Abstract

The generalized analytical quadrature filter from a set of interferograms with arbitrary phase shifts is obtained. Both symmetrical and non symmetrical algorithms for any order are reported. The analytic expression is obtained through the convolution of a set of two-frame algorithms and expressed in terms of the combinatorial theory. Finally, the solution is applied to obtain several generalized tunable quadrature filters.

© 2011 OSA

1. Introduction

A myriad of phase stepping algorithms have been reported since the 70’s [114]. However, an analytical generalized expression for arbitrary phase steps is not reported yet. In a previous work [12], it was proved that the design of a quadrature filter can be solved as geometrical problem from any desired conditions. However, the same procedure was reported as a new method in [13,14]. In a recent work, from a set of two-frame filters a quadrature filter is obtained and several novel tunable algorithms were reported [15]. Therefore, in this paper the generalized analytical expression for a quadrature filter for arbitrary phase steps is finally obtained instead the numerical approach reported in [16]. Thus, this paper is presented as follows: in chapter 2, a general non rotated (symmetric) two-frame algorithm is obtained, and through the rotation matrix the aliased rotated filter (non symmetric algorithm) is derived. In chapter 3, a formalism to obtain any quadrature filter through the convolution of a set of two-frame filters is introduced, and some particular cases for the rotated and non rotated filters are obtained. In chapter 4, the quadrature filters obtained are generalized through the combinatorial theory and two particular cases for symmetric and non symmetric algorithms are reported. Finally in chapter 5, the usefulness of our formalism is used to obtain several novel tunable algorithms.

2. The two-frame algorithm in phase shifting interferometry (PSI)

2.1 The symmetric two-frame filter

In phase shifting interferometry an intensity frame Ik measured at time tk is given by

Ik=a(x,y)+b(x,y)cos[αtk+φ(x,y)]  For k=1 and 2,
where α is the frequency carrier or usually named phase shift, b(x,y) is the amplitude for the position (x,y) and φ(x,y) is the phase to be estimated, meanwhile a(x,y) is the background [115]. In a previous work [12], for t1=1/2, t2=1/2 and a(x,y)=0 the corresponding phase φ(x,y) is obtained from the following tunable two-frame algorithm
tan[φ(x,y)]=NI2DI2=cos(α/2)[11]I2sin(α/2)[1  1] I2=cos(α/2)sin(α/2)[I1I2I1+I2],
where, I2=[I1I2]Tis the intensity column vector formed by the two frames. Then, from [12,15] the Fourier transform H(ω) of this filter that cuts off any frequency ω=α is:
H(ω)=2sin[(ωα)/2].
Finally, from [15] the numerator and denominator that describe the quadrature filter are:
ND=cos(α/2)sin(α/2)[11][1  1] .
In others words, through a two-frame filter it is possible to cut off any specific frequency α over the frequency axes because it Eq. (3) satisfies the quadrature condition H(ω=α)=0. As shown in [12,15], for a(x,y)0, the quadrature condition H(ω=0)=0 must be also satisfied to cancel the DC component and more frame are required to recover the desired phase. That is, two or more individual filters are required to satisfy both quadrature conditions.

2.2 The discrete temporal convolution

For two individual algorithms their corresponding individual phases φ1, φ2 are given by:

tan(φ1)=N1 InD1 In,  and  tan(φ2)=N2 ImD2 Im,
where, In=[I1I2...In] T and Im=[I1I2...Im] T are the column vectors with n and m frames respectively. On the other hand, N1and D1 are two [nx1] row vectors, likewise N2 and D2 are two [mx1] row vectors too. Therefore, the temporal impulse responses h1(t) and h2(t) for both filters are expressed by
h1(t)=(D1+i N1)δn  and  h2(t)=(D2+i N2)   δm,
where, δn and δm are the [nx1] and [mx1] column vectors expressed in terms of the impulse functions such as:
δn=[δ(t),δ(tα),...δ(tnα)]T  and  δm=[δ(t),δ(tα),...δ(tmα)]T.
Therefore, the combined filter between h1(t) and h2(t) is obtained through the following convolution h(t)=h1(t)*h2(t), where ∗ denotes the temporal discrete convolution. Then, from Eq. (6) the temporal response h(t) becomes
h(t)=[D1δn+i N1δn][D2δm+i N2δm]                                     =[D1D2N1N2]δn+m1+i[N1D2+D1N2]δn+m1,
where, δnδm=δm+n1. From the above expression a new (n + m-1) resultant filter is obtained, and from Eq. (8) the desired phase φ(x,y) is recovered as
tan[φ(x,y)]=N In+m1D In+m1=[N1D2+D1N2] In+m1[D1D2N1N2] In+m1.
Foremost, from Eq. (9) the vector In+m1 is dropped to obtain an easier way to evaluate the combined filter.
ND=(N1D1)(N2D2)=N1D2+D1N2D1D2N1N2.
The discrete convolution between the X and Y vectors with n and m orders is obtained from Z(k)=p=1m+n1X(p)Y(kp+1) for k = 1,2 ... m + n-1. The coefficients of the product are given by the convolution of the original coefficient sequences X and Y, extended with zeros where necessary to avoid undefined terms and this is the well known Cauchy product of two polynomials.

2.3 The tunable rotated two-frame algorithm

The numerator Nθ and denominator Dθ and of a rotated filter is expressed as [11]

(NθDθ)=(cos(θ)sin(θ)sin(θ)cos(θ))(ND)=(Ncos(θ)Dsin(θ)Nsin(θ)+Dcos(θ)),
where θ is the rotation angle. That implies that a filter N/D rotated and angle θ=π/2has an associated or aliased filter –D/N. Then, from Eqs. (4),11), the rotated two-frame filter for an arbitrary θ is given by
(NθDθ)=(cos(θ)sin(θ)sin(θ)cos(θ))(cos(α/2)[11]sin(α/2)[1  1])=([cos(θ+α/2)cos(θα/2)][sin(θ+α/2)sin(θα/2)]).
Therefore, the numerator and denominator of this rotated filter becomes
NθDθ=[cos(θ+α/2),cos(θα/2)][sin(θ+α/2),sin(θα/2)].
Consequently, the corresponding phase for this rotated two-frame algorithm is
tan(φ)=NθI2DθI2=cos(θ+α/2)I1cos(θα/2)I2sin(θ+α/2)I1sin(θα/2)I2,
and the Fourier transform Hr(ω) of this filter becomes
Hr(ω)=2exp(iθ)sin[(ωα)/2].
For ω=α the quadrature condition Hr(α)=H(α)=0 is satisfied. That is, both rotated and non rotated filters Hr(ω), H(ω) cut off the same frequency α. Then, Eq. (13) for a specific angle θ=α/2, the ratio between numerator and denominator becomes
Nα/2Dα/2=[cosα1][sinα0].
Thus, for an arbitrary frequency αk the individual phase φk recovered is

tan(φk)=NkI2DkI2=[cosαk1]I2[sinαk0]I2=I1cosαkI2I1sinαk.

