Tomographic Reconstruction in Nuclear Medicine

1

Simple Backprojection

The general goal of reconstruction tomography is to generate a 2-D cross-sectional image of activity from a slice within the object, f ( x,y ), using the sinogram, or set of projection profiles, obtained for that slice. In practice, a set of projection profiles, p ( r ,ϕ _i ), is acquired at discrete angles, ϕ _i , and each profile is sampled at discrete intervals along r. The image is reconstructed on a 2-D matrix of discrete pixels in the ( x,y ) coordinate system. For mathematical convenience, the image matrix size usually is a power of 2 (e.g., 64 × 64 or 128 × 128 pixels). Pixel dimensions Δ x and Δ y can be defined somewhat arbitrarily, but usually they are related to the number of profiles recorded and the width of the sampling interval along r.

The most basic approach for reconstructing an image from the profiles is by simple backprojection. The concepts will be illustrated for a point source object. Figure 16-5A shows projection profiles acquired from different angles around the source. An approximation for the source distribution within the plane is obtained by projecting (or distributing) the data from each element in a profile back across the entire image grid ( Fig. 16-5B ). The counts recorded in a particular projection profile element are divided uniformly amongst the pixels that fall within its projection path. * This operation is called backprojection. When the backprojections for all profiles are added together, an approximation of the distribution of radioactivity within the scanned slice is obtained. Mathematically, the backprojection of N profiles is described by

where ϕ _i denotes the i ^th projection angle and f′ ( x,y ) denotes an approximation to the true radioactivity distribution, f ( x,y ).

* In practice, counts are assigned to a pixel in proportion to the fraction of the pixel area contained within the line of response for the projection element. However, owing to the complexity of the notation, this part of the algorithm is not included here.

As illustrated in Figure 16-5B , the image built up by simple backprojection resembles the true source distribution. However, there is an obvious artifact in that counts inevitably are projected outside the true location of the object, resulting in a blurring of its image. The quality of the image can be improved by increasing the number of projection angles and the number of samples along the profile. This suppresses the “spokelike” appearance of the image but, even with an infinite number of views, the final image still is blurred. No matter how finely the data are sampled, simple backprojection always results in some apparent activity outside the true location for the point source. Figure 16-6 shows an image reconstructed by simple backprojection for a somewhat more complex object and more clearly illustrates the blurring effect.

Mathematically, the relationship between the true image and the image reconstructed by simple backprojection is described by

where the symbol * represents the process of convolution described in Appendix G . A profile taken through the reconstructed image for a point source that is reconstructed from finely sampled data decreases in proportion to (1/ r ), in which r is the distance from the center of the point-source location. Because of this behavior, the effect is known as 1/r blurring. Simple backprojection is potentially useful only for very simple situations involving isolated objects of very high contrast relative to surrounding tissues, such as a tumor with avid uptake of a radiopharmaceutical that in turn has very low uptake in normal tissues. For more complicated objects, more sophisticated reconstruction techniques are required.

2

Direct Fourier Transform Reconstruction

One approach that avoids 1/r blurring is Fourier transform (FT) reconstruction, sometimes called direct Fourier transform reconstruction or direct FT. Although direct FT is not really a backprojection technique, it is presented here as background for introducing the filtered backprojection (FBP) technique in the next section.

Basic concepts of FTs are discussed in Appendix F . Briefly, in the context of nuclear medicine imaging, the FT is an alternative method for representing spatially varying data. For example, instead of representing a 1-D image profile as a spatially varying function, f ( x ), the profile is represented as a summation of sine and cosine functions of different spatial frequencies, k. The amplitudes for different spatial frequencies are represented in the FT of f ( x ), which is denoted by F ( k ). The operation of computing the FT is symbolized by

The function f ( x ) is a representation of the image profile in image space (or “object space”), whereas F ( k ) represents the profile in “spatial frequency space,” also called k-space. FTs can be extended to 2-D functions, f ( x,y ), such as a 2-D image. In this case, the FT also is 2-D and represents spatial frequencies along the x- and y-axes, F ( k _x , k _y ), in which k _x and k _y represent orthogonal axes in 2-D k-space. Symbolically, the 2-D FT is represented as

Mathematically, a function and its FT are equivalent in the sense that either one can be derived from the other. The operation of converting the FT of a function back into the original function is called an inverse FT and is denoted by

FTs can be calculated quickly and conveniently on personal computers, and many image and signal-processing software packages contain FT routines. The reader is referred to Appendix F for additional information about FTs.

