Signal processors and methods for estimating geometric transformations of images for digital data extraction

A non-transitory computer readable medium, on which is stored instructions, which when executed by a processor, perform a method of determining a geometric transform of a reference signal in an image for extracting digital data, the method comprising:

• transforming the image into a frequency domain to produce a discrete frequency domain representation of the image at integer coordinates;
• estimating phase at non-integer coordinates within the discrete frequency domain representation of the image according to a point spread function;
• for plural reference signal components of a reference signal:
• transforming coordinates of a reference signal component according to a candidate geometric transform;
• updating the coordinates of the reference signal component to a location within a neighborhood of the discrete frequency domain representation around the coordinates of the transformed reference signal component, the location corresponding to a highest frequency magnitude value of frequency magnitude values within the neighborhood, wherein frequency magnitude values at non-integer locations in the neighborhood are computed using the phase estimated at non-integer locations;
• determining a new geometric transform that transforms the reference signal components to the updated coordinates;
• and
• applying the new geometric transform to extract encoded digital data from the image.