The MultiDrizzle Handbook
3.3 Weight Maps and Correlated Noise
When images are combined with drizzle a weight map can be specified for each input image. This image minimally contains information on the bad pixels in the image. When the final output science image is generated, an output weight map which combines the information from all the input weighting images is also saved. When a drop of value and user defined weight is added to an output image , with weight and a fractional pixel overlap of , the resulting value of the image and is:
Drizzle has a number of advantages over standard linear reconstruction methods. Since the area of the pixels scales with the Jacobian of the geometric distortion, it is preserved for surface and absolute photometry. Therefor the flux can be measured with an aperture that is independent of its position on the image. Since the method anticipates that a given output pixel might not receive any information from an input pixel, missing data does not cause a substantial problem as long as the observer has taken enough dither samples to fill in the missing information.
The output pixels in the final drizzled image are not independent of one another, causing the noise in the output image to be correlated to some degree. In principle, the correlated noise can be fully described by creating a correlation image. However, the implementation of such schemes becomes complicated when images are shifted at sub-pixel scales. A more practical approach is to use the weight maps generated by drizzle to calculate the expected r.m.s. noise. The weight appropriate to a given value of the scale parameter (expressed in terms of the ratio of the output to input pixel size), can be calculated in the following way (as described by Casertano et al. 2000):
Where D and B are the counts per pixel (in DN) due to the dark current and background, respectively, averaged over the entire image. t is the exposure time in seconds, g is the gain of the detector (users should be aware of the units of their image and use the appropriate gain value), and is the readnoise in DN/pixel. The quantity f represents the inverse flat field, corresponding to the way in which the HST pipeline flats are defined.
A more in-depth discussion of noise in drizzled images can be found in Section 3.3.1 Correlated Noise Details.
3.3.1 Correlated Noise Details
Drizzle frequently divides the power from a given input pixel between several output pixels. As a result, the noise in adjacent pixels will be correlated. Understanding this effect in a quantitative manner is essential for estimating the statistical errors when drizzled images are analyzed using object detection and measurement programs such as SExtractor (Bertin and Arnouts 1996) and DAOPHOT (Stetson 1987).
The correlation of adjacent pixels implies that a measurement of the noise in a drizzled image on the output pixel scale underestimates the noise on larger scales. In particular, if one block sums a drizzled image by pixels, even using a proper weighted sum of the pixels, the per-pixel noise in the block summed image will generally be more than a factor of greater than the per-pixel noise of the original image. The factor by which the ratio of these noise values differ from in the limit as is referred to as the noise correlation ratio, . One can easily see how this situation arises by examining Figure 3.3.Figure 3.3: Schematic View of the Noise Distribution for a Pixel
A schematic view of the distribution of noise from a single input pixel between neighboring output pixels.
In Figure 3.3 we show an input pixel (broken up into two regions, and ) being drizzled onto an output pixel plane. Let the noise in this pixel be and let the area of overlap of the drizzled pixel with the primary output pixel (shown with the heavier border) be , and the areas of overlap with the other three pixels be and where and . Now, the total noise power added to the image variance is ; however, the noise that one would measure by simply adding up the variance of the output image pixel-by-pixel would be:
This inequality exists because all cross terms are missed by summing the squares of the individual pixels. These terms, which represent the correlated noise in a drizzled image, can be significant.
18.104.22.168 The Calculation
In general, the correlation between pixels, and thus R, depends on the choice of drizzle parameters and geometry and orientation of the dither pattern, and often varies across an image. While it is always possible to estimate R for a given set of drizzle parameters and dithers, in the case where all the output pixels receive equivalent inputs (in both dither pattern and noise, though not necessarily from the same input images) the situation becomes far more analytically tractable. In this case, calculating the noise properties of a single pixel gives one the noise properties of the entire image.
Consider then the situation when pixfrac, p, is set to zero. There is then no correlated noise in the output image - since a given input pixel contributes only to the output pixel which lies under its center, and the noise in the individual input pixels is assumed to be independent. Let represent a pixel from any of the input images, and let C be in the set of all whose centers fall on a given output pixel of interest. Then it is simple to show that the expected variance of the noise in that output pixel, when p=0, is simply:
where is the standard deviation of the noise distribution of the input pixel . We term this , as it is the standard deviation calculated with the pixel values added only to the pixels on which they are centered.
Now let us consider a drizzled output image where p > 0. In this case, the set of pixels contributing to an output pixel will not only include pixels whose centers fall on the output pixel, but also those for which a portion of the drop lands on the output pixel of interest even though the center does not. We refer to the set of all input pixels whose drops overlap with a given output pixel as and note that . The variance of the noise in a given output pixel is then:
where is the fractional area overlap of the drop of input data pixel with output pixel o. Here we choose the symbol to represent the standard deviation calculated from all pixels that contribute to the output pixels when pixfrac = p. The degree to which and differ depends on the dither pattern and the values of p and s. However, as more input pixels are averaged together to estimate the value of a given output pixel in P then in C, . When p=0, is by definition equal to .
Now consider the situation where we block average a region of NxN Pixels of the final drizzled image, doing a proper weighted sum of the image pixels. this sum is equivalent to having drizzled onto an output image with a scale size Ns. But as , this approaches the sum over C, or, in the limit of large . However, a prediction of the noise in this region, based solely on a measurement of the pixel-to-pixel noise, without taking into account the correlation between pixels would produce . Thus we see that:
One can therefore obtain R for a given set of drizzle parameters and dither patterns by calculating and and performing the division. However, there is a further simplification that can be made. Because we have assumed that the inputs to each pixel are statistically equivalent, it follows that the weights of the individual output pixels in the final drizzled image are independent of the choice of p. To see this, notice that the total weight of a final image (the sum of the weights of all the pixels in the final image) is independent of the choice of p. Ignoring edge pixels, the number of pixels in the final image with non-zero weight is also independent of the choice of p. Yet as the fraction of pixels within p of the edge scales as 1/N, and the weight of an interior pixel cannot depend on N, we see that the weight of an interior pixel must also be independent of p. As a result,
Therefore, we find that:
Although R must be calculated for any given set of dithers, there is one case that is particularly illustrative when one has many uniformly placed dithers across the pixel - one can approximate the effect of the dither pattern on the noise by assuming that the dither pattern is entirely uniform and continuously fills the output plane. In this case the above sums become integrals over the output pixels, and thus it is not hard (though somewhat tedious) to derive R. If one defines where p=pixfrac and s=scale, then in the case of a filled uniform dither pattern one finds,
Using the relatively typical values of p=0.6 and s=0.5, one finds R=1.662. This formula can also be used when block summing the output image. For example, a weighted block-sum of NxN pixels is equivalent to drizzling into a single pixel of size Ns. The correlated noise in the block summed image can be estimated by replacing s with Ns in the above expressions.
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