3. The design of an M order quadrature filter

In a previous work [15], it was proved that a quadrature filter order M can be obtained from a set of M-1 frequencies αk and a convolution algorithm, which implies that a generalized expression for any order with arbitrary conditions can be achieved.

3.1 The general theory of phase shifting interferometry

The estimated phase ϕ of a quadrature filter order M is expressed as [115]

tan(φ)=k=1MbkIkk=1MakIk=[b1b2...bM] IM[a1a2...aM] IM=N IMD IM,
where IM=[I1I2...IM] T is the column vector containing the frames, while, N=[b1b2...bM] and D=[a1a2...aM] are the numerator and denominator row vectors that describe the filter’s behavior. Fortunately, by using the characteristic polynomial P(x) of the filter defined as [8]:
P(x)=k=1M(ak+ibk)xk1.
Then, there are two sets Ak and Bk with M-1 values such that P(x) can be factorized as
P(x)=k=1M1(xkAkiBk).
From the fundamental theorem of the algebra, this polynomial above have exactly M-1 roots xk=Ak+iBk; for k = 1, 2, … M-1. Thus, for each roots an angle αk can be obtained as
αk=tan1(Bk/Ak).
As showed in [15], each value αk has the geometric interpretation as an interception over the frequency axis. That is, the Fourier transform of the filter has a zero at frequency αk. As a consequence, for each frequency αk an individual Fourier transform Hk(ω) that satisfies Hk(αk)=0 is obtained. Therefore, for the M-1 interceptions αk the Fourier transform H(ω) that meets all the desired conditions is given by:
H(ω)=k=1M1Hk(ωαk).
That is, the Fourier transform H(ω) of a quadrature filter order M is expressed as the product of a set of M-1 individual filters Hk(ωαk). Then, the temporal response of the resultant filter h(t) is obtained through the inverse of the Fourier transform of the filter h(t)=1[H(ω)] and expressed as:
h(t)=1[H1(ω)H2.(ω)...HM1(ω)]=h1(t)h2(t)...hM1(t)=Ωk=1M1hk(t).
For simplicity, the symbol Ω is introduced as a convolution operator that denotes the temporal convolution of a set of M-1 individual filters hk(t). Then, each filter hk(t) has an individual time response given by the two Nkand Dk [2x1] row vectors such as
hk(t)=(Dk+i Nk)δ2.
Thus, through the operator Ω, the temporal response of any quadrature filter is expressed as
h(t)=Ωk=1M1[(Dk+i Nk)δ2]=(D+i N)δM,
where, N and D are the vectors defined in Eq. (18). Then, from Eq. (10) the numerator and the denominator of an algorithm is expressed as
ND=Ωk=1M1(NkDk)=(NkDk)(NkDk)(NkDk) ... (NM1DM1).
By using Eq. (13) into Eq. (26) the filter is expressed as
ND=Ωk=1M1{[cos(θk+αk/2),cos(θkαk/2)][sin(θk+αk/2),sin(θkαk/2)]}.
In other words, in the same way that a polynomial with order M-1 is expressed as the product of M-1 first order polynomials, the Fourier transform of a quadrature filter order M is expressed as the product of M-1 individual filters. Where for each filter a two-frame filter is associated. Therefore, through the convolution of this set of M-1 two-frame filters showed above, the desired filter is obtained. Additionally, from Eqs. (15),23) the Fourier transform of the quadrature filter can be recovered as
H(ω)=(2)M1exp(σi)k=1M1sin[(ωαk)/2]    ,  where   σ=k=1M1θk.
For σ=0, H(ω) is a non rotated filter or symmetric algorithm, because σ becomes the real quantity
H(ω)=(2)M1k=1M1sin[(ωαk)/2].
In this work, this case is so called symmetric algorithm and from Eq. (27) results
ND=Ωk=1M1{cos(αk/2)[11]sin(αk/2)[1  1] }.
Otherwise, for any real value σ0 an aliased or rotated filter is recovered and it is called non symmetric algorithm. Therefore, by using the specific angle θ=α/2 this rotated filter becomes
NrDr=Ωk=1M1{[cosαk1][sinαk0]}.
At this point, three goals are achieved by the introduced formalism. The first one is to obtain an M order filter from a set of M-1 arbitrary frequencies (interceptions). The second goal is to analyze the behavior of any given algorithm. Finally, the last goal is to improve a known algorithm adding new properties through the convolution algorithm.

3.2 The tunable non symmetric three-frame algorithm

From Eqs. (26) and (31), a three-frame filter is obtained from two phases α1, and α2 as

ND=[cosα1,1][sinα1,0][cosα2,1][sinα2,0]=[sinα2,0][cosα1,1]+[cosα2,1][sinα1,0][sinα1,0][sinα2,0][cosα1,1][cosα2,1],
because the convolution of two vectors is simply the product of both polynomials. This way, the use of impulse functions has been eliminated, and from Eq. (10) the filter obtained is
ND=[sin(α1+α2),sin(α1)sin(α2),0][cos(α1+α2),cos(α1)+cos(α2),1].
Then, this filter shifted an angle π/2 is the following aliased filter expressed by
Nπ/2Dπ/2=[cos(α1+α2),cos(α1)cos(α2),1][sin(α1+α2),sin(α1)sin(α2),0].
After that, from [12,15] by using the essential quadrature conditions α2=0 and α1=αin Eq. (34) a novel filter is obtained and given by
ND=[sin(α),sin(α),0][cos(α),cos(α)+1,1].
Finally, the desired phase recovered from this tunable rotated three-frame algorithm is
tan(φ)=sin(α)(I1I2)cos(α)(I1I2)+(I2I3).
It should be noticed that this result was obtained without the usage of the Fourier formalism or temporal response algebra. In the same way, from Eq. (11), the estimated phase for this tunable filter shifted π/2 is given by
tan(φ)=cos(α)(I1I2)(I2I3)sin(α)(I1I2),
by using α=π/2 in Eqs. (37) and (38), both famous three-frame algorithms are recovered [4].

3.3 The tunable non symmetric four frame algorithm

From Eqs. (26), (31), and (33) the rotated four-frame filter for arbitrary conditions α1,α2,α3 is

