Projector Pair Research Articles

Deep-learning-based parallel single-pixel imaging for efficient 3D shape measurement in the presence of strong interreflections by using sampling Fourier strategy

Strong interreflections cause intractable systematic errors to traditional 3D shaping methods, for example, fringe projection profilometry. Parallel single-pixel imaging (PSI) captures the light transport coefficients between the camera projector pair, thus overcoming the influence of interreflections. However, PSI requires a large number of measurements, causing poor capturing efficiency. To achieve efficient 3D shape measurement in the presence of strong interreflections, we present a deep-learning-based parallel single-pixel imaging method (dlPSI), featuring adopting a deep-learning network to achieve accurate light transport coefficient reconstruction and using sampling Fourier strategy to reduce measurements. The deep-learning network is designed to suppress the noise caused by undersampling the Fourier domains. Experiments prove that our dlPSI with the characteristic-based deterministic Fourier sampling strategy achieves high-quality 3D shape measurement while saving 94% of measurements.

GMRES methods for tomographic reconstruction with an unmatched back projector

Unmatched pairs of forward and back projectors are common in X-ray CT computations for large-scale problems; they are caused by the need for fast algorithms that best utilize the computer hardware, and it is an interesting and challenging task to develop fast and easy-to-use algorithms for these cases. Our approach is to use preconditioned GMRES, in the form of the AB- and BA-GMRES algorithms, to handle the unmatched normal equations associated with an unmatched pair. These algorithms are simple to implement, they rely only on computations with the available forward and back projectors, and they do not require the tuning of any algorithm parameters. We show that these algorithms are equivalent to well-known LSQR and LSMR algorithms in the case of a matched projector. Our numerical experiments demonstrate that AB- and BA-GMRES exhibit a desired semi-convergence behavior that is comparable with LSQR/LSMR and that standard stopping rules work well. Hence, AB- and BA-GMRES are suited for large-scale CT reconstruction problems with noisy data and unmatched projector pairs.

A partial ordering approach to characterize properties of a pair of orthogonal projectors

A partial ordering approach to characterize properties of a pair of orthogonal projectors

Scalable and view-independent calibration of multi-projector display for arbitrary uneven surfaces

Most existing systems for calibrating multi-projector display suffered from several important limitations such as dependence on point of view, restriction on the display surface and moreover the number of projectors and using obtrusive markers. In this paper, a new method for view-independent calibration for multi-projector displays is presented. Given that the calibration problem of a multi-projector display is an optimization problem, we compute the calibration parameters by writing the appropriate energy functions for the geometric calibration phase. In this method, the camera and projector are introduced as a pair. In the first step, calibration of the pair of camera and projector was carried out. After that, the calibration problem of the system decreases to estimation of the number of camera positions relative to each other. In this method, there is no particular shape for the screen. In addition, due to the 3D shape of the screen, this method is view independent and eventually the image can be wallpapered on the screen. According to the tests carried out to evaluate the system, the accuracy of the proposed system is sub-pixel, and as a result, no misalignment is observed by the human eye in the overlapping area of projectors.

Self-calibrated surface acquisition for integrated positioning verification in medical applications

This paper presents a novel approach for a position verification system in medical applications. By replacing the already existing cross line laser projectors with galvo- or MEMS-based projectors and utilizing the surveillance cameras, a self-calibration of the system is performed and surface acquisition for positioning verification is demonstrated. The functionality is shown by analyzing the radii of calibration spheres and determining the quality of the captured surface with respect to a reference model. The paper focuses on the demonstration with one pair of camera and projector but can also be extended to a multi-camera-projector system, as present in treatment rooms. Compared to other systems, this approach does not need external hardware and is thus space and cost efficient.

