Short kernel fifth-order interpolation

An explicit analytical formula for a short kernel fifth-order polynomial interpolator is obtained. It is also possible to obtain the explicit forms of even higher order interpolation kernels with the method of calculation used, but it is seen that these local kernels become “remainder-dominated” as the order increases. The frequency domain properties and the accuracies of the obtained kernel and the known convolution kernels are compared. Frequency domain comparison with the cubic B-spline interpolators is also given. Some cases of proper use of the calculated kernels have been pointed out     

Addendum: B-spline interpolation in medical image processing

This paper analyzes B-spline interpolation techniques of degree 2, 4, and 5 with respect to all criteria that have been applied to evaluate various interpolation schemes in a recently published survey on image interpolation in medical imaging (Lehmann et al., 1999). It is shown that high-degree B-spline interpolation has superior Fourier properties, smallest interpolation error, and reasonable computing times. Therefore, high-degree B-splines are preferable interpolators for numerous applications in medical image processing, particularly if high precision is required. If no aliasing occurs, this result neither depends on the geometric transform applied for the tests nor the actual content of images.     

Interpolation artifacts in multimodality image registration based on maximization of mutual information

Mutual information (MI) is an increasingly popular match metric for multimodality image registration. However, its value is affected by interpolation, which may limit registration accuracy. The purpose of this study was to characterize the artifacts from eight interpolators and to investigate efficient strategies to overcome these artifacts. The interpolators were: 1) nearest neighbor; 2) linear; 3) cubic Catmull-Rom; 4) Hamming-windowed sinc; 5) partial volume; 6) NN with jittered sampling (JIT); 7) NN with histogram blurring (BLUR); and 8) NN with JIT and BLUR. The impact of interpolation on MI was evaluated in two dimensions over different translational and rotational misregistration. Interpolation caused spurious fluctuations in MI whenever the voxel grids had coinciding periodicities and were nearly aligned. The artifacts did not lessen by using intensity interpolators with wider support (e.g., cubic Catmull-Rom, Hamming-windowed sinc). PV could lead to either arch artifacts or inverted-arch artifacts, depending on the relative voxel sizes. Several strategies reduced artifacts and improved registration robustness: JIT, BLUR, avoiding an extreme number of intensity bins, and resampling the images in a rotated orientation with different relative voxel sizes (e.g., pi/3). These findings also apply to related methods, including normalized MI, joint entropy, and Hill's third moment.     

Analysis and design of minimax-optimal interpolators

We consider a class of interpolation algorithms, including the least-squares optimal Yen (1956) interpolator, and we derive a closed-form expression for the interpolation error for interpolators of this type. The error depends on the eigenvalue distribution of a matrix that is specified for each set of sampling points. The error expression can be used to prove that the Yen interpolator is optimal. The implementation of the Yen algorithm suffers from numerical ill conditioning, forcing the use of a regularized, approximate solution. We suggest a new, approximate solution consisting of a sinc-kernel interpolator with specially chosen weighting coefficients. The newly designed sinc-kernel interpolator is compared with the usual sinc interpolator using Jacobian (area) weighting through numerical simulations. We show that the sinc interpolator with Jacobian weighting works well only when the sampling is nearly uniform. The newly designed sinc-kernel interpolator is shown to perform better than the sinc interpolator with Jacobian weighting     

Frequency-domain weighted-least-squares design of quadratic interpolators

Existing finite-support interpolators are derived from continuities in the time-domain. In this study, the authors optimally design a quadratic interpolator using two second-degree piecewise polynomials in the frequency-domain. The optimal coefficients of the piecewise polynomials are found by minimising the weighted least-squares error between the ideal and actual frequency responses of the quadratic interpolator subject to a few constraints. Adjusting the weighting functions in different frequency bands can yield accurate frequency response in a specified passband and even can ignore `don`t care` bands so that various quadratic interpolators can be designed for interpolating various discrete signals containing different frequency components. One-dimensional and two-dimensional examples have shown that the quadratic interpolator can achieve much higher interpolation accuracy than the existing interpolators for wide-band signals, and various images have been tested to verify that the quadratic interpolator can achieve comparable interpolation accuracy as the Catmull-Rom cubic for narrow-band signals (images), but the computational complexity can be reduced to about 70%. Therefore both narrow-band and wide-band signals can be interpolated with high accuracy.     

Image interpolation by two-dimensional parametric cubic convolution.