ND=[sin(α1+α2),sin(α1)sin(α2),0][cos(α1+α2),cos(α1)+cos(α2),1][cosα3,1][sinα3,0]=[b1,b2,b3,b4][a1,a2,a3,a4].
Then, from Eq. (10), the coefficients of the numerator and the denominator above become:
b1=cos(α1+α2+α3),
b2=cos(α1+α2)+cos(α1+α3)+cos(α2+α3),
b3=cos(α1)cos(α2)cos(α3),b4=1,
a1=sin(α1+α2+α3),
a2=sin(α1+α2)+sin(α1+α3)+sin(α2+α3),
a3=sin(α1)sin(α2)sin(α3),a4=0.
Thus, by using a3=0 in Eqs. (39)(44) the filter that cuts off the background is given by
ND=[cos(α1+α2),cos(α1+α2)+cos(α1)+cos(α2),1cos(α1)cos(α2),1][sin(α1+α2),sin(α1+α2)+sin(α1)+sin(α2),sin(α1)sin(α2),0],
and from Eq. (11) the filter shifted θ=π/2 is given by
ND=[sin(α1+α2),sin(α1+α2)sin(α1)sin(α2),sin(α1)+sin(α2),0][cos(α1+α2),cos(α1+α2)cos(α1)cos(α2),1+cos(α1)+cos(α2),1].
By means of the values α2=π and α1=α from Eq. (46) the following filter is recovered
ND=[cos(α),1,cos(α),1][sin(α),0,sin(α),0].
According to the reference [11], the above filter should be named as tunable four-frame in cross algorithm [15]. And the estimated phase is
tan(φ)=NIDI=cos(α)(I1I3)I2+I4sin(α)(I1I3).
In the same manner, from Eqs. (11) and (48) the corresponding shifted tunable filter is
tan(φ)=NIDI=sin(α)(I1I3)cos(α)(I1I3)I2+I4
for α=π/2 in Eqs. (48) and (49), the well known four-frames filters are recovered [3,11].

3.4 The tunable non symmetric five-frame algorithm

Likewise, the general five-frame filter for arbitrary conditions α1,α2,α3,α4 is obtained by

ND=Ωk=14{[cosαk,1][sinαk,0]}=[b1,b2,b3,b4,b5][a1,a2,a3,a4,a5].
Then by using the Eq. (33) in Eq. (50) the filter is expressed as
ND=[sin(α1+α2),sin(α1)sin(α2),0][cos(α1+α2),cos(α1)+cos(α2),1][sin(α3+α4),sin(α3)sin(α4),0][cos(α3+α4),cos(α3)+cos(α4),1],
and the coefficients of the filter in Eq. (50) become
b1=sin(α1+α2+α3+α4),
b2=sin(α1+α2+α3)sin(α1+α2+α4)sin(α1+α3+α4)sin(α2+α3+α4),
b3=sin(α1+α2)+sin(α1+α3)+sin(α1+α4)                                                +sin(α2+α3)+sin(α2+α4)+sin(α3+α4),
b4=sin(α1)sin(α2)sin(α3)sin(α4),b5=0,
a1=cos(α1+α2+α3+α4),
a2=cos(α1+α2+α3)cos(α1+α2+α4)cos(α1+α3+α4)cos(α2+α3+α4),
a3=cos(α1+α2)+cos(α1+α3)+cos(α1+α4)                                                           +cos(α2+α3)+cos(α2+α4)+cos(α3+α4),
a4=cos(α1)cos(α2)cos(α3)cos(α4)  and a5=1.
By using α1=α, α2=πα, α3=π and the quadrature condition [12] α4=0 in Eqs. (50)(59) the recovered filter is the well known five-step filter [24]
tan(φ)=NI5DI5=2sin(α)[0,1,0,10]I5[1,0,2,0,1]I5=2sin(α)(I2I4)I1+2I3I5.
From Eq. (50) for the conditions α1=α2=α, α3=πand the quadrature condition [12] α4=0, the tunable five-frame algorithm class A is obtained as
tan(φ)=NI5DI5=sin(2α)(I1I3)2sin(α)(I2I4)cos(2α)(I1I3)+2cos(α)(I2I4)(I3I5).
This filter is insensitive to the linear phase shift detuning error for any phase step α [15].

3.5 The symmetric tunable three-frame algorithm

Repeating the procedure used above a symmetric three-frame filter is expressed as

ND=Ωk=12{cos(αk/2)[1,1]sin(αk/2)[1,  1] }=[sin[(α1+α2)/2],0,sin[(α1+α2)/2]][cos[(α1+α2)/2],2cos[(α1α2)/2],cos[(α1+α2)/2]],
and from the quadrature conditions α2=0 and α1=α the well known three-frame filter is recovered as

tan(φ)=[sin(α/2),0,sin(α/2)]I3[cos(α/2),2cos(α/2),cos(α/2)]I3=tan(α/2)I1I3I1+2I2I3.

3.6 The symmetric tunable four-frame algorithm

From Eq. (31) the symmetric four-frame filter is given by

ND=Ωk=13{cos(αk/2)[11]sin(αk/2)[1  1] }=[b1,b2,b2,b1][a1,a2,a2,a1].
Through the convolution algorithm Eq. (10) the coefficients are given by
b1=cos[(α1+α2+α3)/2],a1=sin[(α1+α2+α3)/2],
b2=cos[(α1+α2α3)/2]+cos[(α1α2+α3)/2]+cos[(α1+α2+α3)/2],
a2=sin[(α1+α2α3)/2]+sin[(α1α2+α3)/2]+sin[(α1+α2+α3)/2].
Then, for the quadrature condition [12] α3=0, the general four-frame filter becomes
b1=cos[(α2+α3)/2],b2=2cos[(α2α3)/2]+cos[(α2+α3)/2]
a1=a2=sin[(α2+α3)/2].
As we expected, for α1=α2=π/2, the four-frame filter class B is recovered [7,15].

3.7 The symmetric and tunable five-frame algorithm

To conclude, from Eq. (30) the expression for a symmetric tunable five-frame filter is

ND=Ωk=14{cos(αk/2)[11]sin(αk/2)[1  1] }=[b1b20b2,b1][a1a2a3a2a1],
b1=sin[(α1+α2+α3+α4)/2],a1=cos[(α1+α2+α3+α4)/2],
b2=sin[(α1+α2+α3α4)/2]+sin[(α1+α2α3+α4)/2]                                      +sin[(α1α2+α3+α4)/2]+sin[(α1+α2+α3+α4)/2],
a2=cos[(α1+α2+α3α4)/2]cos[(α1+α2α3+α4)/2]                                         cos[(α1α2+α3+α4)/2]cos[(α1+α2+α3+α4)/2],
a3=2cos[(α1+α2α3α4)/2]+2cos[(α1α2+α3α4)/2]                                                                                +2cos[(α1+α2+α3α4)/2].
This result comprises all result reported for any symmetric five-frame filter.

For the values α1=α2=α, α3=π and α4=0 the recovered filter is:

tan(φ)=NI5DI5=[cos(α)202,cos(α)]I5sin(α)   [10201]I5=cos(α)(I1I5)+2(I2I4)sin(α)(I12I3+I5).
According to [7] we named this filter above as tunable five-frame algorithm class A. Obviously, for the value α=π/2 becomes the filter reported in [7] and given by
tan(φ)=NI5DI5=[0202,0]I5[10201]I5=2(I2I4)I1+2I3I5.
By using α1=α2=α3=α and α4=0 into Eq. (70), the tunable five-frame filter obtained is
ND=[sin(3α/2)sin(3α/2)+3sin(α/2)0sin(3α/2)3sin(α/2)sin(3α/2)][cos(3α/2)cos(3α/2)3cos(α/2)6sin(α/2)cos(3α/2)3cos(α/2)cos(3α/2)].
Again, according to [7], this filter above should be named as tunable five-frame algorithm class B. Therefore, for α=π/2 the following normalized filter reported in [7] is obtained

tan(φ)=NI5DI5=[-1    4    0    -4    1]I5[-1    -2    6    -2    -1]I5=I14I2+4I4I5I1+2I26I3+2I4+I5.

