# Reconstruction from Uniformly Attenuated SPECT Projection Data Using the DBH Method Page: 7 of 21

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7

position indicated in the figure is truncated due to a small detector.

y Object boundary

Small detector

A typical filtering line

Object can be exactly _(Y)

reconstructed between -

these two lines 0

. . . . . .. . li..s .D detector's field-of-view

Fig. 4: Illustration of a small detector and the area of exact reconstruction by the DBH method.

Summarizing the preceding analysis, the derivative and backprojection are local operations and do not

require projections to be available in the entire FOV. If f(x,y) has a support in the finite region

[-L(y), L(y)], as illustrated in Fig. 4; regardless of the data acquisition geometry, f(x,y) and f(x,y) can be

related by

f(x, y) = cosh(pa)f(x - ,y d.

-L~xyY = z 2

Because of the limited data availability and the limitation in conducting the backprojection, f(x,y) may be

available only in a finite region for the variable x. Intuitively, that region must be large enough to have a

stable inversion. In [28], f(x,y) must be computed in (-3L(y), 3L(y)) . The inversion procedure in [30] uses

f(x,y) over [-L(y), L(y)] and [p(Z /2, s)+ p(-Z /2, s)]/2. The inversion algorithm presented in this work only

needs f(x,y) in [-L(y), L(y)]. All those inversion formulas are mathematically exact, thus f(x,y) can be

completely reconstructed from f(x,y) on [-L(y), L(y)] . Also, there is a direct extension to reconstruction of

fan-beam data, as indicated in (4), (14), and (17).

III. INVERSION OF THE FINITE WEIGHTED HILBERT TRANSFORM

It has been shown that the function f(x,y) can be obtained from the acquisition of both parallel- and fan-

beam data and that f(x,y) and f(x,y) are related by a weighted Hilbert transform. In this section, we show

that f(x,y) can be exactly reconstructed from f(x,y) if f(x,y) is available in the same support region

off(x,y) . Frequently in SPECT imaging the projections are truncated because the FOV is not sufficiently

large enough to image the entire body. However, a smaller region instead of the whole object can be

accurately reconstructed from truncated projection data. In the presence of truncation, a subset of f(x,y),

the region between the two dashed lines in Fig. 4, can be exactly reconstruction.

A function h(t) and its Hilbert transform H(s) are related by:

H(s)=if h(t)dt , h(t)= 1H(s)ds . (18)

Z s - t Z_ s-t

It is assumed that all singular integrals are equal to the Cauchy principal value. The function h(t) is

assumed to be continuously smooth with compact support in [-q, q] , q >0. With the hyperbolic cosine

weighting function, the finite weighted Hilbert transform is defined as

H,(s) = 1 cosh(p(s - t)) h(t)dt . (19)

q_, s-t

It is obvious that when p = 0, the finite weighted Hilbert transform reduces to the finite Hilbert transform.

Notice that H, (s) quickly tends to - as s -* 0 .

Two classic inversion formulas [33] of the finite Hilbert transform (u =0 ) are

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Huang, Qiu; You, Jiangsheng; Zeng, Gengsheng L. & Gullberg, Grant T. Reconstruction from Uniformly Attenuated SPECT Projection Data Using the DBH Method, article, March 20, 2008; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc901800/m1/7/: accessed January 21, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.