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Array algebra of photogrammetry and geodesy unified multi-linear matrix and tensor operators in an expansion of Gaussian adjustment calculus to general matrix inverses and solutions of inverse problems to find all, or some optimal, parametric solutions that satisfy the available observables. By-products in expanding array and tensor calculus to handle redundant observables resulted in general theories of estimation in mathematical statistics and fast transform technology of signal processing. Their applications in gravity modeling and system automation of multi-ray digital image and terrain matching evolved into fast multi-nonlinear differential and integral array calculus. Work since 1980’s also uncovered closed-form inverse Taylor and least squares Newton-Raphson-Gauss perturbation solutions of nonlinear systems of equations. Fast nonlinear integral matching of array wavelets enabled an expansion of the bundle adjustment to 4-D stereo imaging and range sensing where real-time stereo sequence and waveform phase matching enabled data-to-info conversion and compression on-board advanced sensors. The resulting unified array calculus of spacetime sensing is applicable in virtually any math and engineering science, including recent work in spacetime physics. The paper focuses on geometric spacetime reconstruction from its image projections inspired by unified relativity and string theories. The collinear imaging equations of active object space shutter of special relativity are expanded to 4-D Lorentz transform. However, regular passive imaging and shutter inside the sensor expands the law of special relativity by a quantum geometric explanation of 4-D photogrammetry. The collinear imaging equations provide common sense explanations to the 10 (and 26) dimensional hyperspace concepts of a purely geometric string theory. The 11-D geometric M-theory is interpreted as a bundle adjustment of spacetime images using 2-D or 5-D membrane observables of image, string and waveform matching in the unified array calculus of applied mathematics.

Application of unified array calculus to connect 4-D spacetime sensing with string theory and relativity