4. The general non symmetric quadrature filter M order

To express the solutions obtained above the combinatorial theory will be used below.

4.1 Combinatorial theory

For a list of any real frequencies α1,α2,...αn, the notation for sets is standard, and the items are separated by commas and enclosed by curly brackets W = {α1,α2,...αn}, meanwhile the parenthesis denotes a specific combination or arrangement. Then, let W={α1,α2,...αn}rn be a set of different arrangements with r objects chosen from W. For a trivial example, W={a,b,c,d} denotes a set formed by four different numbers a, b, c, and d. From this set W, only four combinations with three objects chosen from W can be made, and they are:

W34={a,b,c,d}34={(a,b,c),(a,b,d),(a,c,d),(b,c,d)}.
In the same way, six different arrangements for two objects chosen from W are:
W24={a,b,c,d}24={(a,b),(a,c),(a,d),(b,c),(b,d),(c,d)}.
The other cases are, W14={a,b,c,d}14={(a),(b),(c),(d)}, and W44={a,b,c,d}44={(a,b,c,d)}. Finally, the arrangement with zero objects chosen from W is the empty set and it is written as: W04={a,b,c,d}04={}. And the number of total combinations is given by
r=0nCnr=r=0nn!(nr)!r!n=2n,
where Cnr is the number of r objects taken from n possibilities.

To express the filters shown in section 3, the usage of combinatorial algebra is required, and through this formalism any quadrature filter can be expressed in an easier and simple expression. Foremost, any filter is easily obtained computationally, even symbolically.

Let Σ be the sum operator, that over a set gives a new set where each element is the sum of each combination in the set. Then, by applying this operator to Eq. (79) the result is

ΣW34=Σ{a,b,c,d}34={(a+b+c),(a+b+d),(a+c+d),(b+c+d)}.
Thus, applying a function over the expression above the following set is obtained:
cos[Σ{a,b,c,d}34]={cos(a+b+c),cos(a+b+d),cos(a+c+d),cos(b+c+d)}.
The operations with a scalar value σ obeys the following algebra rule
σ+(a,b,c,d)=(σ+a,σ+b,σ+c,σ+d).
Combining Eq. (82) into Eq. (84) the set obtained is:
σ+Σ{a,b,c,d}34={(σ+a+b+c),(σ+a+b+d),(σ+a+c+d),(σ+b+c+d)}.
Finally, combining all the rules above the scalar value obtained is:
Σcos(σ+ΣW34)=cos(σ+a+b+c)+cos(σ+a+b+d)+cos(σ+a+c+d)+cos(σ+b+c+d).
That is, through this notation, a filter is expressed analytically in terms of combinatory theory in a simple expression that can easily be evaluated computationally.

4.2 A general tunable non symmetric M-frame algorithm

As showed in section 3, all particular quadrature rotated filters have a well defined model, and from such pattern the generalized expression of an M order rotated filter is expressed as:

ND=Ωk=1M1{[cosαk1][sinαk0]}=[b1b2b3...bM][a1a2a3...aM].
Through the combinatory theory, each element of the numerator is expressed as:
b1=sin[Σ{α1,α2,α3...,αM1}M1M1]=sin(σ),
b2=(1)sin[Σ{α1,α2,α3...,αM1}M2M1]=(1)k=1M1sin(σαk),
and the last element can be expressed as
bM=(1)M+1sin[Σ{α1,α2,α3...,αM1}MMM1]=(1)M+1sin(0)=0,
where the scalar σ is simply the sum of all cut off frequencies (interceptions) and given by
σ=k=1M1αk.
That is, each element of the numerator bk is expressed by:
bk=(1)k+1sin[Σ{α1,α2,α3...,αM1}MkM1]  ;  For 1kM.
In the same way, each element of the denominator ar is expressed by:
a1=cos[Σ{α1,α2,α3...,αM1}M1M1]=cos(σ),
a2=(1)2cos[Σ{α1,α2,α3...,αM1}M2M1]=(1)2k=1M1cos(σαk),
and the last element is:
aM=(1)Mcos[Σ{α1,α2,α3...,αM1}MMM1]=(1)Mcos(0)=(1)M.
Therefore, each element of the denominator ar is generalized as:
ak=(1)kcos[Σ{α1,α2,α3...,αM1}MkM1]  ;  For 1kM.
To conclude, from Eqs. (18), (92), and (96) the desired phase is recovered from the next algorithm
tan(φ)=NIMDIM=k=1M{(1)k+1sin[Σ{α1,α2,...αM1}MkM1]}   Ikk=1M{(1)kcos[Σ{α1,α2,...αM1}MkM1]}   Ik,
and from Eq. (11) the corresponding shifted π/2 filter is,
tan(φ)=NIMDIM=k=1M{(1)k+1cos[Σ{α1,α2,...αM1}MkM1]}   Ikk=1M{(1)k+1sin[Σ{α1,α2,...αM1}MkM1]}   Ik.
That is, the problem of designing a rotated quadrature filter from a set of arbitrary phase steps (any condition) has finally been solved in this paper. From [12], recall that for any filter in this paper α1=0 is the essential condition to obtain a quadrature filter.

4.3 A general symmetric and tunable M-frame algorithm

In the same way, from Eq. (30) any M order symmetric algorithm is expressed by

ND=Ωk=1M1{cos(αk/2)[11]sin(αk/2)[1  1] }=[b1b2b3...bM][a1a2a3...aM].
However, two cases for odd and even order are presented. From section 3, for any M odd order symmetric algorithm, each element of the numerator and the denominator become:
b1=sin[σ/2Σ{α1,α2,α3...,αM1}0M1]=bM=sin(σ/2),
a1=(1)cos[σ/2Σ{α1,α2,α3...,αM1}0M1]=aM=cos(σ/2),
b(m+1)/2=(1)(m+3)/2sin[σ/2Σ{α1,α2,α3...,αM1}(M1)/2M1]=0.
Therefore, each element of the numerator and the denominator are generalized as:
bk=(1)k+1sin[σ/2Σ{α1,α2,α3...,αM1}k1M1]  ;  For 1kM,
ak=(1)kcos[σ/2Σ{α1,α2,α3...,αM1}k1M1]  ;  For 1kM.
As expected, for an M odd order filter the following symmetries br=bMr+1, ar=aMr+1, and b(M+1)/2=0, are satisfied and the phase is recovered from the following algorithm
tan(φ)=NIMDIM=k=1M{(1)k+1sin[σ/2Σ{α1,α2,α3...,αM1}k1M1]}     Ikk=1M{(1)kcos[σ/2Σ{α1,α2,α3...,αM1}k1M1]}     Ik.
On the other hand, although the desired phase for an even M order case can also be obtained from the expression above, but the obtained filter will be a shifted π/2 aliased algorithm. However, from Eq. (11) the symmetric algorithm to recover the phase is obtained as:
tan(φ)=NIMDIM=k=1M{(1)k+1cos[σ/2Σ{α1,α2,α3...,αM1}k1M1]}     Ikk=1M{(1)k+1sin[σ/2Σ{α1,α2,α3...,αM1}k1M1]}     Ik.
That is, for an M even order filter the symmetries br=bMr+1and ar=aMr+1are also satisfied. Where α1=0 is a necessary condition to recover a quadrature filter [12].

