In some situations, PSF and LSF measurements are not ideally suited for a specific imaging application, where the edge spread function (ESF) can be measured. Instead of stimulating the imaging system with a slit image as with the LSF, a sharp edge is presented. The edge gradient that results in the image can then be used to measure the ESF(x). The ESF is particularly useful when the spatial distribution characteristics of glare or scatter phenomenon are the subject of interest—since a large fraction of the field of view is stimulated, low-amplitude effects such as glare or scatter (or both) become appreciable enough in amplitude to be measurable. By comparison, the tiny area of the detector receiving signal in the PSF or LSF measurements would not be sufficient to cause enough optical glare or x-ray scatter to be measured. A sharp, straight edge is also less expensive to manufacture than a point or slit phantom.
An example of spatial domain spread functions is shown in Figure 4-7.
110
■ A. The conventional approach to measuring the LSF involves the acquisition of a slit image along the y-axis, and a profile (gray scale versus position) across it (in the x-axis) is evaluated. This approach is legitimate but is limited by the sampling pitch of the detector. B. An alternative approach to computing the LSF was proposed by Fujita. Here, the slit image is acquired at a slight angle (5 degrees) relative to the y-axis. At any row along the vertical slit, a single LSF can be measured. However, because of the angle, a composite LSF can be synthesized by combining the LSFs from a number of rows in the image. The angle of the slit creates a small differential phase shift of the LSF from row to row—this can be exploited to synthesize an LSF with much better sampling than the pixel pitch. This oversampled LSF is called the presampled LSF.
4.2
Convolution
Convolution is an integral calculus procedure that accurately describes mathematically what the blurring process does physically. The convolution process is also an important mathematical component of image reconstruction, and understanding the basics of convolution is essential to a complete understanding of imaging systems. Below, a basic description of the convolution process is provided. Convolution in 1D is given by
∞
−∞ ⊗
∫
( ) ( ') ( ') ' ( ) ( )
G x H x k xx dx H x k x [4-1]
where is the mathematical symbol for convolution. Convolution can occur in two or three dimensions as well, and the extension to multidimensional convolution is straightforward. Referring to Figure 4-8, a column of numbers H is to be convolved with the convolution kernel, resulting in column G. The function H can also be thought
3 4 5 6 7
■ The three basic spread functions in the spatial domain are shown—the PSF(x,y) is a 2D spread function. The LSF and ESF are both 1D spread functions. There is mathematical relationship between the three spread functions, as discussed in the text.
H
■ The basic operation of discrete convolution is illustrated in this figure. In the three panes (A–C), the function H is convolved with the convolution kernel, resulting in the function G. The process proceeds by indexing one data element in the array, and this is shown in the three different panes. The entire convolution is performed over the entire length of the input array H.
of as a row of pixel data on a 1D image. The kernel in this example is five elements long, and its value sums to unity. In pane A, the first five numbers of H are multiplied by the corresponding elements in the kernel (here, each element of the kernel has a numerical value of 0.20); these five products are summed, resulting in the first entry in column G. In pane B, the convolution algorithm proceeds by shifting down one element in column H, and the same operation is computed, resulting in the second entry in column G. In pane C, the kernel shifts down another element in the array of numbers, and the same process is followed. This shift, multiply, and add procedure is the discrete implementation of convolution—it is the way this operation is performed in a computer. A plot of the values in columns H and G is shown in Figure 4-9, and H in this case is randomly generated noise. The values of G are smoothed relative to H, and that is because the elements of the convolution kernel in this case are designed to perform data smoothing—essentially by averaging five adjacent values in Column H. A plot of various five-element kernels is shown in Figure 4-10. The
■ A plot of the noisy input data H(x), and of the smoothed function G(x), is shown. The first ele-ments in the data shown on this plot correspond to the data illustrated in Figure 4-8. The input function H(x) is simply random noise. The convolution process with the RECT function results in substantial smoothing, and this is evident by the much smoother G(x) function in comparison to H(x).
A
■ Three convolution kernels are illustrated. A is a RECT function defined by [0.2, 0.2, 0.2, 0.2, 0.2], and the convolution process with this function is illustrated in Figures 4-8. Kernel B illustrates another possible smoothing kernel, with values [0.05, 0.20, 0.50, 0.20, 0.05]. Kernel C has the values [−0.15, 0.35, 0.50, 0.35, −0.15]. The negative sidebands of this kernel cause edge enhancement during a convolution pro-cedure, but this can also increase the noise in the resulting function.
kernel illustrated in Figure 4-8 is kernel A in Figure 4-10, and the shape is that of a rectangle (a RECT function), because all of the values in the kernel are the same.
Use of such a kernel in data analysis is also called a boxcar average or running aver-age, with applications in economics and elsewhere. Kernel B in Figure 4-10 has an appearance a bit similar to a bell-shaped curve, to the extent that only five values can describe such a shape. Both kernels A and B have all positive values, and therefore, they will always result in data smoothing. Kernel C has some negative edges on it, and when negative values are found in a kernel, some edge enhancement will result.
In Equation 4-1 or Figure 4-8, it can be seen that negative values in the kernel will result in weighted subtraction of adjacent elements of the input function, and this brings out edges or discontinuities in the input function, and it tends to exacerbate noise in the data as well. It is worth noting that if the kernel was [0, 0, 1, 0, 0], G would be equal to H. This kernel is called a delta function.
Convolution can use kernels of any length, and for image processing, kernels from 3 elements to 511 or more elements are used. Although not required, most kernels that are smaller in length are odd, so that there is a center element. For a kernel of length Nk (odd), convolution is ill defined at the edges of the image for (Nk−1)/2 pixels (Fig. 4-8); however, assuming that the image has zero-valued pixels adjacent to its edges is a common method for dealing with this. Above, only 1D convolution using 1D kernels was discussed, but convolution is routinely performed in two dimensions using 2D kernels. In such case, 3 3, 5 5, and other odd kernel dimensions are routinely used. Examples of 2D convolution kernels are illustrated in Figure 4-11. Three-di-mensional convolution techniques are also used in medical imaging processing. Con-volution is a mathematical process that describes physical blurring phenomena, but convolution techniques can be used to restore (improve) spatial resolution as well—in such cases the process is called deconvolution. While deconvolution can improve spatial resolution in some cases, it also amplifies the noise levels in the image.