Dither algorithm performs an optimal adding of a sequence of images as far as resolution is concerned. The principle is that, at sub-pixel level, shifts between individual input images are nearly randomly distributed. For example, a star in the first image may be centered perfectly in the middle of a pixel, whereas it will be across two pixels in the second one, and so on. Since it is easy to know the exact shift between the images, it is possible to create an output image with a finer sampling, in which resolution may be increased with respected to each input image. In fact, energy from each input pixel is dropped in the output image, and the whole processus may be compared to a drizzle.
Drizzling is adapted to undersampled images, for example when the telescope focal length is too short for the pixel size. One may consider that the system is undersampled when FWHM is smaller than 2 pixels. In this situation much of the information lost in undersample regime can be restored.
Before using drizzling technique, it is necessary to know the exact shift between the images. It is also very important that all the input images are acquired in the same conditions: same exposure time, same sky background level. If this is not the case, you have to adjust offset and gain prior applying drizzling algorithm (NGAIN and NOFFSET commands under Iris for example).
The drizzling algorithm step by step (see figure 8):
Step 1: Reduce
or coarse (by calculation !) the size of the pixels in the starting
image, but preserve the same interval between pixels.
The "shrink" pixel size at the step 1 is crucial. We define pixfrac as the ratio of the linear size of the coarse pixel to the original input pixel linear size. If pixfrac=0 the drizzle algorithm is equivalent to interlacing, while the traditionnal shift-and-add is equivalent to pixfrac=1. One must choose a pixfrac value that is small enough to avoid degrading final image, but large enougth that then all images are dropped, the coverage of the ouput image is fairly uniform. We choose typically pixfrac between 0.5 and 0.7.
Mathematical formulation of drizzling:
W' = a . w + W
The weight w of the pixel can be zero if it is a bad pixel (hot pixels, dead pixels, cosmic rays event, ...), or can be adjusted according to the local noise (the value is then inversely proportional to the variance maps of the input image).
Remenber that algorithm is effective if the images are really undersampled (FWHM of 1 to 2 pixel). The displacement, and more generaly, geometric distortion, between the individual input images must be perfectly well-known (to 1/10 of pixel precision typically). The number of input images must be large (10 or more) to avoid holes in the final image. Most important, displacement between the input images (diphering techniques) must be random on 2 axis. So, it is necessary to shift arbitrary the telescope between each exposure during deep-sky sessions. The amplitude of the shift can be of a fiew fractions of pixels in a random direction. At the processing stage the relative shifts between images is precisely determined by calculation of the centroid of stars (PSF fitting btween common stars or cross-correlation between a reference image and the input images). The registration parameters are fundamental quantities for the drizzling method.
1. Resolution gain can be up to 2.
The Iris software implement a version of the drizzle procedure (DRIZZLE command).The algorithm was developed by Richard Hook and Andrew Fruchter to produce the The Hubble Deep Field, the deepest optical image of the universe yet taken. It is now used for many other field.
For more information about drizzle algorithm:
To show the effectiveness of resolution improvement by combining undersampled multiframes we will now process diphered images carried out from the observatory Pic du Midi Observatory (french Pyrénées) during the summer 1999. The instruments used are simple photographic objectives (55 to 80 mm focal lenght) and an Audine CCD camera.
Figure 10. The Audine camera installed on a Takahashi EM200 german equatorial mount at the Pic du Midi Observatory.