5. Some applications

The proposed formalism brings up a new viewpoint to design and to analyze quadrature filters. However, only some illustrative but novel examples are shown below.

5.1 Improving a known algorithm

The proposed formalism can be used to improve a known filter into another with new desired properties. For example, from the well known five-frame filter [3,4] a new seven-frame filter can be obtained. From Eq. (19), the characteristic polynomial is given by [8]

P(x)=k=15(ak+ibk)xk1=(x1)(x+1)[x+cos(α)+isin(α)][xcos(α)+isin(α)].
From the roots of P(x)=0 and Eq. (21), the set of frequencies W={0,π,α,πα} is obtained. Then, the filter is improved by adding the two new frequencies α5=β, and α6=πβ. Therefore, from Eq. (30) the new seven-frame algorithm obtained is
ND={2sin(α)[01010][10201]}{cos(β/2)[11]sin(β/2)[11]}{sin(β/2)[11]cos(β/2)[11]}.
From Eq. (29) the corresponding Fourier transform of the filter becomes
H(ω)=(2)6sin(ω)   [sin(ω)sin(α)][sin(ω)sin(β)].
Thus, by applying Eq. (10) the numerator and the denominator of the filter are:
ND=[10+ 4 sin(α) sin(β0 4 sin(α) sin(β01][02 sin(β)2 sin(α04 sin(α+4sin(β02 sin(β)2 sin(α0].
Therefore, the estimated phase recovered from this seven-frame algorithm is
tan(φ)=NI7D   I7=(I1I7)+[3+4sin(α)sin(β)](I3I5)2[sin(α)+sin(β)](I22I4+I6).
This is, a novel wide-band tunable seven-frame filter was obtained. Then, for β=0 becomes the tunable filter, which is insensitive to the bias modulation error and given by
tan(φ)=(I1I7)+3(I3I5)2sin(α)(I22I4+I6).
By using α=π/2 in the filter above, the algorithm reported in [10] is obtained as:
tan(φ)=I13I3+3I5I72I24I4+2I6.
In the same way, from Eq. (111) by using the values α=π/2 and β=π/6 a wide-band seven-frame filter that practically cuts off all frequencies between {π/6,  5π/6} is obtained as
tan(φ)=I15I3+5I5I73I26I4+3I6.
Finally, by using β=α, in Eq. (111) a novel seven-frame filter is recovered as
tan(φ)=(I1I7)+[3+4sin2(α)](I3I5)4sin(α)(I22I4+I6)=(I1I7)[52cos(2α)](I3I5)4sin(α)(I22I4+I6),
and for the value α=π/2, the seven-frame algorithm reported in [7,8] is given by:

tan(φ)=I17I3+7I5I74(I22I4+I6).

5.2 Example of a tunable six-frame algorithm

The set W={α1,α,α,α,α} where α1=0 corresponds to a quadrature filter that gives up the maximum insensitivity to the miscalibration error is given by

ND=limα10Ωk=14{[cosα1][sinα0]}[cos(α1)1][sin(α1)0].
From Eq. (29) the corresponding Fourier transform of Eq. (117) for α1=0 is simply
H(ω)=(2)5exp(2αi)sin(ω/2)sin4[(ωα)/2].
From Eqs. (89) and (93) for the subset {α,α,α,α} each element of this partial filter yields:
bk=(1)k+1sin[Σ{α,α,α,α}5k4]=(1)k+1(45k)sin[(5k)α]  ;  For 1k5,
ak=(1)kcos[Σ{α,α,α,α}5k4]=(1)k(45k)cos[(5k)α]  ;  For 1k5.
From Eqs. (10), (117), (119), and (120) for α1=0 the desired numerator and denominator become
ND=[cos(4α), cos(4α)4cos(3α), 4cos(3α)+6cos(2α), 4cos(α)6cos(2α), 4cos(α)+1, 1][sin(4α), sin(4α)4sin(3α), 4sin(3α)+6sin(2α), 4sin(α)6sin(2α), 4sin(α),0].
According to the reference [7], the filter above should be named as tunable six-frame filter class B. Thus, by using α=π/2 the filter reported in [7] is yielded as:

tan(φ)=NI6DI6=[116611]I6[044440]I6=I1I26I3+6I4+I5I64(I2I3I4+I5).

5.3 An example of a tunable eleven-frame algorithm

To conclude, an example of an eleven-frame algorithm is given by the following set W={0,0,π,π,α,α,πα,πα,π+α,π+α}. From Eq. (105) the numerator and the denominator of the filter are given by:

N=sin(α)(I1I11)2sin(2α)(I2I10)sin(3α)(I3I9)                                          +4sin(2α)(I4I8)+[3sin(3α)+5sin(α)](I5I7),
D=cos(α)(I1+I11)+2cos(2α)(I2+I10)+[cos(3α) - 4 cos(α)](I3+I9)            -[4 cos(2a) + 4](I4+I8)[cos(3a)  3cos(a)](I5+I7)+[ 4cos(2a)+8]I6.
Then, by substituting α=π/3 the well known eleven-frame filter reported in [8,10] gives

ND=3  [  1,  2,  0, -4, -5,  0,  5,  4,  0, -2, -1][-1,   2,   6,   4,  -5, -12,  -5,   4,   6,   2,  -1].

6. Conclusions

The problem of designing any quadrature filter from a set of interferograms with arbitrary phase shifts has been solved. The problem is solved through the convolution of a set of two-frame filters and expressed in terms of the combinatory theory. Therefore, both the symmetric and the non-symmetric algorithms are reported. The introduced formalism is extremely versatile to obtain new quadrature filters from other known filters. Foremost, to illustrate the formalism, several tunable examples as five-frame, six-frame and eleven-frame algorithms are easily obtained. Additionally, the introduced expressions to recover the quadrature filters are easily evaluated computationally both numerically and symbolically through practically any mathematical software.

Acknowledgments

This work is dedicated to the memory of P. A. Mosiño Alemán. The authors acknowledge the comments and help granted by MDD. J. J. Lozano in the revision of this work.

