Linear Spline Interpolation Research Articles

Real-time 4D signal processing and visualization using graphics processing unit on a regular nonlinear-k Fourier-domain OCT system

We realized graphics processing unit (GPU) based real-time 4D (3D + time) signal processing and visualization on a regular Fourier-domain optical coherence tomography (FD-OCT) system with a nonlinear k-space spectrometer. An ultra-high speed linear spline interpolation (LSI) method for λ-to-k spectral re-sampling is implemented in the GPU architecture, which gives average interpolation speeds of >3,000,000 line/s for 1024-pixel OCT (1024-OCT) and >1,400,000 line/s for 2048-pixel OCT (2048-OCT). The complete FD-OCT signal processing including λ-to-k spectral re-sampling, fast Fourier transform (FFT) and post-FFT processing have all been implemented on a GPU. The maximum complete A-scan processing speeds are investigated to be 680,000 line/s for 1024-OCT and 320,000 line/s for 2048-OCT, which correspond to 1GByte processing bandwidth. In our experiment, a 2048-pixel CMOS camera running up to 70 kHz is used as an acquisition device. Therefore the actual imaging speed is camera-limited to 128,000 line/s for 1024-OCT or 70,000 line/s for 2048-OCT. 3D Data sets are continuously acquired in real time at 1024-OCT mode, immediately processed and visualized as high as 10 volumes/second (12,500 A-scans/volume) by either en face slice extraction or ray-casting based volume rendering from 3D texture mapped in graphics memory. For standard FD-OCT systems, a GPU is the only additional hardware needed to realize this improvement and no optical modification is needed. This technique is highly cost-effective and can be easily integrated into most ultrahigh speed FD-OCT systems to overcome the 3D data processing and visualization bottlenecks.

Experimental validation of an optimized signal processing method to handle non-linearity in swept-source optical coherence tomography

We evaluate various signal processing methods to handle the non-linearity in wavenumber space exhibited by most laser sources for swept-source optical coherence tomography. The following methods are compared for the same set of experimental data: non-uniform discrete Fourier transforms with Vandermonde matrix or with Lomb periodogram, resampling with linear interpolation or spline interpolation prior to fast-Fourier transform (FFT), and resampling with convolution prior to FFT. By selecting an optimized Kaiser-Bessel window to perform the convolution, we show that convolution followed by FFT is the most efficient method. It allows small fractional oversampling factor between 1 and 2, thus a minimal computational time, while retaining an excellent image quality.

Comparison of different mass transport calculation methods for wind erosion quantification purposes

AbstractQuantitative estimation of the material transported by the wind under field conditions is essential for the study and control of wind erosion. A critical step of this calculation is the integration of the curve that relates the variation of the amount of the material carried by the wind with height. Several mathematical procedures have been proposed for this calculation, but results are scarce and controversial. One objective of this study was to assess the efficiency of three mathematical models (a rational, an exponential, and a simplified Gaussian function) for the calculation of the mass transport, as compared to the linear spline interpolation. Another objective of this study was to compare the mass transport calculated from field measurements obtained from a minimum of three discrete sampling heights with measurements of nine sampling heights. With this purpose, wind erosion was measured under low surface roughness conditions on an Entic Haplustoll during 25 events. The rational function was found to be mathematically limited for the estimation of wind eroded sediment mass flux. The simplified Gaussian model did not fit to the vertical mass flux profile data. Linear spline interpolation generally produced higher mass transport estimates than the exponential equation, and it proved to be a very flexible and robust method. Using different sampling arrangements and different mass flux models can produce differences of more than 45% in mass transport estimates, even under similar field conditions. Under the conditions of this study, at least three points between the soil surface and 1·5 m high, including one point as closest as possible to the surface, should be sampled in order to obtain accurate mass transport estimates. Additionally, the linear spline interpolation and the non‐linear regression using an exponential model, proved to be mathematically reliable methods for calculating the mass transport. Copyright © 2010 John Wiley & Sons, Ltd.

Quadrature formula for approximating the singular integral of Cauchy type with unbounded weight function on the edges

New quadrature formulas (QFs) for evaluating the singular integral (SI) of Cauchy type with unbounded weight function on the edges is constructed. The construction of the QFs is based on the modification of discrete vortices method (MMDV) and linear spline interpolation over the finite interval [ − 1 , 1 ] . It is proved that the constructed QFs converge for any singular point x not coinciding with the end points of the interval [ − 1 , 1 ] . Numerical results are given to validate the accuracy of the QFs. The error bounds are found to be of order O ( h α | ln h | ) and O ( h | ln h | ) in the classes of functions H α ( [ − 1 , 1 ] ) and C 1 ( [ − 1 , 1 ] ) , respectively.