Cubic convolution is a popular method for image interpolation. Traditionally, the piecewise-cubic kernel has been derived in one dimension with one parameter and applied to two-dimensional (2-D) images in a separable fashion. However, images typically are statistically nonseparable, which motivates this investigation of nonseparable cubic convolution. This paper derives two new nonseparable, 2-D cubic-convolution kernels. The first kernel, with three parameters (designated 2D-3PCC), is the most general 2-D, piecewise-cubic interpolator defined on [-2, 2] x [-2, 2] with constraints for biaxial symmetry, diagonal (or 90 degrees rotational) symmetry, continuity, and smoothness. The second kernel, with five parameters (designated 2D-5PCC), relaxes the constraint of diagonal symmetry, based on the observation that many images have rotationally asymmetric statistical properties. This paper also develops a closed-form solution for determining the optimal parameter values for parametric cubic-convolution kernels with respect to ensembles of scenes characterized by autocorrelation (or power spectrum). This solution establishes a practical foundation for adaptive interpolation based on local autocorrelation estimates. Quantitative fidelity analyses and visual experiments indicate that these new methods can outperform several popular interpolation methods. An analysis of the error budgets for reconstruction error associated with blurring and aliasing illustrates that the methods improve interpolation fidelity for images with aliased components. For images with little or no aliasing, the methods yield results similar to other popular methods. Both 2D-3PCC and 2D-5PCC are low-order polynomials with small spatial support and so are easy to implement and efficient to apply.     

High-accuracy interpolator design with reduced first-order derivative constraints

In this article, we propose a new frequency-domain weighted-least-squares method for designing high-accuracy interpolator (interpolation kernel) that reduces the number of the first-order derivative continuity constraints. By reducing the number of the first-order derivative constraints, we can increase the degree of freedom in the design, and thus have more flexibility to get more accurate design results. The interpolator consists of 6 piecewise polynomials of the third degree (cubic), and it is performed in the frequency-domain through minimising the weighted integrated-squared-error of the spectrum (frequency response). The weighting function is adjusted so as to ignore some insignificant frequency bands and put more emphasis on the important frequency bands. By imposing the continuity constraints on the interpolator itself as well as the reduced first-order derivative constraints at the contacting points, we get three free parameters of the interpolator. These three parameters are then optimised in such a way that the weighted integrated-squared-error of the frequency response is minimised. We will utilise a narrow-band example to demonstrate the performance improvement over other existing interpolators.     

Efficient implementation of accurate geometric transformations for 2-D and 3-D image processing

This paper proposes the use of a polynomial interpolator structure (based on Horner's scheme) which is efficiently realizable in hardware, for high-quality geometric transformation of two- and three-dimensional images. Polynomial-based interpolators such as cubic B-splines and optimal interpolators of shortest support are shown to be exactly implementable in the Horner structure framework. This structure suggests a hardware/software partition which can lead to efficient implementations for multidimensional interpolation.     

Block-level parallel processing for scaling evenly divisible images

Image scaling is a frequent operation in medical image processing. This paper presents how two-dimensional (2-D) image scaling can be accelerated with a new coarse-grained parallel processing method. The method is based on evenly divisible image sizes which is, in practice, the case with most medical images. In the proposed method, the image is divided into slices and all the slices are scaled in parallel. The complexity of the method is examined with two parallel architectures while considering memory consumption and data throughput. Several scaling functions can be handled with these generic architectures including linear, cubic B-spline, cubic, Lagrange, Gaussian, and sinc interpolations. Parallelism can be adjusted independent of the complexity of the computational units. The most promising architecture is implemented as a simulation model and the hardware resources as well as the performance are evaluated. All the significant resources are shown to be linearly proportional to the parallelization factor. With contemporary programmable logic, real-time scaling is achievable with large resolution 2-D images and a good quality interpolation. The proposed block-level scaling is also shown to increase software scaling performance over four times.     

Fourier-based reconstruction for fully 3-D PET: optimization of interpolation parameters

Fourier-based approaches for three-dimensional (3-D) reconstruction are based on the relationship between the 3-D Fourier transform (FT) of the volume and the two-dimensional (2-D) FT of a parallel-ray projection of the volume. The critical step in the Fourier-based methods is the estimation of the samples of the 3-D transform of the image from the samples of the 2-D transforms of the projections on the planes through the origin of Fourier space, and vice versa for forward-projection (reprojection). The Fourier-based approaches have the potential for very fast reconstruction, but their straightforward implementation might lead to unsatisfactory results if careful attention is not paid to interpolation and weighting functions. In our previous work, we have investigated optimal interpolation parameters for the Fourier-based forward and back-projectors for iterative image reconstruction. The optimized interpolation kernels were shown to provide excellent quality comparable to the ideal sinc interpolator. This work presents an optimization of interpolation parameters of the 3-D direct Fourier method with Fourier reprojection (3D-FRP) for fully 3-D positron emission tomography (PET) data with incomplete oblique projections. The reprojection step is needed for the estimation (from an initial image) of the missing portions of the oblique data. In the 3D-FRP implementation, we use the gridding interpolation strategy, combined with proper weighting approaches in the transform and image domains. We have found that while the 3-D reprojection step requires similar optimal interpolation parameters as found in our previous studies on Fourier-based iterative approaches, the optimal interpolation parameters for the main 3D-FRP reconstruction stage are quite different. Our experimental results confirm that for the optimal interpolation parameters a very good image accuracy can be achieved even without any extra spectral oversampling, which is a common practice to decrease errors caused by interpolation in Fourier reconstruction.