References and links

1. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13(11), 2693–2703 (1974). [CrossRef]   [PubMed]  

2. J. C. Wyant, “Use of an ac heterodyne lateral shear interferometer with real-time wavefront correction systems,” Appl. Opt. 14(11), 2622–2626 (1975). [CrossRef]   [PubMed]  

3. P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26(13), 2504–2506 (1987). [CrossRef]   [PubMed]  

4. J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf ed., (Elsevier, 1990).

5. K. G. Larkin and B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9(10), 1740–1748 (1992). [CrossRef]  

6. P. Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34(22), 4723–4730 (1995). [CrossRef]   [PubMed]  

7. J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34(19), 3610–3619 (1995). [CrossRef]   [PubMed]  

8. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51–60 (1996). [CrossRef]   [PubMed]  

9. D. W. Phillion, “General methods for generating phase-shifting interferometry algorithms,” Appl. Opt. 36(31), 8098–8115 (1997). [CrossRef]  

10. K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A 14(4), 918–930 (1997). [CrossRef]  

11. H. Schreiber, J. H. Brunning, and J. E. Greivenkamp, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara ed., (John Wiley & Sons, Inc., 2007).

12. J. F. Mosiño, D. M. Doblado, and D. M. Hernández, “A method to design tunable quadrature filters in phase shifting interferometry,” Opt. Express 17(18), 15772–15777 (2009). [CrossRef]   [PubMed]  

13. M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009). [CrossRef]   [PubMed]  

14. A. Téllez-Quiñones and D. Malacara-Doblado, “Inhomogeneous phase shifting: an algorithm for nonconstant phase displacements,” Appl. Opt. 49(32), 6224–6231 (2010). [CrossRef]   [PubMed]  

15. J. F. Mosiño, J. C. Gutiérrez-García, T. A. Gutiérrez-García, and J. M. Macías-Preza, “Two-frame algorithm to design quadrature filters in phase shifting interferometry,” Opt. Express 18(24), 24405–24411 (2010). [CrossRef]   [PubMed]  

16. Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29(14), 1671–1673 (2004). [CrossRef]   [PubMed]  

References

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  1. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13(11), 2693–2703 (1974).
    [Crossref] [PubMed]
  2. J. C. Wyant, “Use of an ac heterodyne lateral shear interferometer with real-time wavefront correction systems,” Appl. Opt. 14(11), 2622–2626 (1975).
    [Crossref] [PubMed]
  3. P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26(13), 2504–2506 (1987).
    [Crossref] [PubMed]
  4. J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf ed., (Elsevier, 1990).
  5. K. G. Larkin and B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9(10), 1740–1748 (1992).
    [Crossref]
  6. P. Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34(22), 4723–4730 (1995).
    [Crossref] [PubMed]
  7. J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34(19), 3610–3619 (1995).
    [Crossref] [PubMed]
  8. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51–60 (1996).
    [Crossref] [PubMed]
  9. D. W. Phillion, “General methods for generating phase-shifting interferometry algorithms,” Appl. Opt. 36(31), 8098–8115 (1997).
    [Crossref]
  10. K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A 14(4), 918–930 (1997).
    [Crossref]
  11. H. Schreiber, J. H. Brunning, and J. E. Greivenkamp, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara ed., (John Wiley & Sons, Inc., 2007).
  12. J. F. Mosiño, D. M. Doblado, and D. M. Hernández, “A method to design tunable quadrature filters in phase shifting interferometry,” Opt. Express 17(18), 15772–15777 (2009).
    [Crossref] [PubMed]
  13. M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009).
    [Crossref] [PubMed]
  14. A. Téllez-Quiñones and D. Malacara-Doblado, “Inhomogeneous phase shifting: an algorithm for nonconstant phase displacements,” Appl. Opt. 49(32), 6224–6231 (2010).
    [Crossref] [PubMed]
  15. J. F. Mosiño, J. C. Gutiérrez-García, T. A. Gutiérrez-García, and J. M. Macías-Preza, “Two-frame algorithm to design quadrature filters in phase shifting interferometry,” Opt. Express 18(24), 24405–24411 (2010).
    [Crossref] [PubMed]
  16. Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29(14), 1671–1673 (2004).
    [Crossref] [PubMed]

2010 (2)

2009 (2)

2004 (1)

1997 (2)

1996 (1)

1995 (2)

1992 (1)

1987 (1)

1975 (1)

1974 (1)

Brangaccio, D. J.

Bruning, J. H.

Creath, K.

Doblado, D. M.

Eiju, T.

Estrada, J. C.

Farrant, D. I.

Gallagher, J. E.

Groot, P.

Gutiérrez-García, J. C.

Gutiérrez-García, T. A.

Han, B.

Hariharan, P.

Hernández, D. M.

Herriott, D. R.

Hibino, K.

Larkin, K. G.

Macías-Preza, J. M.

Malacara-Doblado, D.

Mosiño, J. F.

Oreb, B. F.

Phillion, D. W.

Quiroga, J. A.

Rosenfeld, D. P.

Schmit, J.

Servin, M.

Surrel, Y.

Téllez-Quiñones, A.

Wang, Z.

White, A. D.

Wyant, J. C.

Appl. Opt. (8)

J. Opt. Soc. Am. A (2)

Opt. Express (3)

Opt. Lett. (1)

Other (2)

H. Schreiber, J. H. Brunning, and J. E. Greivenkamp, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara ed., (John Wiley & Sons, Inc., 2007).

J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf ed., (Elsevier, 1990).

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Equations (125)

