Sharpness

The sharpness method calculates Full Width Half Maximum (FWHM) values along and across track using a blur kernel estimator. This methodology is still under development.

Theoretical background

Let us consider the following model that relates a sharp image \(u\) with the image \(v\) acquired by an optical system:

\[ v = u * k + n, \]

where \(k\) is an intrinsec blur kernel and \(n\) is the acquisition noise. Given a working image \(v\), our objective here is to find \(k\). Notice this is an ill-posed problem, because both \(u\) and \(k\) are unknown and, in general, there is not a unique solution.

Paper [1] Jérémy Anger , Gabriele Facciolo, Mauricio Delbracio (2019) ‘Blind Image Deblurring using the l0 Gradient Prior'. Image Processing On Line (IPOL) proposes to estimate \(k\) in an iterative fashion: the first step is to produce a sharp image \(u\) by deconvolving the blurry image using the kernel \(k\) estimated from the previous iteration (starting with an initial seed). Then a new \(k\) is estimated by solving a linear equation that involves the gradient of the image \(u\) we have just obtained.

Algorithm implementation

We assume our working image is a grayscale (1 channel) array. In case our working image has multiple channels, each of these channels is measured independently.

1- Working image is divided in windows (sub-images) of size \(w x w\) pixels.

Initial window grid covering working image.

2- For each window, a set of statistics are computed. These values are used to:

• Discard those windows based on a threshold \(t_{i}\) value for each statistic \(s_{i}\).

• Order those windows that passed the previous filter. The first window is a priori the most suitable for the sharpness measurement of the working image. The purpose of this filtering is to get rid of those windows that are not feature rich, where it is known that the blur kernel estimation will give an inaccurate or directly erroneous result.

Green: windows that passed the filtering; red: windows that were filtered out.

3- The first \(N\) windows are selected. If less than \(N\) windows passed the filtering process, all of these windows are considered. If no window passed the filtering process, the image is tagged as invalid to measure sharpness onto.

For each of the selected windows:

4- A blur kernel is estimated by means of the algorithm described in [1] (see Theoretical background section above for an overview). The main parameters of this algorithm are: the number of iterations \(p_{n_{iter}}\), and the size of the kernel \(p_{kernel_{size}}\) in pixels.

Working window (sub image) and its estimated blur kernel. Note: this is not a real example, only for explanatory purposes.

5- The obtained blur kernel is modeled as a 2D gaussian function using a non-linear least squares approach. In particular, the dispersion values along its main axis \({\sigma}_{x}\), \({\sigma}_{y}\) are estimated.

Estimated blur kernel and its modeling as a 2D Gaussian. Note: this is not a real example, only for explanatory purposes.

6- The values for the FWHM along each major axis of the modeled 2D gaussian distribution are reported. They are related with the dispersion (sigma) value \({\sigma}\) by the following expression, [Tavernier, Stefaan. (2010). Experimental Techniques in Nuclear and Particle Physics. 10.1007/978-3-642-00829-0].

\[ FWHM_{x,j} = \sqrt{8ln2} · {\sigma}_{x,j}, FWHM_{y,j} = \sqrt{8ln2} · {\sigma}_{y,j} \]

7- Results for x axis for all \(N\) windows are aggregated using the harmonic mean \(H\). And the same for the results for y axis. This gives us two values \(FWHM_{x}\), \(FWHM_{y}\) that are further aggregated into a single value \(FWHM\) using the geometric mean. That is:

\[ FWHM_{x,j} = \sqrt{FWHM_{x} · FWHM_{y}}, \]

where

\[ FWHM_{x} = H(FWHM_{x,1}, ...., FWHM_{x,N}), FWHM_{y} = H(FWHM_{y,1}, ...., FWHM_{y,N}). \]

Caution: Experimental Metric

This metric is currently experimental. Its values are preliminary and require further characterization. Use with discretion.