The concept of k-space will be familiar to readers who have studied magnetic resonance imaging (MRI), because this is the coordinate system in which MRI data are acquired. To reconstruct an image from its 2-D FT, the full 2-D set of k-space data must be available ( Equation 16-7 ). In MRI, data are acquired point-by-point for different ( k _x , k _y ) locations in a process known as “scanning in k-space.” There is no immediately obvious way to directly acquire k-space data in nuclear medicine imaging. Instead, nuclear medicine CT relies on the projection slice theorem , or Fourier slice theorem. In words, this theorem says that the FT of the projection of a 2-D object along a projection angle ϕ [in other words, the FT of a profile, p ( r ,ϕ)], is equal to the value of the FT of the object measured through the origin and along the same angle, ϕ, in k-space (note, the value of the FT, not the projection of the FT). Figure 16-7 illustrates this concept. Mathematically, the general expression for the projection slice theorem is

where F ( k _r , ϕ) denotes the value of the FT measured at a radial distance k _r along a line at angle ϕ in k-space.

The projection slice theorem provides a means for obtaining 2-D k-space data for an object from a series of 1-D measurements in object space. Figure 16-7 and Equation 16-8 provide the basis for reconstructing an object from its projection profiles as follows:

• 1

  Acquire projection profiles in object space at N projection angles, ϕ _i , i = 1, 2, …, N as previously described.

• 2

  Compute the 1-D FT of each profile.

• 3

  Insert the values of these FTs at the appropriate coordinate locations in k-space. Note that values are inserted in polar coordinates along radial lines through the origin in k-space. For a specific value of k _r in the FT of the projection acquired at a rotational angle ϕ, the data are inserted at rectangular coordinates given by

  $\begin{array}{c}{k^{\prime }}_x=k_r\cos \varphi \\ {k^{\prime }}_y=k_r\sin \varphi \end{array}$

  where primed notation is used to indicate that the coordinate locations do not correspond exactly to points on a rectangular grid. The inserted values are closely spaced near the origin and more widely spaced farther away from the origin. This “over-representation” of data near the origin in k-space is one explanation for the 1/r blurring that occurs in simple backprojection, as was discussed in Section B.1.

• 4

  Using the values inserted in polar coordinates, interpolate values for k _x and k _y on a rectangular grid in k-space.

• 5

  Use the interpolated values in k-space and a standard 2-D (inverse) FT ( Equation 16-7 ) to compute the image of the object.

With noise-free data, perfect projection profiles (i.e., line integrals that represent precisely the sum of activity along a line measured through the object) and perfect interpolation, the direct FT reconstruction technique is capable of producing an exact representation of the object. Additional criteria regarding the required numbers of projection profiles and sampled points across each profile are discussed in Section C.

A major drawback of direct Fourier reconstruction is that the interpolation from polar to rectangular coordinates in k-space is computationally intensive. As well, it can lead to artifacts in the image, if not done carefully. A more elegant (and practical) approach, called filtered backprojection (FBP), is described in the next section.

3

Filtered Backprojection

Like the direct FT algorithm, FBP employs the projection slice theorem but uses the theorem in combination with backprojection in a manner that eliminates 1/r blurring. The steps are as follows:

• 1

  Acquire projection profiles at N projection angles (same as direct FT).

• 2

  Compute the 1-D FT of each profile (same as direct FT). In accordance with the projection slice theorem (see Fig. 16-7 and Equation 16-8 ), this provides values of the FT for a line across k-space.

• 3

  Apply a “ramp filter” to each k-space profile. Mathematically, this involves multiplying each projection FT by | k _r |, the absolute value of the radial k-space coordinate at each point in the FT. Thus the value of the FT is increased (amplified) linearly in proportion to its distance from the origin of k-space. Figure 16-8 illustrates the profile of a ramp filter, with filter amplitude denoted by H ( k _r ). Applying the ramp filter produces a modified FT for each projection, given by

  $P^{\prime }(k_r,\varphi )=\left|k_r\right|P(k_r,\varphi )$
  

  where P ( k _r , ϕ) is the unfiltered FT.

• 4

  Compute the inverse FT of each filtered FT profile to obtain a modified (filtered) projection profile. This is given by

  $\begin{array}{c}{p}^{\prime }(r,\varphi )=F^{-1}\left[P^{\prime }\right(k_r,\varphi \left)\right]\\ =F^{-1}\left[\left|k_r\right|P\right(k_r,\varphi \left)\right]\end{array}$

• 5

  Perform conventional backprojection using the filtered profiles. Mathematically, the result is

  $f(x,y)=\frac{1}{N}\sum _{i=1}^N{p}^{\prime }(x\cos {\varphi }_i+y\sin {\varphi }_i,{\varphi }_i)$

Step 5 is essentially the same as simple backprojection, but with filtered profiles. However, unlike Equation 16-3 , in which f′ ( x,y ) is only an approximation of the true distribution, FBP, when applied with perfectly measured noise-free data, yields the exact value of the true distribution, f ( x,y ). Figure 16-9 schematically illustrates the process of FBP for a pointlike object.

