If the image has any background, it must be subtracted before applying the transform, such that the image intensity goes to zero near the edge (however, the Direct and Hansen–Law methods effectively disregard a constant background). In other words, only localized objects can be transformed, and the object must be contained entirely within the image frame. Thus the intensity distributions \(f\) and \(F\) must be zero outside these ranges, otherwise the transforms cannot be performed correctly. Note that the forward and inverse Abel transforms are defined on the whole space, with infinite integration limits, but in reality, experimental data are limited to finite ranges of \(r\) or \(y\). The following comparisons focus on the results of the inverse Abel transform, because it is the inverse Abel transform that is used most frequently to interpret experimental data. PyAbel incorporates numerous algorithms for the inverse Abel transform, and some of these algorithms also support the forward Abel transform. Various algorithms have been developed to address these issues. Indeed, a simple numerical integration of the above equations has been shown to provide unreliable results. Finally, a simple approach requires a large number of sampling points in order to provide an accurate transform. Additionally, \(y = r\) presents a singularity where the denominator goes to zero and the integrand goes to infinity. Such nested loops can be computationally expensive. First, the simplest computational interpretation of inverse Abel transform equation involves three loops: over \(z\), \(r\), and \(y\), respectively. Several issues arise when attempting to evaluate the Abel transform numerically. While the transform equations can be evaluated analytically for some mathematical functions, experiments typically generate discrete data (e.g., images collected with a digital camera), which must be evaluated numerically. However, some methods are orders-of-magnitude more efficient than others. In general, all of the methods provide reasonably quality results, with some methods providing options for additional smoothing of the data. The following sections present several comparisons of the quality and speed of the various Abel-transform algorithms presented in PyAbel.
Fortunately, PyAbel provides a consistent interface for the Abel-transform methods via the Python programming language, which allows for a straightforward, quantitative comparison of the output. Each algorithm was originally implemented with somewhat different mathematical conventions and with often conflicting requirements for the size and format of the input data. Each method was created with specific goals in mind, with some taking advantage of pre-existing knowledge about the shape of the object, some prioritizing robustness to noise, and others offering enhanced computational efficiency. Consequently, many numerical methods have been developed to provide approximate solutions to the Abel transform (for example:, ,, ,, ,, , ). While the forward and inverse Abel transforms may be written as simple, analytical expressions, attempts to naively evaluate them numerically for experimental images does not yield reliable results. As indicated by the Abel transform equations (below), the 3D object is described in terms of \((r, z)\) coordinates, while the 2D projection is recorded in \((y, z)\) coordinates. The inverse Abel transform takes the 2D projection and mathematically reconstructs the 3D object.
For example, an X-ray image of the object on the left would produce the projection shown on the right. 1 The forward Abel transform maps a cylindrically symmetric three-dimensional (3D) object to its two-dimensional (2D) projection, a physical process that occurs in many experimental situations.
0 kommentar(er)
