A method of determining a geometric transform of a reference signal in an image for extracting digital data, the method comprising: with a programmed processor or digital logic circuit: transforming the image into a frequency domain to produce a discrete frequency domain representation of the image at integer coordinates; estimating complex frequency components at non-integer coordinates within the discrete frequency domain representation of the image according to a point spread function, whereas the complex frequency components comprise both phase and magnitude; 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 of the frequency magnitudes within the neighborhood, wherein magnitude values at non-integer locations in the neighborhood are computed using the complex frequency components 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. with a programmed processor or digital logic circuit: transforming the image into a frequency domain to produce a discrete frequency domain representation of the image at integer coordinates; estimating complex frequency components at non-integer coordinates within the discrete frequency domain representation of the image according to a point spread function, whereas the complex frequency components comprise both phase and magnitude; transforming the image into a frequency domain to produce a discrete frequency domain representation of the image at integer coordinates; estimating complex frequency components at non-integer coordinates within the discrete frequency domain representation of the image according to a point spread function, whereas the complex frequency components comprise both phase and magnitude; 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 of the frequency magnitudes within the neighborhood, wherein magnitude values at non-integer locations in the neighborhood are computed using the complex frequency components estimated at non-integer locations; determining a new geometric transform that transforms the reference signal components to the updated coordinates; and 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 of the frequency magnitudes within the neighborhood, wherein magnitude values at non-integer locations in the neighborhood are computed using the complex frequency components 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. <EOS>