Lumbar muscle electromyographic dynamic topography during flexion-extension

The objective of this study is to introduce dynamic topography of surface electromyography (SEMG) to visualize lumbar muscle myoelectric activity and provides a new view to analyze muscle activity in vivo. A total of 20 healthy male subjects and 15 males LBP were enrolled. An electrode-array was applied to the lumbar region to collect SEMG. The root mean square (RMS) value was calculated for each channel, and then a 160×120 matrix was constructed using a linear cubic spline interpolation of each scan to create a 2-D color topographic image. Along a definite interval of action, a series of RMS topography matrices was concatenated as a function of position and time, to form a dynamic topographical video of lumbar muscle activity. Relative area (RA), relative width (RW), relative height (RH) and Width-to-Height Ratio ( W/ H) were chosen as the four quantitative parameters in measuring topographic features. Normal RMS dynamic topography was found to have a consistent, symmetric pattern with a high intensity area in the paraspinal area. LBP patients had a different RMS dynamic topography, with an asymmetric, broad, or disorganized distribution. Quantitative SEMG features were found significantly different between normal control and LBP. After physiotherapy rehabilitation, the dynamic topography images of LBP tended towards the normal pattern. There are obvious differences in lumbar muscle coordination between healthy subjects and LBP patients. The dynamic topography allows the continuous visualization of the distribution of surface EMG signals and the coordination of muscular contractions.

Energy Absorption Buildup Factors and Energy Conservation

The ANSI/ANS energy absorption buildup factors (ANSI/ANS-6.4.3–1991) were used to determine if they conserve absorbed energy from a point source of monoenergetic photons in an infinite, absorbing medium. Air, water, concrete, iron, lead, and uranium were considered for photon energies ranging from 0.015 to 15 MeV. The primary calculations were based on fitting natural cubic splines to the energy absorption buildup factors and analytical integration of the energy absorbed using the cubic spline curve fit. This method was supplemented by linear spline interpolation. For air, water, concrete, and iron, the ANSI/ANS energy absorption buildup factors were determined using the method of moments with theenergy attenuation and deposition coefficients of Hubbell (NSRDS-NBS 29). The results of this study essentially confirmed energy conservation for these four materials using the energy attenuation and deposition coefficients of Hubbell. For lead and uranium, the ANSI/ANS energy absorption buildup factors were determined using the discrete ordinates-integral transport method with PHOTX energy attenuation and deposition coefficients that are essentially the same as those in the ANSI/ANS report. The calculations in this study could not confirm energy conservation for these two materials.

Energy Absorption Buildup Factors and Energy Conservation

The ANSI/ANS energy absorption buildup factors (ANSI/ANS-6.4.3–1991) were used to determine if they conserve absorbed energy from a point source of monoenergetic photons in an infinite, absorbing medium. Air, water, concrete, iron, lead, and uranium were considered for photon energies ranging from 0.015 to 15 MeV. The primary calculations were based on fitting natural cubic splines to the energy absorption buildup factors and analytical integration of the energy absorbed using the cubic spline curve fit. This method was supplemented by linear spline interpolation. For air, water, concrete, and iron, the ANSI/ANS energy absorption buildup factors were determined using the method of moments with theenergy attenuation and deposition coefficients of Hubbell (NSRDS-NBS 29). The results of this study essentially confirmed energy conservation for these four materials using the energy attenuation and deposition coefficients of Hubbell. For lead and uranium, the ANSI/ANS energy absorption buildup factors were determined using the discrete ordinates-integral transport method with PHOTX energy attenuation and deposition coefficients that are essentially the same as those in the ANSI/ANS report. The calculations in this study could not confirm energy conservation for these two materials.

Refined-mesh interpolation method for functions with a boundary-layer component

Linear and quadratic spline interpolation methods for a one-variable function with a boundary-layer component are examined. It is shown that the interpolation method for such a function leads to considerable errors when applied on a uniform mesh. The error of linear and quadratic spline interpolations on meshes that are refined in the boundary layer is estimated. Numerical results are presented.

Velocity structure of the lithosphere on the 2003 Mongolian-Baikal transect from SV waves

The S wave velocity distribution in the Earth’s crust and the first two hundred kilometers of the upper mantle is inferred from data of a seismological linear network including 18 broadband stations installed in the framework of the international teleseismic experiment carried out in 2003 in the south of Siberia and in Mongolia. Models were constructed by using P-to-S received function inversion beneath each station. Vertical cross sections of S wave velocities from the surface to depths of 65 and 270 km covering the entire 1000-km profile are constructed by the linear spline interpolation of individual velocity models. The vertical sections are also represented as anomalies relative to the standard velocity model. The most intense low velocity anomalies (from −3 to −6%) in the crust and upper mantle are identified beneath the Sayan, Khamar-Daban, and Khangai highlands and the Djida fold zone and agree both with other geophysical data and with the distribution of Late Cenozoic volcanic fields. The results of this work suggest that the activation of Mongolian-Siberian highlands is largely connected with uplift of the asthenosphere to the base of the crust.

