Totagraphy for superresolution imaging and signal processing of positive, real-valued images and signals

A method for recovering missing spectral amplitude and phase information of an input signal, the method comprising:

• providing at least one transformation unit with an input and an output;
• providing at least three masks or illumination patterns, comprising a first transparent mask and a complementary pair of unipolar masks or illumination patterns, each of the at least three masks or illumination patterns configured to be disposed at the input;
• separately multiplying the input with each of the at least three masks or illumination patterns to generate a modified input from each of the masks or illumination patterns;
• convolving the modified inputs with a transfer function to yield outputs comprising positive real values to be detected by an output sensor, the transfer function comprising at least one invertible region and at least one non-invertible region;
• passing the outputs through the at least one transformation unit, the at least one transformation unit configured to produce a spectral representation of each output corresponding to each modified input;
• defining two regions of a recovered output comprising a plurality of points, divided into Region 1 in which the transfer function has known values which are invertible, and Region 2 in which the transfer function has very small, unknown or corrupted values;
• computing amplitude and phase values of the points corresponding to Region 1 in each spectral representation;
• setting an amplitude at each point in Region 2 of each spectral representation to be the mean amplitude computed over all points in Region 1 of the corresponding spectral representation, and setting a corresponding phase to a random value in [0,2π] for each point in Region 2 of the corresponding spectral representation to produce a set of modified spectral representations;
• calculating an Inverse Fourier Transform of each of the modified spectral representations to obtain a set of candidate intermediate outputs;
• averaging the candidate intermediate outputs to yield a single averaged candidate intermediate output;
• providing the single averaged candidate intermediate output at an input plane and applying each of the at least three masks or illumination patterns to generate modified inputs from each of the masks or illumination patterns;
• repeating the above steps, iteratively producing new candidate intermediate outputs by recalculating the values in Region 2 and replacing Region 1 with the original Region 1 until the candidate intermediate output values converge, yielding the recovered output.