Constrained Least-squares Solution Research Articles

CALIBRATION OF SMARTPHONE’S REAR DUAL CAMERA SYSTEM

This paper aims to calibrate smartphone’s rear dual camera system which is composed of two lenses, namely; wide-angle lens and telephoto lens. The proposed approach handles large sized images. Calibration was done by capturing 13 photos for a chessboard pattern from different exposure positions. First, photos were captured in dual camera mode. Then, for both wide-angle and telephoto lenses, image coordinates for node points of the chessboard were extracted. Afterwards, intrinsic, extrinsic, and lens distortion parameters for each lens were calculated. In order to enhance the accuracy of the calibration model, a constrained least-squares solution was applied. The applied constraint was that the relative extrinsic parameters of both wide-angle and telephoto lenses were set as constant regardless of the exposure position. Moreover, photos were rectified in order to eliminate the effect of lens distortion. For results evaluation, two oriented photos were chosen to perform a stereo-pair intersection. Then, the node points of the chessboard pattern were used as check points.

Optimization of end‐members used in multiple linear regression geochemical mixing models

AbstractTracking marine sediment provenance (e.g., of dust, ash, hydrothermal material, etc.) provides insight into contemporary ocean processes and helps construct paleoceanographic records. In a simple system with only a few end‐members that can be easily quantified by a unique chemical or isotopic signal, chemical ratios and normative calculations can help quantify the flux of sediment from the few sources. In a more complex system (e.g., each element comes from multiple sources), more sophisticated mixing models are required. MATLAB codes published in Pisias et al. () solidified the foundation for application of a Constrained Least Squares (CLS) multiple linear regression technique that can use many elements and several end‐members in a mixing model. However, rigorous sensitivity testing to check the robustness of the CLS model is time and labor intensive. MATLAB codes provided in this paper reduce the time and labor involved and facilitate finding a robust and stable CLS model. By quickly comparing the goodness of fit between thousands of different end‐member combinations, users are able to identify trends in the results that reveal the CLS solution uniqueness and the end‐member composition precision required for a good fit. Users can also rapidly check that they have the appropriate number and type of end‐members in their model. In the end, these codes improve the user's confidence that the final CLS model(s) they select are the most reliable solutions. These advantages are demonstrated by application of the codes in two case studies of well‐studied datasets (Nazca Plate and South Pacific Gyre).

Network-based Collaborative Navigation in GPS-Denied Environment

Global Positioning System (GPS) has been used as a primary source of navigation in land and airborne applications. However, challenging environments cause GPS signal blockage or degradation, and prevent reliable and seamless positioning and navigation using GPS only. Therefore, multi-sensor based navigation systems have been developed to overcome the limitations of GPS by adding some forms of augmentation. The next step towards assured robust navigation is to combine information from multiple ground-users, to further improve the chance of obtaining reliable navigation and positioning information. Collaborative (or cooperative) navigation can improve the individual navigation solution in terms of both accuracy and coverage, and may reduce the system's design cost, as equipping all users with high performance multi-sensor positioning systems is not cost effective. Generally, ‘Collaborative Navigation’ uses inter-nodal range measurements between platforms (users) to strengthen the navigation solution. In the collaborative navigation approach, the inter-nodal distance vectors from the known or more accurate positions to the unknown locations can be established. Therefore, the collaborative navigation technique has the advantage in that errors at the user's position can be compensated by other known (or more accurate) positions of other platforms, and may result in the improvement of the navigation solutions for the entire group of users. In this paper, three statistical network-based collaborative navigation algorithms, the Restricted Least-Squares Solution (RLESS), the Stochastic Constrained Least-Squares Solution (SCLESS) and the Best Linear Minimum Partial Bias Estimation (BLIMPBE) are proposed and compared to the Kalman filter. The proposed statistical collaborative navigation algorithms for network solution show better performance than the Kalman filter.

Constrained least-squares solution and regularization in inverse boundary element analysis of photoelastic models involving contact

Existing work by the present team has involved the development of an inverse boundary element approach for fitting unknown boundary conditions to full-field stress data obtained using photoelasticity, treating the displacements on the unknown region as the solution variables. However, the recovered boundary conditions sometimes show unrealistic features such as negative contact pressures or shear tractions which exceed realistic frictional values; noisy or incomplete input data make these problems worse and can make the results oscillatory. The work described here exploits available constrained least-squares techniques in order to express the contact conditions as linear inequality constraints upon the displacement solution. A variant of Tikhonov regularization is also used to penalize against oscillatory traction distributions on the unknown boundary. The chosen implementation, solved using the LSSOL algorithm, can handle the issue of rigid-body motion without the need for manual imposition of restraints. The technique is illustrated using simulated and real examples, demonstrating that the technique correctly reconstructs the individual principal stresses.

