In this paper we review 30 years of developments and applications of the variable projection method for solving separable nonlinear least-squares problems. These are problems for which the model function is a linear combination of nonlinear functions. Taking advantage of this special structure, the method of variable projections eliminates the linear variables obtaining a somewhat more complicated function that involves only the nonlinear parameters. This procedure not only reduces the dimension of the parameter space but also results in a better-conditioned problem. The same optimization method applied to the original and reduced problems will always converge faster for the latter. We present first a historical account of the basic theoretical work and its various computer implementations, and then report on a variety of applications from electrical engineering, medical and biological imaging, chemistry, robotics, vision, and environmental sciences. An extensive bibliography is included. The method is particularly well suited for solving real and complex exponential model fitting problems, which are pervasive in their applications and are notoriously hard to solve.

https://iopscience.iop.org/article/10.1088/0266-5611/19/2/201/pdf

Separable nonlinear least squares: the variable projection method and its applications

This paper discusses a proposed processing technique for combining video imagery with auxiliary sensor information. The latter greatly simplifies image processing by reducing complexity of the transformation model. The mosaics produced by this technique are adequate for many applications, in particular habitat mapping. The algorithm is demonstrated through simulations and hardware configuration is described.

/pdf/sensor-assisted-video-mosaicing-for-seafloor-mapping-2sniip24bs.pdf

Sensor-assisted video mosaicing for seafloor mapping

Given a set of synchronized video sequences of a dynamic scene taken by different cameras, we address the problem of creating a virtual video of the scene from a novel viewpoint. A key aspect of our algorithm is a method for recursively propagating dense and physically accurate correspondences between the two video sources. By exploiting temporal continuity and suitably constraining the correspondences, we provide an efficient framework for synthesizing realistic virtual video. The stability of the propagation algorithm is analyzed, and experimental results are presented.

/pdf/efficiently-synthesizing-virtual-video-109fa591z4.pdf

Efficiently synthesizing virtual video

This paper deals with a new registration method based on a specific level-line grouping. Because of its contrast-change invariance, our approach is an appropriate method for matching outdoor image sequences. Moreover, it does not require any estimation of the unknown transformation between images and handle well the critical cases that usually lead to pairing ambiguities, such as repetitive patterns in the images. This study focuses on invariants primitive construction under projective transformation, using level-lines. The registration by itself is performed through an efficient level-line cumulative matching based on a multi-stage primitive election procedure. Each stage provides a coarse estimate of the transformation that the next stage gets to refine. Experiments on real outdoor road scene show the accuracy and efficiency of this approach, using several image sequences covering different pertaining cases with different type of motion.

Matching Images Using Invariant Level-line Primitives under Projective Transformation

This paper addresses several estimation problems involving correspondence in digital video. We present three cases, in order of increasing complexity: affine transformations, projective transformations, and general correspondence. We review some recent research in the first two cases and present a new framework for discussing the third case. We also discuss how to use the formalism of correspondence graphs to ensure that any estimated correspondence is physically valid.

/pdf/estimating-correspondence-in-digital-video-27qa1amy0w.pdf

Estimating correspondence in digital video

The estimation of the parameters of a projective transformation that relates the coordinates of two image planes is a standard problem that arises in image and video mosaicking, virtual video, and problems in computer vision This problem is often posed as a least squares minimization problem based on a finite set of noisy point samples of the underlying transformation While in some special cases this problem can be solved using a linear approximation, in general, it results in an 8-dimensional nonquadratic minimization problem that is solved numerically using an 'off-the-shelf' procedure such as the Levenberg-Marquardt algorithm We show that the general least squares problem for estimating a projective transformation can be analytically reduced to a 2-dimensional nonquadratic minimization problem Moreover, we provide both analytical and experimental evidence that the minimization of this function is computationally attractive We propose a particular algorithm that is a combination of a projection and an approximate Gauss-Newton scheme, and experimentally verify that this algorithm efficiently solves the least squares problem

/pdf/efficiently-estimating-projective-transformations-1b340yeaz0.pdf

S. Iisaku

Papers

Efficiently estimating projective transformations