A Comparative Study of the Use of Linear and Modified Cubic Spline Interpolation for Image Reconstruction

Abstract

Image reconstruction is the process of recovering a function of two variables from experimentally obtained estimates of its integrals alone certain lines. An important version in medicine is the recovery of the density distribution within a cross-section of the human body from a number of X-ray projections.A computationally efficient technique for image reconstruction is the so-called convolution method.It consists of two steps: (i) data obtained by each of the orojections of the cross-section are separately (discrete) convolved with a fixed function; (ii) the density of the function at any point in the crosssection is estimated as the sum of values (one from each projection) of the convolved projection data.A difficulty is that part (ii) usually requires values of the convolved orojection data at points other than where they have been calculated during part (i). This is usually resolved by interpolation between the calculated values.In this paper we report on a computer experimental study which compares the efficacy of two methods of interpolation (linear interoolation and a modified cubic spline interpolation) when used with the convolution reconstruction method. The two interoolation techniques are examined for their mathematical properties and are compared from the points of view of resolution of fine details, smoothness of the reconstructed cross-sections, sensitivity to noise in the data, the overall nearness of the oriqinal and reconstructed objects, and the cost of implementation. Both methods are illustrated on reconstructions of a mathematically described cross-section of the human head from computer simulated X-ray data.

A chronology of interpolation: from ancient astronomy to modern signal and image processing

“…However, whereas the improvement of linear over nearest-neighbor interpolation was considerable, the further improvement of cubic convolution interpolation was found to be negligible in this application. Similar observations had been made earlier by Herman et al [191] in the context of image reconstruction from projections.…”

“…In the fields of computer graphics and visualization, the third-order cubic convolution kernel is therefore usually referred to as the Catmull-Rom spline. It has also been called the (modified or cardinal) cubic spline [191]- [197]. Finally, this cubic convolution kernel is precisely the kernel implicitly used in the previously mentioned osculatory interpolation scheme proposed around 1900 by Karup and King.…”

Quantitative evaluation of convolution-based methods for medical image interpolation

“…In fact, the aforementioned constraints were adopted from the literature on cubic convolution. In the literature on visualization and computer graphics, the cubic convolution kernel resulting from the flatness constraint is also known as the Catmull-Rom spline [4] and the modified cubic spline [16], and is sometimes erroneously referred to as the cardinal cubic spline [34,35].…”

Sparsely Sampled Wideband Radar Holographic Imaging for Detection of Concealed Objects

“…A third-order cubic spline interpolator is usually adequate for most applications [23]. Noise experiments show that the linear interpolation is less sensitive than cubic splines to Gaussian white noise [26]. Its local support and shift-invariant properties offer very attractive computational procedures [27].…”