Mapping impervious surfaces with the integrated use of Landsat Thematic Mapper and radar data: A case study in an urban–rural landscape in the Brazilian Amazon

This research explored the integrated use of Landsat Thematic Mapper (TM) and radar (i.e., ALOS PALSAR L-band and RADARSAT-2 C-band) data for mapping impervious surface distribution to examine the roles of radar data with different spatial resolutions and wavelengths. The wavelet-merging technique was used to merge TM and radar data to generate a new dataset. A constrained least-squares solution was used to unmix TM multispectral data and multisensor fusion images to four fraction images (high-albedo, low-albedo, vegetation, and soil). The impervious surface image was then extracted from the high-albedo and low-albedo fraction images. QuickBird imagery was used to develop an impervious surface image for use as reference data to evaluate the results from TM and fusion images. This research indicated that increasing spatial resolution by multisensor fusion improved spatial patterns of impervious surface distribution, but cannot significantly improve the statistical area accuracy. This research also indicated that the fusion image with 10-m spatial resolution was suitable for mapping impervious surface spatial distribution, but TM multispectral image with 30 m was too coarse in a complex urban–rural landscape. On the other hand, this research showed that no significant difference in improving impervious surface mapping performance by using either PALSAR L-band or RADARSAT C-band data with the same spatial resolution when they were used for multi-sensor fusion with the wavelet-based method.

Modeling and rendering of impossible figures

This article introduces an optimization approach for modeling and rendering impossible figures. Our solution is inspired by how modeling artists construct physical 3D models to produce a valid 2D view of an impossible figure. Given a set of 3D locally possible parts of the figure, our algorithm automatically optimizes a view-dependent 3D model, subject to the necessary 3D constraints for rendering the impossible figure at the desired novel viewpoint. A linear and constrained least-squares solution to the optimization problem is derived, thereby allowing an efficient computation and rendering new views of impossible figures at interactive rates. Once the optimized model is available, a variety of compelling rendering effects can be applied to the impossible figure.

3D Local Earthquake Hypocenter Determination with an Irregular Shortest-Path Method

A simple but robust method for joint (multiple) earthquake location in arbitrary 3D velocity media is presented in this article. It exploits the advantages of the irregular shortest-path method (irregular spm) for the raytracing, and uses a damped minimum norm, constrained least-squares solution for the source parameter (hypocenter and origin time) update. Special features of the scheme, made possible by the irregular spm, are that the raytracing needs to be carried out only once and the elements of the Jacobian matrix are calculated directly. Six supplementary sources are added around a trial (to be updated) source and trilinear interpolation used to form the travel times and the derivatives. Comparison tests against the regular spm and the finite-difference method show that the newly developed procedure is able to achieve comparable accuracy in both the travel-time calculation and in the hypocentral location solution, but at considerable computational economy. Further numerical tests for a complex 3D model indicate that the method is accurate, with a location error in depth comparable to that in the horizontal plane, at least for the station distribution used. In a practical sense, there is no requirement to have a close initial guess of the source coordinates and origin time before the location process is undertaken if the sources are not too far from the seismic network. The results show that this approach is fairly insensitive to modest levels of random noise, but like all hypocenter determination algorithms, it strongly depends on having the correct velocity model.

Discriminant image resolution: a novel multivariate image analysis method utilizing a spatial classification constraint in addition to bilinear nonnegativity

