OPTICAL IMAGING AND SPECTROSCOPY Phần 3 potx

A function ff(x) [ V(f) may be represented as

ff(x) ¼ X

n[Z

cnfn(x) (3:124)

In contrast with the sampling theorem and with the Haar wavelet expansion, the

expansion coefficients are not samples of ff or inner products between ff and the

basis vectors. For the B-splines it turns out that we can derive complementary

functions f

n(x) for each fn(x) ¼ bm(x n) such that hfnjfn0i ¼ dnn0 . The comp￾lementary functions can be used to produce a continuous estimate for f(x) that is

completely consistent with the discrete measurements. This interpolated function is

fest(x) ¼ X

n[Z

hfnj fifn(x) (3:125)

Given the orthogonality relationship between the sampling functions and the comp￾lementary functions, fest is by design consistent with the measurements. We can

further state that fest ¼ ff if the complementary functions are such that f [ V(f),

in which case there exist discrete coefficients p(k) such that

f(x) ¼ X

k[Z

p(k)f(x k) (3:126)

Using the convolution theorem, the Fourier transform of f(x) is

f

^(u) ¼ f^(u) X

k[Z

p(k)ei2pku " # (3:127)

The orthogonality between the dual bases may be expressed as

hf

njfn0i ¼ dnn0

¼ X

k[Z

p(k)a(n0 k n) (3:128)

where a(n) ¼ hf(x)jf(x n)i. Without loss of generality, we set n ¼ 0 and sum both

sides of Eqn. (128) against the discrete kernel ei2pn0

u to obtain

X

n0

[Z

d0n0ei2pn0

u ¼ 1

¼ X

k[Z

X

n0

[Z

p(k)a(n0 k)ei2pn0

u

¼ X

k[Z

p(k)ei2pku " # X

n00[Z

a(n00)ei2pn00u

" # (3:129)

where we use the substitution of variables n00 ¼ n0 k.

3.9 B-SPLINES 91

Poisson’s summation formula is helpful in analyzing the sums in Eqn. (3.129).

The summation formula states that for g(x) [ L1(R)

X

n[Z

g(n)ei2pnu ¼ X

k[Z

^g(u þ k) (3:130)

where ^g(u) is the Fourier transform of g(x). To prove the summation formula,

note that

h(u) ¼ X

k[Z

^g(u þ k) (3:131)

is periodic in u with period 1. The Fourier series coefficients for h(u) are

^hn ¼

ð

1

0

h(u)e2pinudu

¼ X

k[Z

ð

1

0

^g(u þ k)e2pinudu

¼ X

k[Z

k

ð

þ1

k

^g(u)e2pinudu

¼

1ð

1

^g(u)e2pinudu

¼ g(n) (3:132)

Since a(x) is the autocorrelation of f, its Fourier transform is jf^(u)j

2

. Thus by the

Poisson summation formula

X

n[Z

a(n)ei2pnu ¼ X

k[Z

jf^(u þ k)j

2 (3:133)

Reconsidering Eqn. (3.129), we find

X

k[Z

p(k)ei2pku ¼ 1

P

n[Z a(n)ei2pnu

¼ 1

P

k[Z jf^(u þ k)j

2 (3:134)

92 ANALYSIS

Substitution in Eqn. (3.127) yields

f

^(u) ¼ f^(u)

P

k[Z

jf^(u þ k)j

2 (3:135)

We can evaluate Eqn. (3.135) to determine f

^(u) and f(x) if P

k[Z jf^(u þ k)j

2 is

finite. The requirement that there exist positive constants A and B such that

A  X

k[Z

jf^(u þ k)j

2  B (3:136)

is the defining feature of a Riesz basis. A Riesz basis may be considered as a gener￾alization of an orthonormal basis. In the case that P

k[Z jf^(u þ k)j

2 ¼ 1, Eqn. (135)

reduces to f

^(u) ¼ f^(u) and an orthonormal basis may be obtained.