The only difference between simple and filtered backprojection is that in the latter method, the profiles are modified by a reconstruction filter applied in k-space before they are backprojected across the image. The effect of the ramp filter is to enhance high spatial frequencies (large k _r ) and to suppress low spatial frequencies (small k _r ). The result of the filtering is to eliminate 1/r blurring. * One way to visualize the effect is to note that, unlike unfiltered profiles (see Fig. 16-5 ), the filtered profiles have both positive and negative values (see Fig. 16-9 ). The negative portions of the filtered profiles near the central peak “subtract out” some of the projected intensity next to the peak that otherwise would create 1/r blurring.

* More precisely, 1/r blurring is the convolution of the true image with a blurring function, b ( r ) = 1/ r ( Equation 16-4 ). As discussed in Appendix G , convolution in image space is equivalent to multiplying by the FT of the blurring function in k-space, which for b ( r ) is B ( k _r ) = 1/| k _r |. Thus multiplying by | k _r | in k-space is equivalent to deconvolving the blurring function in image space, thereby eliminating the blurring effect.

Amplification of high spatial frequencies in FBP also leads to amplification of high-frequency noise. Because there usually is little signal in the very highest frequencies of a nuclear medicine image, whereas statistical noise is “white noise” with no preferred frequency, this also leads to degradation of signal-to-noise ratio (SNR). For this reason, images reconstructed by FBP appear noisier than images reconstructed by simple backprojection. (This is a general result of any image filtering process that enhances high frequencies to “sharpen” images.) In addition, filters that enhance high frequencies sometimes have edge-sharpening effects that lead to “ringing” at sharp edges. This is an unwanted byproduct of the positive-negative oscillations introduced by the filter, illustrated in the filtered profile at the top of Figure 16-9 .

To minimize these effects on SNR and artifacts at sharp edges, the ramp filter usually is modified so as to have a rounded shape to somewhat suppress the enhancement of high spatial frequencies. Figure 16-10 illustrates a ramp filter and two other commonly used reconstruction filters. Also shown are the equations describing these filters. A variety of reconstruction filters have been developed, each with its own theoretical rationale. Filters also play a role in the suppression of artifacts caused by aliasing in FT-based reconstruction techniques, as discussed in Section C. Additional discussions of these filters can be found in reference .

Because of its speed and relative ease of implementation, FBP became a widely used reconstruction method in nuclear medicine. A single 2-D image slice can be reconstructed in a fraction of a second on a standard computer. Under idealized conditions (noise-free data acquisition, completely sampled data, and so forth), FBP produces an accurate representation of the distribution of radioactivity within the slice. However, FBP is not without its limitations. First, it is susceptible to major artifacts if data from the object are measured incompletely (possibly because of collimator defects, portions of the object outside the FOV of the camera for some projections, etc.). Second, in datasets that have poor counting statistics or random “noise” spikes (perhaps caused by instrument malfunction), FBP produces annoying “streak artifacts.” These artifacts can be suppressed by employing a k-space filter with a strong roll-off at high spatial frequencies, but this results in loss of image resolution as well.

Finally, the FBP algorithm cannot readily be modified to take into account various physical aspects of the imaging system and data acquisition, such as limited spatial resolution of the detector, scattered radiation, and the fact that the sensitive volume of the detector collimator holes actually is a cone rather than a cylinder, as assumed for the reconstruction process. These factors require additional preprocessing or postprocessing data manipulations that work with varying degrees of success. They are discussed further in Chapter 17 , Section B.

By contrast, another set of reconstruction methods, known as iterative reconstruction techniques, can build these steps directly into the reconstruction algorithm and are less prone to the artifacts described in the preceding paragraph. These techniques are described in Section D.

4

Multislice Imaging

The analysis presented earlier for backprojection and Fourier-based reconstruction techniques applies to single-slice images. In practice, as described in Chapter 17, Chapter 18 , both SPECT and PET imaging are performed with detectors that acquire data simultaneously for multiple sections through the body. Projection data originating from each individual section through the object can be reconstructed as described. Individual image slices then are “stacked” to form a 3-D dataset, which in turn can be “resliced” using computer techniques to obtain images of planes other than those that are directly imaged. Thus 2-D multislice imaging of contiguous slices can be used to generate 3-D volumetric images. In many SPECT systems, the distance between image slices and slice thickness can be adjusted to achieve different axial resolutions, much the same as the sampling interval, Δ r , across the image profile can be adjusted to vary the in-plane resolution. In PET systems, the distance between image slices and the slice thickness often is fixed by the axial dimensions of the segmented scintillator crystals typically used in the detectors (See Chapter 18 , Sections B and C).