Wavelet-based interpolation algorithm for MRI images

Three-dimensional (3D) visualization is a common tool that is used to make a correct diagnostic in medicine. However, in most medical data acquisition systems the information is obtained in a two-dimensional (2D) form. In this paper, an interpolation algorithm that works with a set of 2D images to create a reliable 3D model of a desired part of the body is presented. In most cases, the popular interpolation methods produce noticeable smoothing in the data generated that may cause the loss of important details. It is also important to perform noise reduction from the images that will serve as reference points to the interpolation. In order to deal with these two problems, we propose the use of the wavelet transform. One of the principal characteristics of the wavelet transform is the space–frequency decomposition that can be used to differentiate between the high detail areas and the smooth ones. The proposed approach then decompose the reference images into desired wavelet level to perform the analysis. Afterwards, the approximation coefficients are used to define the location of the interpolation area. The wavelet coefficients related to high frequency (vertical, horizontal, and diagonal) are used to locate the detail areas mentioned above. It is important to note that this is achieved by the use of a dynamic thresholding method that varies with the wavelet level and type of coefficients. Finally, we apply a linear or cubic spline interpolation depending on the number of reference images.

Image reconstruction of steel reinforcing bars in concrete using Fourier-domain interpolation applied to a sparsely populated data set

High-resolution image generation of bars in concrete using a single coil scanning inductive sensor is a time consuming process. This paper presents a method of generating the image using a sparsely populated data set (SPDS), obtained from a reduced number of scan lines, whose use is justifies by an analysis of the sensor spatial frequency response. Three methods are discussed: linear and cubic spline interpolation in the spatial domain, and Fourier interpolation, all of which are applied to an SPDS. Data for the SPDS are provided from a widely spaced scanning regime implemented in both the x and y-axes. Interpolated values are then synthesized to obtain high-resolution images. The Fourier-based method employs zero padding in the spatial frequency domain and inverse Fourier transformation to obtain higher resolution data in the time/spatial domain. In general, the results obtained by the linear interpolation algorithm are unacceptable since they do not represent the point-spread function of the sensor. The results obtained by the cubic spline and Fourier methods are very satisfactory, with the deviation from the results obtained by the standard high-resolution image generation process being very small. However, the cubic spline method is cumbersome to implement, requiring the computation of a large number of unique polynomials. In contrast, the Fourier algorithm is efficient, straightforward to code and yields an ideal band-limited interpolation. Experiments show that this new methodology is faster than the traditional scanning protocol by at least a factor of 10; a large area scan of 0.5 m2 can thus be produced in 12 min, rather than 2 h. In the future, this technique could be applied to widely spaced solid-state arrays, requiring a fraction of a second for image synthesis.

Sculpting a sound space with information properties

The design and construction of an organised sound space to support information representations in the human–computer interface is described. The design of the sound space is guided by four principles which match perceptual structure with data structure to improve natural comprehension of an auditory display. These principles – completeness, comprehensibility, consistency, and cohesiveness – have been generalised from the use of colour displays in scientific visualisation. The choice of perceptual parameters to represent different types of data is informed by the body of psychoacoustic literature. The raw material for the construction of the sound space is the McGill University Master Samples (MUMS) palette of musical instrument samples. This is an important choice because this reference resource enables reproduction and confirmation of the results. The construction was carried out in four stages – the ‘pedestal’, the ‘skin’, the ‘skeleton’ and the ‘flesh’. The pedestal consists of eight equally discriminable timbres organised in a circle by perceptual similarity. The skin is the boundary of variation in the space, defining the limits of dynamic range for pitch and brightness at each timbre. The skeleton characterises the internal behaviour of the space at a number of perceptually measured points. The flesh is a continuous medium moulded to the skeleton and skin, realised by a three-dimensional (3D) regularised linear spline interpolation. The concrete realisation of the sound space can be investigated through a user interface, called the GamutExplorer. Colour visualisations of slices and wireframe views of the 3D space can be chosen, and sounds can be picked with a mouse.

Monotone linear rational spline interpolation

A method for interpolating a monotone data sequence (values and derivatives) with a C 1 monotone rational B-spline of degree 1 is presented, along with some test results. This method is tailor-made for reparameterizing B-spline curves in Geometric Modeling.

Positional accuracy of isodose lines as a function of dose matrix resolution

The authors calculated the accuracy of isodose lines for a variety of clinical treatment techniques and beam energies using linear, quadratic and cubic spline interpolation methods. For a single beam, the greatest positional errors occur on either side of the steepest part of the penumbra (Niemierko and Goitein 1989). This is the region where the spatial second derivative of the dose is greatest. For plans with multiple beams, overall accuracy of isodose lines depends on the dose pattern as well as on the size of the geometric penumbra of the individual beams. Since computing the dose at a point is much faster using interpolation than physical simulation, the authors considered whether dose matrix calculation could be speeded by using higher order interpolation with fewer grid points for the same overall positional accuracy.

Linear and quadratic spline interpolation at knot averages

For linear and quadratic spline interpolation at nodes which are weighted avearges of the spline knots, conditions on the weights are developed which guarantee that the corresponding spline interpolation operators are norm-bounded independently of the knot locations.

Sampling for Spline Reconstruction

The use of a prefilter prior to sampling or resampling can lead to reduced mean square error after reconstruction. This means that the samples should be obtained from the original data by convolution with an optimal local weighting function, whose form depends upon the reconstruction method to be used. We display and discuss these weighting functions for the most common reconstruction methods, namely nearest neighbor, linear and cubic spline interpolation.