# Resolution Limit

From the author of Sizing Up The Universe

Shown below are two versions of an image of Jupiter taken on August 31, 2010. The left image is a stack of about 2200 raw frames from a 4 minute video taken with my 3.5" aperture Questar telescope. The right image is the same image after wavelet-sharpening.

 Unsharpened Image Wavelet Sharpened Image

With it's 89 mm aperture, the (Rayleigh criterion) resolution limit of the 3.5" Questar is 1.5 arcseconds. According to the Nyquist-Shannon Sampling Theorem, the pixel resolution is about one-half the Rayleigh resolution, in other words, 0.75 arcseconds. We can test to see if this is correct.

At the time the photograph shown above was taken, Jupiter was 49 arcseconds in equatorial diameter. In the pictures, Jupiter is 180 pixels wide. Hence, each pixel is 0.27 arcseconds.

But, what is the resolution achieved in this image? One way to answer this question, suggested to me by Rus Belikov, is to downsample the sharpened image to a specific pixel resolution and then upsample it back to the original resolution. If the resulting upsampled image looks worse than the original, then the resolution of the original image was better than the resolution of the downsampled image.

Four resolution resamplings are shown below. Shown next to each one is the original (wavelet sharpened) image for comparison.

The 0.6 arcsecond resampling looks almost unchanged but the 0.8 arcsecond resampling is clearly degraded. Hence, the resolution of the sharpened image is about 0.7 arcseconds. This is the pixel resolution and it agrees quite well with the predicted pixel resolution of 0.75 arcseconds.

 1.0 Arcsecond Wavelet Sharpened Image 0.8 Arcsecond Wavelet Sharpened Image 0.6 Arcsecond Wavelet Sharpened Image 0.5 Arcsecond Wavelet Sharpened Image