Yusuke Kameda

Variational method for super-resolution optical flow

The motion fields in an image sequence observed by a car-mounted imaging system depend on the positions in the imaging plane. Since the motion displacements in the regions close to the camera centre are small, for accurate optical flow computation in this region, we are required to use super-resolution of optical flow fields. We develop an algorithm for super-resolution optical flow computation. Super-resolution of images is a technique for recovering a high-resolution image from a low-resolution image and/or image sequence. Optical flow is the appearance motion of points on the image. Therefore, super-resolution optical flow computation yields the appearance motion of each point on the high-resolution image from a sequence of low-resolution images. We combine variational super-resolution and variational optical flow computation in super-resolution optical flow computation. Our method directly computes the gradient and spatial difference of high-resolution images from those of low-resolution images, without computing any high-resolution images used as intermediate data for the computation of optical flow vectors of the high-resolution image.

A convergence proof for the Horn-Schunck optical-flow computation scheme using neighborhood decomposition

In this paper, we prove the convergence property of the Horn-Schunck optical-flow computation scheme. Horn and Schunck derived a Jacobi-method-based scheme for the computation of optical-flow vectors of each point of an image from a pair of successive digitised images. The basic idea of the Horn-Schunck scheme is to separate the numerical operation into two steps: the computation of the average flow vector in the neighborhood of each point and the refinement of the optical flow vector by the residual of the average flow vectors in the neighborhood. Mitiche and Mansouri proved the convergence property of the Gauss-Seidel- and Jacobi-method-based schemes for the Horn-Schunck-type minimization using algebraic properties of the matrix expression of the scheme and some mathematical assumptions on the system matrix of the problem. In this paper, we derive an alternative proof for the original Horn-Schunck scheme. To prove the convergence property, we develop a method of expressing shift-invariant local operations for digital planar images in the matrix forms. These matrix expressions introduce the norm of the neighborhood operations. The norms of the neighborhood operations allow us to prove the convergence properties of iterative image processing procedures.

Continuity order of local displacement in volumetric image sequence

We introduce a method for volumetric cardiac motion analysis using variational optical flow computation involving the prior with the fractional order differentiations. The order of the differentiation of the prior controls the continuity class of the solution. Fractional differentiations is a typical tool for edge detection of images. As a sequel of image analysis by fractional differentiation, we apply the theory of fractional differentiation to a temporal image sequence analysis. Using the fractional order differentiations, we can estimate the orders of local continuities of optical flow vectors. Therefore, we can obtain the optical flow vector with the optimal continuity at each point.

Classification of optical flow by constraints

In this paper, we analyse mathematical properties of spatial optical-flow computation algorithm. First by numerical analysis, we derive the convergence property on variational optical-flow computation method used for cardiac motion detection. From the convergence property of the algorithm, we clarify the condition for the scheduling of the regularisation parameters. This condition shows that for the accurate and stable computation with scheduling the regularisation coefficients, we are required to control the sampling interval for numerical computation.

Computing the Local Continuity Order of Optical Flow Using Fractional Variational Method

We introduce variational optical flow computation involving priors with fractional order differentiations. Fractional order differentiations are typical tools in signal processing and image analysis. The zero-crossing of a fractional order Laplacian yields better performance for edge detection than the zero-crossing of the usual Laplacian. The order of the differentiation of the prior controls the continuity class of the solution. Therefore, using the square norm of the fractional order differentiation of optical flow field as the prior, we develop a method to estimate the local continuity order of the optical flow field at each point. The method detects the optimal continuity order of optical flow and corresponding optical flow vector at each point. Numerical results show that the Horn-Schunck type prior involving the n + *** order differentiation for 0 < *** < 1 and an integer n is suitable for accurate optical flow computation.

Decomposition and construction of neighbourhood operations using linear algebra

In this paper, we introduce a method to express a local linear operated in the neighbourhood of each point in the discrete space as a matrix transform. To derive matrix expressions, we develop a decomposition and construction method of the neighbourhood operations using algebraic properties of the noncommutative matrix ring. This expression of the transforms in image analysis clarifies analytical properties, such as the norm of the transforms. We show that the symmetry kernels for the neighbourhood operations have the symmetry matrix expressions.

The William Harvey Code: Mathematical Analysis of Optical Flow Computation for Cardiac Motion

For the non-invasive imaging of moving organs, in this chapter, we investigate the generalisation of optical flow in three-dimensional Euclidean space. In computer vision, optical flow is dealt with as a local motion of pixels in a pair of successive images in a sequence of images. In a space, optical flow is defined as the local motion of the voxel of spatial distributions, such as x-ray intensity and proton distributions in living organs. Optical flow is used in motion analysis of beating hearts measured by dynamic cone beam x-ray CT and gated MRI tomography. This generalisation of optical flow defines a class of new constraints for optical-flow computation. We first develop a numerically stable optical-flow computation algorithm. The accuracy of the solution of this algorithm is guaranteed by Lax equivalence theorem which is the basis of the numerical computation of the solution for partial differential equations. Secondly, we examine numerically the effects of the divergence-free condition, which is required from linear approximation of infinitesimal deformation, for the computation of cardiac optical flow from images measured by gated MRI. Furthermore, we investigate the relation between the vector-spline constraint and the thin plate constraint. Moreover, we theoretically examine the validity of the error measure for the evaluation of computed optical flow.

Hierarchical properties of multi-resolution optical flow computation

Most of the methods to compute optical flows are variational-technique-based methods, which assume that image functions have spatiotemporal continuities and appearance motions are small. In the viewpoint of the discrete errors of spatial- and time-differentials, the appropriate resolution for optical flow depends on both the resolution and the frame rate of images since there is a problem with the accuracy of the discrete approximations of derivatives. Therefore, for low frame-rate images, the appropriate resolution for optical flow should be lower than the resolution of the images. However, many traditional methods estimate optical flow with the same resolution as the images. Therefore, if the resolution of images is too high, down-sampling the images is effective for the variational-technique-based methods. In this paper, we analyze the appropriate resolutions for optical flows estimated by variational optical-flow computations from the viewpoint of the error analysis of optical flows. To analyze the appropriate resolutions, we use hierarchical structures constructed from the multi-resolutions of images. Numerical results show that decreasing image resolutions is effective for computing optical flows by variational optical-flow computations in low frame-rate sequences.

An Iterative Method for Superresolution of Optical Flow Derived by Energy Minimisation

Super resolution is a technique to recover a high resolution image from a low resolution image. We develop a variational super resolution method for the subpixel accurate optical flow computation using variational optimisation. We combine variational super resolution and the variational optical flow computation for the super resolution optical flow computation.

Two Step Variational Method for Subpixel Optical Flow Computation

We develop an algorithm for the super-resolution optical flow computation by combining variational super-resolution and the variational optical flow computation. Our method first computes the gradient and the spatial difference of a high resolution images from these of low resolution images directly, without computing any high resolution images. Second the algorithm computes optical flow of high resolution image using the results of the first step.