Hello Dr. Talley,
Thank you very much for your attention and thorough analysis! Please allow me to respond to your questions one by one.
1) Yes, this study assume that there is no or only low noises. To be frank, we do not believe this paper can solve all the problems perfectly. We treated it as just a beginning instead of a perfect ending. Actually, noise is a hard issue in all kinds of field, and it is a relatively independent research topic (or even area). On the one hand:
In our algorithms, the only input data is the observed image and the PSF image, any other factors (e.g., shot noise, wavefront distortion, detector limitation, etc.) can be attribute to the noises or distortion of these images. Thereby, our paper treats it as an important future direction. Given the progressive advancement of denoising technology, we believe it can be solved or improved gradually.
On the other hand:
Based on the above considerations, input data are assumed to be accurate in this study. Then, our major concern is: if the distance of two points is smaller than the diffraction-limit, and they are imaged simultaneously by a conventional light microscope, are they resolvable in the same image? According classic theories (about the diffraction-limit and Rayleigh criterion), the answer would be no. Of course, existing super-resolution techniques have already achieved resolutions beyond the diffraction-limit. But to our understanding, adjacent points (or different frequency components) are imaged in different time. However, this study finds an exceptional condition (termed “resolvable condition”), such points can be resolved directly. In such a “resolvable condition”, neither profile nor detail information is damaged by diffraction. Thereby, it can be recovered reversibly from a diffraction-blurred image (i.e., an image without high frequency components). This condition is tightly relevant to the imaging condition of existing super-resolution techniques. Then, a method is proposed based on the condition which can achieve unlimited high resolutions in principle.
2) First of all, please allow me to explain this question specifically. In our section “2.1. Background analysis” (page 3), we write: “In many other cases, observed signals are used to carry information. Example 2: in a Single-Molecule-Localization microscope, the observed image of individual molecules is blurred, and the pixel values do not show the molecules’ detailed structure directly.” These sentences are not associated to images. Then in section “2.2. Method for spatial domain” (page 8), we use the figure: “Fig. 3. The 2D situation of the spatial domain method. (a) Before convolution (the ideal image). (b) After convolution (the observed image). The two dashed-line rectangles indicate the ROI.” It is a general example of a ground truth image and the corresponding blurred image. We do not mention that it is a fluorescence image. It is only used for illustration of our method, and not used in the experiment section. Actually, random images are used in our simulation experiments so as to assure our methods works for any possible structures.
Then, please note that our finding and methods apply to various light microscopes including fluorescence microscopes. The imaging of fluorescent molecules is one of their possible applications on fluorescence microscopes. On the one hand, we admit that the extracted images do not include full structure of protein molecules in this way. Thereby, we are glad to modify the sentence in our paper: “For example, further resolve the inner details of individual molecules, fluorescent probes or tiny light sources after localization them”. On the other hand, our methods can extract the inner structures of illuminated ROIs. Such ROIs could actually be various objects as long as it fulfills the “resolvable condition”. In the future, maybe we can try structures comprised of adjacent molecules, or multiple chromophores? Maybe we can also try to extract the inner structure of chromophores? Or, if a molecule is illuminated by light directly (not in a fluorescent manner), is it possible to extract its inner structure better? These are just guesses, but may worth exploration because they are in accordance with our method’s principle.
Edward Y. Sheffield