ilarly, all pixels in the acquisition matrix are smoothed, and the profiles are
then backprojected.
The spatial kernel described above with all positive weighting factors
reduces noise but degrades spatial resolution of the image. Sharp edges
in the original image become blurred in the smoothed image as a result
of averaging the counts of the edge pixels with those of the neighboring
pixels.
Another filter kernel commonly used in the spatial domain consists of a
narrow central peak with both positive and negative values on both sides
of the peak, as shown in Figure 12.6. When this so-called edge-sharpening
filter is applied centrally to a pixel for correction, the negative values in
effect cancel or erase all neighboring pixel count densities, thus creating a
corrected central pixel value. This is repeated for all pixels in each projec-
tion and the corrected projections are then backprojected. This technique
reproduces the original image with better spatial resolution but with
increasing noise. Note that blurring due to simple backprojection is
removed by this technique but the noise inherent in the data acquisition
due to the limitations of the spatial resolution of the imaging device is not
removed but rather enhanced.
The Fourier Method
Nuclear medicine data obtained in the spatial domain (Fig. 12.7A) can be
expressed in terms of a Fourier series in the frequency domain as the sum
160 12. Single Photon Emission Computed Tomography
Fig. 12.5. The smoothing technique in the spatial domain using a 9-point smooth-
ing kernel. The thick-lined pixel with value 5 is smoothed by first assuming a 3 × 3
acquisition matrix (same size as the smoothing matrix) centered at this pixel and
multiplying each pixel value of the matrix by the corresponding weighting factor,
followed by summing the products. The weighting factor is calculated by dividing
the individual pixel value by the sum of all pixel values of the smoothing matrix.
After smoothing the value of the pixel is changed from 5 to 3. Similarly all pixel
values of the acquisition matrix are smoothed by the 9-point smoothing kernel, to
give a smoothed image.