Spectroscopic image analysis can be characterized as two different and distinct problems depending on the kind of information required from the solution. In its simplest form, the data can be decomposed into two submatrices, each of which carries different aspects of pure component information. Typically, this means information about pure spectra and pure intensities are obtained from the solution. This is the well-known and well-characterized bilinear form that by itself cannot guarantee a unique solution due to the rotational ambiguity inherent in the mathematical solution. This problem has been addressed in a number of ways by different authors using novel constraints applied to the least-squares solution. This usually takes the form of natural constraints as suggested by Tauler as the standard methodology to improve the resolution of data during alternative least squares (ALS) iterative process. A second type of multivariate image analysis problem is proposed in this paper that is quite different from the tradition methods and in some ways potentially is more useful. This involves the solution as a class problem in which the relevant information is not necessarily contained in pure component information, but rather, in unique combinations of the pure components that are allowed to be spatially collocated. This discriminant image resolution (DIR) method theoretically can be treated as a more generalized solution to the problem because the distribution of components is allowed to freely mix in simplified combinations of solutions. The result is a constrained least-squares solution where the constraints are more limited and therefore less restrictive. The constraints in this case employ the results of probabilistic class partition information by applying Bayesian discriminant clustering to the intensity submatrix. This amounts to a spatial constraint because the probability of class association is used as a way of limiting the components that are allowed to appear in a given pixel. This is a unique modification of the original modified alternating least squares (MALS) concept and to the authors' knowledge, the first time this type of constraint has been combined with an ALS algorithm for analysis of multivariate or hyperspectral data. Modified alternating least squares (MALS) is used as the computational engine to drive good convergence while imposing the constraints in this modified ALS model. This kind of analysis produces single and simple multicomponent images and spectra based on classification from spectral information. The present constraint is most useful in resolving image data when the true images are not severely overlapped but it will also perform well under conditions of more severe collinearity.

Linear mixture model applied to Amazonian vegetation classification

Many research projects require accurate delineation of different secondary succession (SS) stages over large regions/subregions of the Amazon basin. However, the complexity of vegetation stand structure, abundant vegetation species, and the smooth transition between different SS stages make vegetation classification difficult when using traditional approaches such as the maximum likelihood classifier (MLC). Most of the time, classification distinguishes only between forest/non-forest. It has been difficult to accurately distinguish stages of SS. In this paper, a linear mixture model (LMM) approach is applied to classify successional and mature forests using Thematic Mapper (TM) imagery in the Rondônia region of the Brazilian Amazon. Three endmembers (i.e., shade, soil, and green vegetation or GV) were identified based on the image itself and a constrained least-squares solution was used to unmix the image. This study indicates that the LMM approach is a promising method for distinguishing successional and mature forests in the Amazon basin using TM data. It improved vegetation classification accuracy over that of the MLC. Initial, intermediate, and advanced successional and mature forests were classified with overall accuracy of 78.2% using a threshold method on the ratio of shade to GV fractions, a 7.4% increase over the MLC. The GV and shade fractions are sensitive to the change of vegetation stand structures and better capture biophysical structure information.

Adaptive regularized constrained least squares image restoration

In noisy environments, a constrained least-squares (CLS) approach is presented to restore images blurred by a Gaussian impulse response, where instead of choosing a global regularization parameter, each point in the signal has its own associated regularization parameter. These parameters are found by constraining the weighted standard deviation of the wavelet transform coefficients on the finest scale of the inverse signal by a function r which is a local measure of the intensity variations around each point of the blurred and noisy observed signal. Border ringing in the inverse solution is proposed decreased by manipulating its wavelet transform coefficients on the finest scales close to the borders. If the noise in the inverse solution is significant, wavelet transform techniques are also applied to denoise the solution. Examples are given for images, and the results are shown to outperform the optimum constrained least-squares solution using a global regularization parameter, both visually and in the mean squared error sense.

Model-based image reconstruction by means of a constrained least-squares solution

The problem of the optimal use of object model information in image reconstruction is addressed. A closed-form solution for the estimated object spectrum is derived with the Lagrange multiplier technique, which assumes a measured image, knowledge of the optical transfer function, statistical information about the measurement noise, and a model of the object. This reconstruction algorithm is iterative in nature because the optimal Lagrange multiplier is not generally known at the start of the problem. We derive the estimator, describe one technique for determining the optimal Lagrange multiplier, demonstrate a stopping criterion based on the mean-square error between a noise-free image and the photon-limited version of the image, and show representative results for both filled- and sparse-aperture imaging applications.

Restoration of bilinearly distorted images 3: Constrained least-squares method

A constrained least-squares solution based on a second difference smoothing measure is obtained for images which are distorted by a bilinear (quadratic and with nonzero memory) system in the presence of additive signal-independent noise. Results are applied to a partially coherent diffraction-limited imaging system. It is found that the optimum weight given to the smoothness factor is larger for a coherent (nonlinear) system than for an incoherent (linear) system. However, the restoration quality improves as the imaging system approaches incoherence.