Complex-valued backprogation (CVBP) is an essential algorithm parsing a large amount of complex data for statistical inferences to deep optics neural networks. However, the operating optical transformations are almost unitary, which require the unitary backpropagation algorithms to meet the hetero-equivalence from their mathematical to optical mechanisms. This paper mainly compares two compact formulations for a common CVBP and its unitary-variants under holomorphic and compatible conditions, for which composite mechanisms are invented. For unitary weights-update, essentially, the Riemannian gradient of a CVBP is expanded from Euclidean gradient by executing an exponential mapping onto its unitary manifold. The convergent speed and accuracy, as well as the evolutions of their energies and averaging-phases at each layer are progressively investigated and compared, which reflect the dynamic behaviors on the network convergences. We find that not all the nonlinear activations under the respective conditions are always ensured to convergences, and the averaging-phases still remain several jumps although the energies at current layer situate at the convergent districts.
To address the camera calibration problems on multi-cameras photography without any common field of views (FOVs), an over-the-horizon camera calibration on smartphones with planar mirror reflections is proposed. The liquid crystal display (LCD) target is placed suitably in the FOV of front cameras assembled on smartphones, and the chessboard images are captured from different views to obtain the intrinsic and extrinsic parameters for the front cameras. Then the target is reflected into the FOV of rear cameras with planar mirror reflections, which present parallel characteristics with respect to the front cameras, and then the absolute poses between the LCD target and the rear cameras are estimated. The over-the-horizon camera calibration experiments on a single and double smartphones show that the avenue of planar mirror reflections is simple and delicate. The Kappa’s poses between the front and the rear cameras are exhibited in the three-dimensional diagrams along with the mirrored images, in which the poses transfer between a series of cameras and a pubic shared target are implemented. The method behaves significantly potentials to resolve some intractable problems on a multi-camera holography without any common FOVs.
To address the camera calibration problems on multi-camera photography without any common field of views (FOVs), an over-the-horizon camera calibration method on smartphones with planar mirror reflections is proposed. The liquid crystal display (LCD) target is placed suitably in the FOV of front cameras assembled on smartphones, and the chessboard images are captured from different views to obtain the intrinsic and extrinsic parameters for the front cameras. Then the target is reflected into the FOVs of rear cameras with planar mirror reflections, which presents parallel characteristics with respect to the front cameras, and then the absolute poses among the LCD target and the rear cameras are estimated. The over-the-horizon camera calibration experiments on a single and double smartphone show that the avenue of planar mirror reflections is simple and delicate. The Kappa’s poses among the front and the rear cameras are exhibited in the three-dimensional diagrams along with the mirrored images, in which the poses transfer among a series of cameras and a public shared target are implemented. The method behaves significantly potentials to resolve some intractable problems on a multi-camera holography without any common FOVs.
Since liquid crystal display (LCD) screen locates outside of the camera’s field of view in fringe-reflection photogrammetry, fringes displayed on LCD screen are obtained through specular reflection by a fixed camera. Thus, the pose calibration between camera and LCD screen is one of the main challenges in fringe-reflection photogrammetry. A markerless planar mirror is used to reflect the LCD screen more than three times, and the fringes are mapped into the fixed camera. The geometrical calibration can be accomplished by estimating the pose between the camera and virtual image of fringes. With the help of the relation between their pose, incidence and reflection ray can be unified in the camera frame, forward triangulation intersection can be operated in the camera frame to measure 3D coordinate of specular surface. In the final optimization, constraint bundle adjustment is operated to refine simultaneously the camera intrinsic parameters including distortion coefficients, estimated geometrical pose between LCD screen and camera, 3D coordinate of specular surface, with the help of absolute phase collinear constraint. Results of simulations and experiments demonstrate that the pose calibration with planar mirror reflection is simple, feasible and constraint bundle adjustment can enhance the three-dimensional coordinate measurement accuracy in fringe-reflection photogrammetry.
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