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I k = a ( x , y ) + b ( x , y ) cos [ α t k + φ ( x , y ) ]   For  k = 1 and 2,
tan [ φ ( x , y ) ] = N I 2 D I 2 = cos ( α / 2 ) [ 1 1 ] I 2 sin ( α / 2 ) [ 1    1 ]   I 2 = cos ( α / 2 ) sin ( α / 2 ) [ I 1 I 2 I 1 + I 2 ] ,
H ( ω ) = 2 sin [ ( ω α ) / 2 ] .
N D = cos ( α / 2 ) sin ( α / 2 ) [ 1 1 ] [ 1    1 ]   .
tan ( φ 1 ) = N 1   I n D 1   I n ,  and   tan ( φ 2 ) = N 2   I m D 2   I m ,
h 1 ( t ) = ( D 1 + i   N 1 ) δ n   and   h 2 ( t ) = ( D 2 + i   N 2 )     δ m ,
δ n = [ δ ( t ) , δ ( t α ) , ... δ ( t n α ) ] T   and   δ m = [ δ ( t ) , δ ( t α ) , ... δ ( t m α ) ] T .
h ( t ) = [ D 1 δ n + i   N 1 δ n ] [ D 2 δ m + i   N 2 δ m ]                                       = [ D 1 D 2 N 1 N 2 ] δ n + m 1 + i [ N 1 D 2 + D 1 N 2 ] δ n + m 1 ,
tan [ φ ( x , y ) ] = N   I n + m 1 D   I n + m 1 = [ N 1 D 2 + D 1 N 2 ]   I n + m 1 [ D 1 D 2 N 1 N 2 ]   I n + m 1 .
N D = ( N 1 D 1 ) ( N 2 D 2 ) = N 1 D 2 + D 1 N 2 D 1 D 2 N 1 N 2 .
( N θ D θ ) = ( cos ( θ ) sin ( θ ) sin ( θ ) cos ( θ ) ) ( N D ) = ( N cos ( θ ) D sin ( θ ) N sin ( θ ) + D cos ( θ ) ) ,
( N θ D θ ) = ( cos ( θ ) sin ( θ ) sin ( θ ) cos ( θ ) ) ( cos ( α / 2 ) [ 1 1 ] sin ( α / 2 ) [ 1    1 ] ) = ( [ cos ( θ + α / 2 ) cos ( θ α / 2 ) ] [ sin ( θ + α / 2 ) sin ( θ α / 2 ) ] ) .
N θ D θ = [ cos ( θ + α / 2 ) , cos ( θ α / 2 ) ] [ sin ( θ + α / 2 ) , sin ( θ α / 2 ) ] .
tan ( φ ) = N θ I 2 D θ I 2 = cos ( θ + α / 2 ) I 1 cos ( θ α / 2 ) I 2 sin ( θ + α / 2 ) I 1 sin ( θ α / 2 ) I 2 ,
H r ( ω ) = 2 exp ( i θ ) sin [ ( ω α ) / 2 ] .
N α / 2 D α / 2 = [ cos α 1 ] [ sin α 0 ] .
tan ( φ k ) = N k I 2 D k I 2 = [ cos α k 1 ] I 2 [ sin α k 0 ] I 2 = I 1 cos α k I 2 I 1 sin α k .
tan ( φ ) = k = 1 M b k I k k = 1 M a k I k = [ b 1 b 2 ... b M ]   I M [ a 1 a 2 ... a M ]   I M = N   I M D   I M ,
P ( x ) = k = 1 M ( a k + i b k ) x k 1 .
P ( x ) = k = 1 M 1 ( x k A k i B k ) .
α k = tan 1 ( B k / A k ) .
H ( ω ) = k = 1 M 1 H k ( ω α k ) .
h ( t ) = 1 [ H 1 ( ω ) H 2. ( ω ) ... H M 1 ( ω ) ] = h 1 ( t ) h 2 ( t ) ... h M 1 ( t ) = Ω k = 1 M 1 h k ( t ) .
h k ( t ) = ( D k + i   N k ) δ 2 .
h ( t ) = Ω k = 1 M 1 [ ( D k + i   N k ) δ 2 ] = ( D + i   N ) δ M ,
N D = Ω k = 1 M 1 ( N k D k ) = ( N k D k ) ( N k D k ) ( N k D k )   ...   ( N M 1 D M 1 ) .
N D = Ω k = 1 M 1 { [ cos ( θ k + α k / 2 ) , cos ( θ k α k / 2 ) ] [ sin ( θ k + α k / 2 ) , sin ( θ k α k / 2 ) ] } .
H ( ω ) = ( 2 ) M 1 exp ( σ i ) k = 1 M 1 sin [ ( ω α k ) / 2 ]     ,  where    σ = k = 1 M 1 θ k .
H ( ω ) = ( 2 ) M 1 k = 1 M 1 sin [ ( ω α k ) / 2 ] .
N D = Ω k = 1 M 1 { cos ( α k / 2 ) [ 1 1 ] sin ( α k / 2 ) [ 1    1 ]   } .
N r D r = Ω k = 1 M 1 { [ cos α k 1 ] [ sin α k 0 ] } .
N D = [ cos α 1 , 1 ] [ sin α 1 , 0 ] [ cos α 2 , 1 ] [ sin α 2 , 0 ] = [ sin α 2 , 0 ] [ cos α 1 , 1 ] + [ cos α 2 , 1 ] [ sin α 1 , 0 ] [ sin α 1 , 0 ] [ sin α 2 , 0 ] [ cos α 1 , 1 ] [ cos α 2 , 1 ] ,
N D = [ sin ( α 1 + α 2 ) , sin ( α 1 ) sin ( α 2 ) , 0 ] [ cos ( α 1 + α 2 ) , cos ( α 1 ) + cos ( α 2 ) , 1 ] .
N π / 2 D π / 2 = [ cos ( α 1 + α 2 ) , cos ( α 1 ) cos ( α 2 ) , 1 ] [ sin ( α 1 + α 2 ) , sin ( α 1 ) sin ( α 2 ) , 0 ] .
N D = [ sin ( α ) , sin ( α ) , 0 ] [ cos ( α ) , cos ( α ) + 1 , 1 ] .
tan ( φ ) = sin ( α ) ( I 1 I 2 ) cos ( α ) ( I 1 I 2 ) + ( I 2 I 3 ) .
tan ( φ ) = cos ( α ) ( I 1 I 2 ) ( I 2 I 3 ) sin ( α ) ( I 1 I 2 ) ,
N D = [ sin ( α 1 + α 2 ) , sin ( α 1 ) sin ( α 2 ) , 0 ] [ cos ( α 1 + α 2 ) , cos ( α 1 ) + cos ( α 2 ) , 1 ] [ cos α 3 , 1 ] [ sin α 3 , 0 ] = [ b 1 , b 2 , b 3 , b 4 ] [ a 1 , a 2 , a 3 , a 4 ] .
b 1 = cos ( α 1 + α 2 + α 3 ) ,
b 2 = cos ( α 1 + α 2 ) + cos ( α 1 + α 3 ) + cos ( α 2 + α 3 ) ,
b 3 = cos ( α 1 ) cos ( α 2 ) cos ( α 3 ) , b 4 = 1 ,
a 1 = sin ( α 1 + α 2 + α 3 ) ,
a 2 = sin ( α 1 + α 2 ) + sin ( α 1 + α 3 ) + sin ( α 2 + α 3 ) ,
a 3 = sin ( α 1 ) sin ( α 2 ) sin ( α 3 ) , a 4 = 0.
N D = [ cos ( α 1 + α 2 ) , cos ( α 1 + α 2 ) + cos ( α 1 ) + cos ( α 2 ) , 1 cos ( α 1 ) cos ( α 2 ) , 1 ] [ sin ( α 1 + α 2 ) , sin ( α 1 + α 2 ) + sin ( α 1 ) + sin ( α 2 ) , sin ( α 1 ) sin ( α 2 ) , 0 ] ,
