Filtered Backprojection
The simple backprojection has the problem of “star pattern” artifacts (Fig.
12.3C) caused by “shining through” radiations from adjacent areas of
increased radioactivity resulting in the blurring of the object. Because the
blurring effect decreases with distance (r) from the object of interest, it can
be described by a 1/rfunction (Fig. 12.3D). It can be considered as a
spillover of certain counts from a pixel of interest into neighboring pixels,
and the spillover decreases from the nearest pixels to the farthest pixels.
This blurring effect is minimized by applying a “filter” to the acquisition
data, and the filtered projections are then backprojected to produce an
image that is more representative of the original object. Such methods are
called the filtered back projection. There are in general two methods of fil-
tered backprojection: the convolution method in the spatial domain and the
Fourier method in the frequency domain, both of which are described
below.
The Convolution Method
The blurring of reconstructed images caused by simple backprojection is
eliminated by the convolution method in which a function, termed “kernel,”
is convolved with the projection data, and the resultant data are then back-
projected. The application of a kernel is a mathematical operation that
essentially removes the l/rfunction by taking some counts from the neigh-
boring pixels and putting them back into the central pixel of interest. Math-
ematically, a convolved image f′(x,y) can be expressed as
(12.1)where fij(x−i,y−j) is the pixel count density at the x−i,y−jlocation in
the acquired projection, the hijvalues are the weighting factors of the con-
volution kernel, and indicates the convolution operation. The arrange-
ment of hijis available in many forms.
A familiar “nine-point smoothing” kernel (i.e., 3 ×3 size), also called
smoothing filter, has been widely used in nuclear medicine to decrease sta-
tistical variation. The essence of this technique is primarily to average the
counts in each pixel with those of the neighboring pixels in the acquisition
matrix. An example of the application of nine-point smoothing to a section
of an image is given in Figure 12.5. Let us assume that the thick-lined pixel
with value 5 in the acquisition matrix is to be smoothed. First, we assume
a 3 ×3 acquisition matrix (same as 3 ×3 kernel matrix) centered at the pixel
to be convolved. Each pixel datum of this matrix is multiplied by the cor-
responding weighting factor, followed by the summation of the products.
The weighting factors are calculated by dividing the individual pixel values
of the kernel matrix by the sum of all pixel values of the matrix. The result
of this operation is that the value of the pixel has changed from 5 to 3. Sim-
′()=−()−
=−=−fxy ∑∑ h fxiy jij
j NN
ij
iNN
,,Single Photon Emission Computed Tomography 159