On relationships between two linear subspaces and two orthogonal projectors

AbstractSum and intersection of linear subspaces in a vector space over a field are fundamental operations in linear algebra. The purpose of this survey paper is to give a comprehensive approach to the sums and intersections of two linear subspaces and their orthogonal complements in the finite-dimensional complex vector space. We shall establish a variety of closed-form formulas for representing the direct sum decompositions of them-dimensional complex column vector space 𝔺mwith respect to a pair of given linear subspaces 𝒨 and 𝒩 and their operations, and use them to derive a huge amount of decomposition identities for matrix expressions composed by a pair of orthogonal projectors onto the linear subspaces. As applications, we give matrix representation for the orthogonal projectors onto the intersections of a pair of linear subspaces using various matrix decomposition identities and Moore–Penrose inverses; necessary and su˚cient conditions for two linear subspaces to be in generic position; characterization of the commutativity of a pair of orthogonal projectors; necessary and su˚cient conditions for equalities and inequalities for a pair of subspaces to hold; equalities and inequalities for norms of a pair of orthogonal projectors and their operations; as well as a collection of characterizations of EP-matrix.

ВСЕГДА НЕВЫРОЖДЕННЫЕ МНОГОЧЛЕНЫ ОТ ДВУХ ПРОЕКТОРОВ

This paper discusses the polynomials of two projectors that with any selection of these projectors have the value of the nonsingular matrix. Results of work [1] about block-triangular form pair of projectors apply to deduce equations, that the coecients of always nonsingular polynomials satisfy to. From the equations is obtained the main result, namely always nonsigular polynomial can be decomposed into a product of special polynomials. Special polynomial of two projectors P; Q is a linear binomial I + P; I + Q, or a polynomial like this I + x1(PQP PQ) + x2(PQPQP PQPQ) + ::: . It is proved that special polynomials are irreducible. It turns out that linear binomials can be rearranged with some other special polynomials. If in a product of special polynomials the linear binomials are rearranged as much as possible to the left, you will get a product of special polynomials, called standard. It is proved that the standard form of product by special polynomials is unigue. The obtained results have provided a description of the structure of all polynomials of two projectors that with any selection of these projectors are nilpotent matrices (nilpotent polynomials). Similar results were obtained for the involute polynomials and polynomialsprojectors.

Further characterizations of functions of a pair of orthogonal projectors

Further characterizations of functions of a pair of orthogonal projectors

GPU-accelerated iterative reconstruction from Compton scattered data using a matched pair of conic projector and backprojector

Background and ObjectiveIterative reconstruction from Compton scattered data is known to be computationally more challenging than that from conventional line-projection based emission data in that the gamma rays that undergo Compton scattering are modeled as conic projections rather than line projections. In conventional tomographic reconstruction, to parallelize the projection and backprojection operations using the graphics processing unit (GPU), approximated methods that use an unmatched pair of ray-tracing forward projector and voxel-driven backprojector have been widely used. In this work, we propose a new GPU-accelerated method for Compton camera reconstruction which is more accurate by using exactly matched pair of projector and backprojector. MethodsTo calculate conic forward projection, we first sample the cone surface into conic rays and accumulate the intersecting chord lengths of the conic rays passing through voxels using a fast ray-tracing method (RTM). For conic backprojection, to obtain the true adjoint of the conic forward projection, while retaining the computational efficiency of the GPU, we use a voxel-driven RTM which is essentially the same as the standard RTM used for the conic forward projector. ResultsOur simulation results show that, while the new method is about 3 times slower than the approximated method, it is still about 16 times faster than the CPU-based method without any loss of accuracy. ConclusionsThe net conclusion is that our proposed method is guaranteed to retain the reconstruction accuracy regardless of the number of iterations by providing a perfectly matched projector–backprojector pair, which makes iterative reconstruction methods for Compton imaging faster and more accurate.

