Clearly, the bulk of the noise is in wavelet Scale 1. If we can figure out a way to isolate those pixels in the image which contain no significant image information in Scale 1—in astronomical images, these are the starless parts of the sky background—we can determine the noise in Scale 1 and use the value in the table to find the contribution due to Gaussian noise at every other scale.

To do this, we create a special type of image called a significance image or multiresolution support image that encodes significant features—that is, non-noise features—of an image. The process begins with a wavelet transform and a best-guess estimate, a, of the noise in the image. At each scale in the wavelet transform, we treat a wavelet coefficient as significant if its absolute value is equal to or greater than 3 Gj :

Next, compute the significance image. Using the Greek letter gamma, T, to represent the significance image, with r}(jc, y) representing the value of T at scale j, compute pixel values for the significance image:

If T(x, y)= 0 , the pixel in the image has no structure at any scale exceeding the estimated noise a, so we compute the standard deviation for all pixels that satisfy this condition, to produce a new estimate of o . Because the initial best-guess estimate may have been wrong, the significance image may have inaccurate—but the new value of a should be closer to the correct value than the initial estimate. To get the best estimate, we use the new value of a to compute a new significance image, and iterate until the value of o converges, usually with a few iterations.

• Tip: While significance images are decidedly ugly, they are useful "inside features" that AIP4Win uses to measure and filter noise.

18.2.4 Rejecting Noise and Retaining Significant Features

The third obstacle in separating noise from statistically significant features in an image is to insure that even as noise is cut, significant image features are not cut. The best way to accomplish this comes neither from finely tuning a one-pass filter nor from some kind of brute-force technique, but by iteratively correcting and testing the image using a method similar to that used to determine the noise.

Suppose, for a moment, that you did have a filter that separated noise from signal perfectly: What would you see if you took an image, filtered it, and then subtracted the filtered image from the original noisy image? That residual image would have nothing but noise, free of any trace of the original image. This suggests that an effective method for removing noise will be an iterative process that filters, tests, and corrects the filtered image.

18.2.5 The Solution: An Iterative Wavelet Noise Filter

We have now seen that the three obstacles to effective wavelet noise filters can be overcome. Images containing Poisson noise and mixtures of Poisson noise and Gaussian noise can be transformed into pure Gaussian noise. We also have an effective method for measuring noise in images, and finally, we've seen that an iterative procedure that filters, tests, and corrects should be able to cleanly separate statistically significant image features from noise.

Here then is a procedure that combines the predictable behavior of Gaussian noise, an accurate method for measuring noise, and wavelet filtering with active feedback:

1. Convert an image with a mixture of Poisson noise and Gaussian noise to Gaussian noise using the Anscombe transform. Although this requires

Figure 18.10 Here are the images from Figure 18.9 as they appear after iterative wavelet filtering, with the original noise-free image (upper left) for comparison. To a good approximation, the filtered images now appear the same as the original—yet before wavelet filtering they contained large amounts of noise.

Figure 18.10 Here are the images from Figure 18.9 as they appear after iterative wavelet filtering, with the original noise-free image (upper left) for comparison. To a good approximation, the filtered images now appear the same as the original—yet before wavelet filtering they contained large amounts of noise.

knowing the gain and readout noise in your CCD camera, as well as the bias value for the original image, it insures accurate noise removal.

2. Measure the amplitude of the noise in the transformed image using the iterative noise measurement procedure described in the previous section. From the total noise, compute the noise expected at every wavelet scale, and use this information to create a significance image.

3. Create a solution image and set all pixel values in it to zero. This image begins filled with zeros, but it will eventually contain the filtered image.

4. Subtract the solution image from the original image to produce a residual image. The residual image contains the difference between the ideal filtered image and the original. Initially, it will contain a copy of the original image, but that is about to change!

5. Perform a wavelet transform on the residual image. Using the noise values computed for each scale, truncate the wavelet coefficients, then perform an inverse wavelet transform.

6. Add the inverse wavelet output to the existing solution image. The first time through this process, the solution image begins full of zeros; on later passes, it contains the solution from the previous iteration.

7. Subtract the solution image from the original to create a new residual image. If the noise filter were perfect, the residual image would contain only noise, but chances are that some image features were incorrectly removed and some noise remains in the solution image.

8. Measure the noise in the residual image, and use the noise figure to create a new significance image based on the residual image.

9. Iterate by returning to step 5. At each pass, significant features that remain in the residual image will be added to the solution image. When the noise you measure in the residual image stabilizes at a steady value, all significant image structure has been transferred to the solution image and all noise under 3 o has been left in the residual image—and filtering is complete.

This procedure requires that you know your camera's gain and readout noise, and the output bias of the image so that you can convert the noise in the image to Gaussian noise. Applying an iterative wavelet filter to images in which Poisson noise is dominant results in image artifacts and oversmoothed output images.

Iterative wavelet filtering works best for critically sampled and oversampled images. In undersampled images, a background filled with small star images can appear as noise, and produce overfiltering artifacts. In critically sampled and oversampled images, noise is determined correctly, and the image is correctly filtered.

Wavelet noise filtering solves the problem of separating real features from image noise—like a lawn mower that cuts the grass without cutting the daisies. Attempting to cut noise ("grass") in frequency space with the Fourier Transform always cuts image detail ("daisies"), and in image space, smoothing and averaging with convolution methods removes both daisies and grass. The hybrid nature of wavelets allows cutting noise without losing statistically significant image detail.

• Tip: The Iterative Wavelet Filter in AIP4Win removes "grain" from images that have been transformed to Gaussian noise using Constant Sigma Scaling. Figures 18.9 and 18.10 show how effective wavelet filtering can be on images that have small, medium, and large amounts of noise.

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