

• As we reject noise in the filtering process, we must test the rejected wavelet coefficients to insure that we have not accidentally rejected genuine features with the noise.

Fortunately, it is possible to surmount each of these obstacles. In the following sections, we discuss each in turn and show how to overcome it.

18.2.2 Transforming Poisson Noise to a Gaussian Deviate

In images from CCDs and digital cameras, Poisson noise is usually the dominant noise source. Poisson noise differs from Gaussian noise because the standard in an image with Poisson noise depends on the pixel value. In addition to the Poisson noise, CCDs and digital cameras have additive Gaussian from readout.

Poisson noise arises from photon-counting statistics. Given a photon flux having an average count rate of n photons per unit time, the standard deviation is Jfi, thus the noise rises with the square root of the photon count. To a good approximation, if we divide the photon count by its own square root, the resulting value has a standard deviation of 1.0 regardless of the photon count. (This occurs because the noise must also be divided by the square root of the photon count, so Jfi/ Jfi = 1 .) This forms the basis for the Anscombe transform that converts Poisson noise to a signal with a constant Gaussian deviation.

In CCD images, a pixel value, 5", is the sum of a bias level plus a photoelec-

tron count, n, divided by the amplifier gain:

where the bias, 5, is in ADUs and the amplifier gain, g, has units of photoelectrons per ADU. Solving for the photon count gives us:

Two other factors will enter: the readout noise, R, which is composed of Gaussian noise, and a small correction term that compensates for the slight asymmetry of the Poisson distribution relative to the Gaussian distribution. Combining these gives an equation that transforms each pixel value in an image into a new pixel value with a standard deviation of 1:

To transform a CCD image into a form suitable for wavelet noise filtering, it is necessary to know:

• the amplifier gain in photoelectrons per ADU,

Running wavelet noise filtering on an image that has not been transformed normally results in too little noise filtering in the brighter areas of the image relative to the sky background, or too little noise filtering altogether.

•Tip: AIP4Win's Constant Sigma Scaling Tool performs the Anscombe transform. The transform has little effect on the visual appearance of the image, but Anscombe-scaled images work much better with Wavelet Noise Filtering.

18.2.3 Measuring Noise in Images

When you measure the root mean square variation of the pixel values an image, the result reflects not only noise, but also variations due to the structure of the image itself. For the synthetic star images shown in Figure 18.6, the noise, a, was known to be 1.0, so the clipping threshold could be set at 3a, or 3.0. To use wavelets to remove noise, its is extremely useful to know the noise in order to set the clipping threshold correctly. Fortunately, it is possible to use wavelets to isolate and measure noise, and then to threshold wavelet scales to remove it.

The distribution of Gaussian noise follows a regular and repeatable pattern. Each successive wavelet scale in an image consisting of pure Gaussian noise with 0 = 1 has lower noise:

Scale

Noise, Oy image 1

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