The Fourier transform of the mth-order B-spline is

b^m

(u) ¼ [sinc(u)](mþ1)eipju

¼ b^0

(u)

h i(mþ1)

eip(mþ1j)u (3:137)

where j ¼ 0 if m is odd and j ¼ 1 if m is even. For the B-spline basis, we obtain

Qm(u) ¼ X

k[Z

jf^(u þ k)j

2 ¼ X

k[Z

jsinc(u þ k)j

2(mþ1) (3:138)

Since the zeroth-order B-spline produces an orthogonal basis, we know that

Q0(u) ¼ 1. For higher orders we note that jsinc(u þ k)j

2(mþ1)  jsinc(u þ k)j

2

,

meaning that Qm(u)  Qo(u). Thus, 0 , Qm(u) , 1 and the B-spline functions of

all orders satisfy the Riesz basis condition.

In contrast with the B-splines themselves, the complementary functions f(x) do

not have finite support. It is possible, nevertheless, to estimate f(x) over a finite

interval for each B-spline order by numerical methods. Estimation of Qm(u) from

Eqn. (3.138) is the first step in numerical analysis. This objective is relatively

easily achieved because Qm(u) is periodic with period 1 in u. Evaluation of the

sum over the first several thousand orders for closely spaced values of 0  u  1

takes a few seconds on a digital computer.

Given Qm(u), we may estimate f(x) by using a numerical inverse Fourier trans￾form of Eqn. (3.135) or by calculating p(k) from Eqn. (3.134). Since p(k) must be

real and since Qm(u) is periodic, we obtain

p(k) ¼

ð

1

0

cos (2pku)

Qm(u) du (3:139)

Estimation of p(k) was the approach taken to calculate f(x) for Fig. 3.16.

3.9 B-SPLINES 93

Given f(x) ¼ bm(x n) and f(x), we can calculate ff(x) for target functions. For

example, Fig. 3.17 shows the signals of Figs. 3.8 and 3.9 projected onto the V(f) sub￾spaces for B-splines of orders 0–3. Higher-order splines smoothly represent signals

with higher-order local polynomial curvature. Note that higher-order splines are not

more localized than the lower-order functions, however, and thus do not immediately

translate into higher signal resolution. Notice also the errors at the edges of the signal

windows in Fig. 3.17. These arise from the boundary conditions used to truncate the

infinite time signal f(x). In the case of these figures, f(x) was assumed to be periodic

in the window width, such that sampling and interpolation functions extending

beyond the window could be wrapped around the window.

The interpolated signals plotted in Fig. 3.17 are the projections ff(x) [ V(f) of

f(x) onto the corresponding subspaces V(f). The consistency requirement designed

into the interpolation strategy means that these functions, despite their obvious discre￾pancies relative to the actual signals, would yield the same sample projections.

Corrections that map the interpolated signals back onto the actual signal lie in

V?(f). Strategies for sampling and interpolation to take advantage of known

constraints on f(x) to so as to infer correction components f?(x) are discussed in

Chapter 7.

Figure 3.16 Complementary interpolation functionsf(x) for the B-splines of orders 0–3.

The zeroth-order B-spline is orthonormal such that f(x) ¼ b0(x).

94 ANALYSIS

Use of Eqn. (3.125) to estimate f(x) is somewhat unfortunate given that f

n(x) does

not have finite support. A primary objection to the use of the original sampling

theorem [Eqn. (3.92)] for signal estimation is that sinc(x) has infinite support and

decays relatively slowly in amplitude. While f(x) is better behaved for low-order

B-splines, it is is still true that accurate estimation of f(x) may be computationally

expensive if a large window is used for the support of f. As the order of the

B-spline tends to infinity, f(x) converges on sinc(x) [235]. If we remove the require￾ment that f(x) [ V(f), it is possible to generate a biorthogonal dual basis for bm(x)

with compact support [49]. The compactly supported biorthogonal wavelets in this

case introduce a complementary subspace V spanned by f(x).

The goal of the current section has been to consider how one might use a set of

discrete B-spline inner products to estimate a continuous signal. This problem is

central to imaging and optical signal analysis. We have already encountered it in

the coded aperture and tomographic systems considered in Chapter 2, and we will

encounter it again in the remaining chapters of the text. We leave this problem for

now, however, to consider the use of sampling functions and multiscale represen￾tations in signal and system analysis. One may increase the resolution and fidelity

Figure 3.17 Projection of f(x) ¼ x2=10 and the signal of Fig. 3.9 onto the V(f) subspace for

B-splines of orders 0–3.

3.9 B-SPLINES 95