Parallelizing a Matched Pair of Ray-Tracing Projector and Backprojector for Iterative Cone-Beam CT Reconstruction

Iterative reconstruction methods in X-ray CT can provide better image quality than analytical methods but their applications in practice are still limited due to computationally expensive calculations of repeated projection and backprojection operations. In the past decade, GPU-accelerated methods have been successfully used to reduce the computation time for projection and backprojection. However, it has been of a difficult problem to overcome a trade-off between the accuracy of reconstructed images and the efficiency of parallel computations. For example, when the size of the voxel in the reconstructed volume is larger than that of the detector bin, the use of the conventional unmatched projector-backprojector pair can lower the accuracy of reconstructed images due to the error caused by the mismatch between the projector and backprojector. In this paper, we propose a new GPU-accelerated scheme for the most widely used ray-tracing method (RTM) to perform projection and backprojection operations. Unlike the previous works that accelerate the computation of backprojection by using approximations, our method does not use any approximations for parallelizing the projection and backprojection operations. Since our method is exact, the results are as accurate as those obtained from the nonaccelerated method. We apply our method to iterative reconstruction for dental cone-beam CT systems and test its performance using both the simulated data using a 3-D digital phantom and the real data acquired from an offset flat-panel X-ray CT system. Our experimental results show that, the proposed method achieves a substantially high acceleration rate while retaining the accuracy of the RTM for both projection and backprojection.

Property of polynomials in two projectors

It is proved that a polynomial in two projectors vanishes for all pairs of projectors only if all the coefficients of this polynomial are equal to zero.

IMU and Multiple RGB-D Camera Fusion for Assisting Indoor Stop-and-Go 3D Terrestrial Laser Scanning

Autonomous Simultaneous Localization and Mapping (SLAM) is an important topic in many engineering fields. Since stop-and-go systems are typically slow and full-kinematic systems may lack accuracy and integrity, this paper presents a novel hybrid “continuous stop-and-go” mobile mapping system called Scannect. A 3D terrestrial LiDAR system is integrated with a MEMS IMU and two Microsoft Kinect sensors to map indoor urban environments. The Kinects’ depth maps were processed using a new point-to-plane ICP that minimizes the reprojection error of the infrared camera and projector pair in an implicit iterative extended Kalman filter (IEKF). A new formulation of the 5-point visual odometry method is tightly coupled in the implicit IEKF without increasing the dimensions of the state space. The Scannect can map and navigate in areas with textureless walls and provides an effective means for mapping large areas with lots of occlusions. Mapping long corridors (total travel distance of 120 m) took approximately 30 minutes and achieved a Mean Radial Spherical Error of 17 cm before smoothing or global optimization.

On subspace distances determined by the Frobenius norm

Two measures of separation between two subspaces of a finite dimensional complex vector space are introduced and investigated. The measures were inspired by known expressions for the cosines of the angle and minimal angle between subspaces and are based on the Frobenius norm. By using partitioned representations of a pair of orthogonal projectors, several properties of the measures are identified and their links with other notions used to characterize a separation between subspaces are pointed out.

On column and null spaces of functions of a pair of oblique projectors

A number of characterizations involving column and null spaces of various functions of a pair of oblique projectors are established. Particular attention is paid to properties of a product of the two projectors, but other functions of the pair (such as sum, difference or sum and difference of products) are considered as well. In many cases, the results obtained generalize those already known to be valid for orthogonal projectors.

Dynamic exit pupil trackers for autostereoscopic displays

This paper describes the first demonstrations of two dynamic exit pupil (DEP) tracker techniques for autostereoscopic displays. The first DEP tracker forms an exit pupil pair for a single viewer in a defined space with low intraocular crosstalk using a pair of moving shutter glasses located within the optical system. A display prototype using the first DEP tracker is constructed from a pair of laser projectors, pupil-forming optics, moving shutter glasses at an intermediate pupil plane, an image relay lens, and a Gabor superlens based viewing screen. The left and right eye images are presented time-sequentially to a single viewer and seen as a 3D image without wearing glasses and allows the viewer to move within a region of 40 cm × 20 cm in the lateral plane, and 30 cm along the axial axis. The second DEP optics can move the exit pupil location dynamically in a much larger 3D space by using a custom spatial light modulator (SLM) forming an array of shutters. Simultaneous control of multiple exit pupils in both lateral and axial axes is demonstrated for the first time and provides a viewing volume with an axial extent of 0.6-3 m from the screen and within a lateral viewing angle of ± 20° for multiple viewers. This system has acceptable crosstalk (< 5%) between the stereo image pairs. In this novel version of the display the optical system is used as an advanced dynamic backlight for a liquid crystal display (LCD). This has advantages in terms of overall display size as there is no requirement for an intermediate image, and in image quality. This system has acceptable crosstalk (< 5%) between the stereo image pairs.

