Fig. 2: The working principle of the conjugate-CLASS algorithm.

a A set of recorded electric field images \(E\left({{{{{{\bf{r}}}}}}}_{{{{{{\rm{o}}}}}}}-{{{{{{\bf{r}}}}}}}_{{{{{{\rm{in}}}}}}};{{{{{{\bf{r}}}}}}}_{{{{{{\rm{in}}}}}}}\right)\) of the reflected wave in the camera plane for various \({{{{{{\bf{r}}}}}}}_{{{{{{\rm{in}}}}}}}\) (only the amplitude maps are shown). These images constitute the reflection matrix \({{{{{\boldsymbol{R}}}}}}\). A USAF resolution target was placed under the skull. The inset images indicate the illumination points \({{{{{{\bf{r}}}}}}}_{{{{{{\rm{in}}}}}}}\) within the field of illumination. Color bar: normalized amplitude. b Each image in a was converted into \(E\left({{{{{{\bf{w}}}}}}}_{{{{{{\rm{o}}}}}}};{{{{{{\bf{w}}}}}}}_{{{{{{\rm{in}}}}}}}\right)\) after the basis conversion (only the amplitude maps are shown). These images constitute the \({{{{{{\boldsymbol{R}}}}}}}_{{{{{{\rm{con}}}}}}}\). Color bar: log-scale of the normalized amplitude. c Aberration correction map \({\phi }_{{{{{{\rm{in}}}}}}}^{{{{{{\rm{c}}}}}}}\) obtained by conjugate-CLASS algorithm. It is decomposed into the quadratic phase \({\phi }_{{{{{{\rm{quad}}}}}}}=\left(k/2z\right){\left|{{{{{{\bf{w}}}}}}}_{{{{{{\rm{in}}}}}}}\right|}^{2}\) and the skull-induced aberration \({\phi }_{{{{{{\rm{s}}}}}}}\). Color bar: phase in radians. d \({\widetilde{O}}_{{{{{{\rm{M}}}}}}}\) before conjugate-CLASS correction. e \({\widetilde{O}}_{{{{{{\rm{M}}}}}}}\) after conjugate-CLASS correction. Color bars in d, e: log-scale of the normalized intensity with respect to the maximum intensity in e. f Time-gated confocal image. g Aberration-corrected image by the pupil-CLASS method. h conjugate-CLASS image obtained by the inverse Fourier transform of \({\widetilde{O}}_{{{{{{\rm{M}}}}}}}\) in e. Color bars in f–h: normalized intensity. i PSFs before and after aberration correction obtained from the reflection matrix. They correspond to the convolution of input and output PSFs, thereby accounting for the roundtrip aberrations. Line plots along the center pixel line in the PSF images are shown on the lefthand. c-CLASS conjugate-CLASS. Scale bar: 20 μm (a, f, i), 100 μm (b, c).