基于局部基函数的CT重建算法优化

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"Local Basis-Function Approach to Computed Tomography: A Comprehensive Overview" 本文讨论了一种针对计算机断层扫描(Computed Tomography, CT)的创新方法，即局部基函数方法。该方法将图像重建过程视为一系列基函数的线性组合，这些基函数在矩形网格上分布，并具有局部支持区域。这种方法的关键在于所使用的基函数是正交且可能相互重叠的，其中立方B-spline基函数表现出显著的优势，尤其是在通过迭代重建算法实现计算精度提升方面。在局部基函数方法中，每一个重建值被表示为一组系数的集合，这使得在应用诸如上限和下限这样的局部约束时更加便捷。由于每个位置都由一组系数精确指定，即使图像的粗粒度表示也不再需要依赖于任意的插值函数，从而提高了图像显示的灵活性和一致性。使用重复基函数的一个关键优势在于，这允许通过预先计算单个基函数上的线或条带积分，显著减少算法中的计算工作量。这样做的好处是提高了效率，降低了计算成本，并简化了实际操作中的复杂性。此外，这种方法对于处理包含物体形状变化和空间分辨率要求高的情况特别有用，因为局部支持的特性有助于保留图像细节，并能够更好地适应不同部位的特性。局部基函数方法为CT图像重建提供了一个强大的工具，它在提高重建质量、优化计算效率和适应复杂约束条件等方面展现出了巨大的潜力。随着技术的发展，这种方法有望在未来的CT成像领域得到广泛应用。

specified by the set of

coefficients

aj.

Combining

Eqs.

(5)

and (6), we see that the coefficients must

satisfy

ajS(bj;rj,B,w)

gi,

. .

,M.

(7)

j=1

This approach has been considered before in CT re-

con~truction.~.~ It is also referred to as collocation

when applied to the solution of integral and differen-

tial

equation^.^^

The projection operation, defined by

Eq. (I), becomes a sum over contributions from the

individual basis functions. Backprojection, used as

the principle means of updating reconstructions,

amounts to adding a quantity to the

that is propor-

tional to

S(b,;r&,w).

If we define a matrix

whose entry in the ith row

and jth column is given by

Eq.

(7)

can be written in matrix form as

(9)

where

is the vector of

unknown coefficients

a,,

and

is the vector of

measurements

g,.

If Eq.

(9)

can be

solved for

the values off at any point

(x,y)

can be

obtained using Eq. (6). The essential problem, then, is

to invert the

measurement matrix

When

either

or when

is a singular matrix, the

inverse of

does not exist. When the problem is

underdetermined, as it must be when

for

example, a plethora of solutions exists. Finding a

solution is not the dilemma; rather, it is to decide

which of the many solutions to choose. To make the

solution in some sense unique, it might be required

that out of the many possible solutions the one with

minimum norm be chosen. This amounts to employ-

ing the pseudoinverse or generalized inverse2' of

(or

P'IP).

The ambiguous nature of the solutions to this

~roblem has its roots in the existence of a null s~ace~

the matrix

When any vector lying wbollyAin the

null space is multiplied by

the result is zero. The

components of

that lie in this null space cannot be

determined from the measurements a1one.I When

reconstruction is formulated as a least-square error

problem, the matrix that must be inverted is

PTP,

which is square but may be singular nevertheless.

Consider the magnitude of this inversion task when

one has 10 views with 100 samples in each view and the

reconstruction image is to be defined on a 100

100

grid, which is a CT problem of only moderate size. In

this case,

is a 1000

10,000 matrix. Fortunately,

when the basis functions are local,

is sparse. For

example, in the above problem,

might contain only

200,000 nonzero elements, i.e., 2% of the total. It is

known3 that in such a case iterative reconstruction

algorithms may be be used to solve Eq.

(9)

with a small

number (between

and 20) of iterations, where one

iteration includes

all

the measurements.

At this point it becomes necessary to choose the

basis functions

bj.

Desirable properties of a suitable

basis-function set include the following:

(A)

strong linear independence;

(B)

power of approximation;

(D)

efficient computation of projections and back-

projections;

(E) efficient implementation of reconstruction con-

straints;

(F)

fidelity of visual appearance.

It is quite possible that no single set of local basis

functions can meet all these conditions ideally, but

some do much better than others. Let us discuss these

criteria in turn. Linear independence of functions in a

basis set is necessary to specify the coefficients corre-

sponding to a given reconstruction function uniquely.

Moreover, we prefer strong linear independence so

that a pair of distant points in the coefficient space

corresponds to significantly different reconstruction

'functions.

Ideally we want a basis-function set to be complete

in the class of all acceptable reconstructions so that

any reconstruction can be represented. In practice,

we can only ask that any reconstruction function be

accurately approximated by a linear combination of

basis functions. For a fixed number of basis functions,

some sets provide better approximations than others.

For example, we expect a piecewise-linear function to

furnish a better approximation to a smooth function

than a piecewise-constant one. The number of coeffi-

cients needed to represent the

reconstmction is often a

dominant consideration in the selection of a basis-

function set because of the need to limit the amount of

computer storage and computation speed required to

perform a reconstruction. Hence it is desirable to

select a representation that gives the best approxima-

tion with a fixed minimal number of coefficients. It is

expected that, for any form of baais function, the de-

gree of approximation will improve as the number of

basis functions is increased. In particular, it is desir-

able to have agood representation of both the constant

andlinearly varyingfunctions. It is oftenthe case that

the objective of reconstruction is to detect the presence

of a low contrast anomaly against a simple slowly vary-

ing background. If the representation is incapable of

properly portraying such a background, it might be

difficult to distinguish the actual anomaly from de-

fects in representation. Similarly, it is undesirable for

a step function to evoke oscillations in the reconstruc-

tion, as in the Gibbs phenomenon.

It would be unfortunate if a shift of the basis func-

tions resulted in a much different approximation to

the same function. Such behavior could only be a

boon if the basis functions were used to enforce a

certain structure in the reconstructed function that

was known beforehand. Generally such information is

not available, and the choice of the position of the

basis-function set is arbitrary and random relative to

the object.

Computational speed is very important for iterative

CT reconstruction algorithms, because many itera-

tions are usually required

obtain an acceptable re-

sult. In these algorithms it is the projection and back-

4030

APPLIED

OPTICS

Vol.

24.

Nor23

December

1985

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