Figure 4.

Deconvolution of fluorescence images using the reconstructed x-ray probe provided by ptychography and modified deconvolution filters. (a) Power spectra of Ca, P, K, S fluorescence images shown along with their respective signal trends $S\propto f^{-a_Z}$, and the half-period length scales x _cutoff marking the transition to a noise floor N. Also shown is the azimuthally-averaged inverse modulation transfer function 1/MTF(f) that one would use as a Fourier deconvolution filter with a noise-free image (Eq. 2). (b) The deconvolution filters D _Z(f) for the elements Z = Ca, P, K, and S obtained using Eq. 4. (c) The intensity point spread function (PSF) of the reconstructed probe, or p(x, y), provided by ptychography; its inverse Fourier transform, azimuthally-averaged, leads to the modulation transfer function (MTF) plotted in (a) and contributes to the deconvolution filters D _Z(f) in (b). The fluorescence maps i(x, y) before deconvolution are shown in (d) (these are the same images as shown in Fig. 2(a)), and the deconvolved images o(x, y) are shown in (e). The deconvolved images show improved detail, and a reduction of a poorly-spatially-resolved fluorescence “haze” on the images.