Let us do the twist again

Krämer (Sankhy $$\bar{\mathrm{a }}$$ 42:130–131, 1980) posed the following problem: “Which are the $$\mathbf{y}$$ , given $$\mathbf{X}$$ and $$\mathbf{V}$$ , such that OLS and Gauss–Markov are equal?”. In other words, the problem aimed at identifying those vectors $$\mathbf{y}$$ for which the ordinary least squares (OLS) and Gauss–Markov estimates of the parameter vector $$\varvec{\beta }$$ coincide under the general Gauss–Markov model $$\mathbf{y} = \mathbf{X} \varvec{\beta } + \mathbf{u}$$ . The problem was later called a “twist” to Kruskal’s Theorem, which provides conditions necessary and sufficient for the OLS and Gauss–Markov estimates of $$\varvec{\beta }$$ to be equal. The present paper focuses on a similar problem to the one posed by Krämer in the aforementioned paper. However, instead of the estimation of $$\varvec{\beta }$$ , we consider the estimation of the systematic part $$\mathbf{X} \varvec{\beta }$$ , which is a natural consequence of relaxing the assumption that $$\mathbf{X}$$ and $$\mathbf{V}$$ are of full (column) rank made by Krämer. Further results, dealing with the Euclidean distance between the best linear unbiased estimator (BLUE) and the ordinary least squares estimator (OLSE) of $$\mathbf{X} \varvec{\beta }$$ , as well as with an equality between BLUE and OLSE are also provided. The calculations are mostly based on a joint partitioned representation of a pair of orthogonal projectors.

Quantum state tomography using successive measurements

We describe a quantum state tomography scheme which is applicable to a system described in a Hilbert space of arbitrary finite dimensionality and is constructed from sequences of two measurements. The scheme consists of measuring the various pairs of projectors onto two bases—which have no mutually orthogonal vectors. The two members of each pair are measured in succession. We show that this scheme implies measuring the joint quasi-probability of any pair of non-degenerate observables having the two bases as their respective eigenbases. The model Hamiltonian underlying the scheme makes use of two meters initially prepared in an arbitrary given quantum state, following the ideas that were introduced by von Neumann in his theory of measurement.

Investigation of phase error correction for digital sinusoidal phase-shifting fringe projection profilometry

Digital sinusoidal phase-shifting fringe projection profilometry (DSPFPP) is a powerful tool to reconstruct three-dimensional (3D) surface of diffuse objects. However, a highly accurate profile is often hindered by nonlinear response, color crosstalk and imbalance of a pair of digital projector and CCD/CMOS camera. In this paper, several phase error correction methods, such as Look-Up-Table (LUT) compensation, intensity correction, gamma correction, LUT-based hybrid method and blind phase error suppression for gray and color-encoded DSPFPP are described. Experimental results are also demonstrated to evaluate the effectiveness of each method.

Functions of orthogonal projectors involving the Moore–Penrose inverse

Several results scattered in the literature express an oblique projector having given onto and along spaces in terms of a pair of orthogonal projectors. The results were established in various settings, including finite and infinite dimensional vector spaces over either real or complex numbers, but their common feature is that they are valid merely under the assumption of the nonsingularity of certain functions of the involved projectors. In the present paper, these results are unified and reestablished in a generalized form in a complex Euclidean vector space, with the generalization obtained by relaxing the nonsingularity assumption and use of the Moore–Penrose inverse instead of the ordinary inverse. Additionally, several new formulae of the type are provided.

Eigenvalues of functions of orthogonal projectors

By representing two orthogonal projectors in a finite dimensional vector space as partitioned matrices, several characterizations concerning eigenvalues of various functions of the pair are obtained. These results substantially extend the ones already available in the literature. Additionally, some related results dealing with the functions of a pair of orthogonal projectors are provided, with the emphasis laying on the problem of invertibility.