N D = [ sin ( α 1 + α 2 ) , sin ( α 1 + α 2 ) sin ( α 1 ) sin ( α 2 ) , sin ( α 1 ) + sin ( α 2 ) , 0 ] [ cos ( α 1 + α 2 ) , cos ( α 1 + α 2 ) cos ( α 1 ) cos ( α 2 ) , 1 + cos ( α 1 ) + cos ( α 2 ) , 1 ] .
N D = [ cos ( α ) , 1 , cos ( α ) , 1 ] [ sin ( α ) , 0 , sin ( α ) , 0 ] .
tan ( φ ) = N I D I = cos ( α ) ( I 1 I 3 ) I 2 + I 4 sin ( α ) ( I 1 I 3 ) .
tan ( φ ) = N I D I = sin ( α ) ( I 1 I 3 ) cos ( α ) ( I 1 I 3 ) I 2 + I 4
N D = Ω k = 1 4 { [ cos α k , 1 ] [ sin α k , 0 ] } = [ b 1 , b 2 , b 3 , b 4 , b 5 ] [ a 1 , a 2 , a 3 , a 4 , a 5 ] .
N D = [ sin ( α 1 + α 2 ) , sin ( α 1 ) sin ( α 2 ) , 0 ] [ cos ( α 1 + α 2 ) , cos ( α 1 ) + cos ( α 2 ) , 1 ] [ sin ( α 3 + α 4 ) , sin ( α 3 ) sin ( α 4 ) , 0 ] [ cos ( α 3 + α 4 ) , cos ( α 3 ) + cos ( α 4 ) , 1 ] ,
b 1 = sin ( α 1 + α 2 + α 3 + α 4 ) ,
b 2 = sin ( α 1 + α 2 + α 3 ) sin ( α 1 + α 2 + α 4 ) sin ( α 1 + α 3 + α 4 ) sin ( α 2 + α 3 + α 4 ) ,
b 3 = sin ( α 1 + α 2 ) + sin ( α 1 + α 3 ) + sin ( α 1 + α 4 )                                                  + sin ( α 2 + α 3 ) + sin ( α 2 + α 4 ) + sin ( α 3 + α 4 ) ,
b 4 = sin ( α 1 ) sin ( α 2 ) sin ( α 3 ) sin ( α 4 ) , b 5 = 0 ,
a 1 = cos ( α 1 + α 2 + α 3 + α 4 ) ,
a 2 = cos ( α 1 + α 2 + α 3 ) cos ( α 1 + α 2 + α 4 ) cos ( α 1 + α 3 + α 4 ) cos ( α 2 + α 3 + α 4 ) ,
a 3 = cos ( α 1 + α 2 ) + cos ( α 1 + α 3 ) + cos ( α 1 + α 4 )                                                             + cos ( α 2 + α 3 ) + cos ( α 2 + α 4 ) + cos ( α 3 + α 4 ) ,
a 4 = cos ( α 1 ) cos ( α 2 ) cos ( α 3 ) cos ( α 4 )   and  a 5 = 1.
tan ( φ ) = N I 5 D I 5 = 2 sin ( α ) [ 0 , 1 , 0 , 1 0 ] I 5 [ 1 , 0 , 2 , 0 , 1 ] I 5 = 2 sin ( α ) ( I 2 I 4 ) I 1 + 2 I 3 I 5 .
tan ( φ ) = N I 5 D I 5 = sin ( 2 α ) ( I 1 I 3 ) 2 sin ( α ) ( I 2 I 4 ) cos ( 2 α ) ( I 1 I 3 ) + 2 cos ( α ) ( I 2 I 4 ) ( I 3 I 5 ) .
N D = Ω k = 1 2 { cos ( α k / 2 ) [ 1 , 1 ] sin ( α k / 2 ) [ 1 ,    1 ]   } = [ sin [ ( α 1 + α 2 ) / 2 ] , 0 , sin [ ( α 1 + α 2 ) / 2 ] ] [ cos [ ( α 1 + α 2 ) / 2 ] , 2 cos [ ( α 1 α 2 ) / 2 ] , cos [ ( α 1 + α 2 ) / 2 ] ] ,
tan ( φ ) = [ sin ( α / 2 ) , 0 , sin ( α / 2 ) ] I 3 [ cos ( α / 2 ) , 2 cos ( α / 2 ) , cos ( α / 2 ) ] I 3 = tan ( α / 2 ) I 1 I 3 I 1 + 2 I 2 I 3 .
N D = Ω k = 1 3 { cos ( α k / 2 ) [ 1 1 ] sin ( α k / 2 ) [ 1    1 ]   } = [ b 1 , b 2 , b 2 , b 1 ] [ a 1 , a 2 , a 2 , a 1 ] .
b 1 = cos [ ( α 1 + α 2 + α 3 ) / 2 ] , a 1 = sin [ ( α 1 + α 2 + α 3 ) / 2 ] ,
b 2 = cos [ ( α 1 + α 2 α 3 ) / 2 ] + cos [ ( α 1 α 2 + α 3 ) / 2 ] + cos [ ( α 1 + α 2 + α 3 ) / 2 ] ,
a 2 = sin [ ( α 1 + α 2 α 3 ) / 2 ] + sin [ ( α 1 α 2 + α 3 ) / 2 ] + sin [ ( α 1 + α 2 + α 3 ) / 2 ] .
b 1 = cos [ ( α 2 + α 3 ) / 2 ] , b 2 = 2 cos [ ( α 2 α 3 ) / 2 ] + cos [ ( α 2 + α 3 ) / 2 ]
a 1 = a 2 = sin [ ( α 2 + α 3 ) / 2 ] .
N D = Ω k = 1 4 { cos ( α k / 2 ) [ 1 1 ] sin ( α k / 2 ) [ 1    1 ]   } = [ b 1 b 2 0 b 2 , b 1 ] [ a 1 a 2 a 3 a 2 a 1 ] ,
b 1 = sin [ ( α 1 + α 2 + α 3 + α 4 ) / 2 ] , a 1 = cos [ ( α 1 + α 2 + α 3 + α 4 ) / 2 ] ,
b 2 = sin [ ( α 1 + α 2 + α 3 α 4 ) / 2 ] + sin [ ( α 1 + α 2 α 3 + α 4 ) / 2 ]                                        + sin [ ( α 1 α 2 + α 3 + α 4 ) / 2 ] + sin [ ( α 1 + α 2 + α 3 + α 4 ) / 2 ] ,
a 2 = cos [ ( α 1 + α 2 + α 3 α 4 ) / 2 ] cos [ ( α 1 + α 2 α 3 + α 4 ) / 2 ]                                           cos [ ( α 1 α 2 + α 3 + α 4 ) / 2 ] cos [ ( α 1 + α 2 + α 3 + α 4 ) / 2 ] ,
a 3 = 2 cos [ ( α 1 + α 2 α 3 α 4 ) / 2 ] + 2 cos [ ( α 1 α 2 + α 3 α 4 ) / 2 ]                                                                                  + 2 cos [ ( α 1 + α 2 + α 3 α 4 ) / 2 ] .
tan ( φ ) = N I 5 D I 5 = [ cos ( α ) 2 0 2 , cos ( α ) ] I 5 sin ( α )     [ 1 0 2 0 1 ] I 5 = cos ( α ) ( I 1 I 5 ) + 2 ( I 2 I 4 ) sin ( α ) ( I 1 2 I 3 + I 5 ) .
tan ( φ ) = N I 5 D I 5 = [ 0 2 0 2 , 0 ] I 5 [ 1 0 2 0 1 ] I 5 = 2 ( I 2 I 4 ) I 1 + 2 I 3 I 5 .
N D = [ sin ( 3 α / 2 ) sin ( 3 α / 2 ) + 3 sin ( α / 2 ) 0 sin ( 3 α / 2 ) 3 sin ( α / 2 ) sin ( 3 α / 2 ) ] [ cos ( 3 α / 2 ) cos ( 3 α / 2 ) 3 cos ( α / 2 ) 6 sin ( α / 2 ) cos ( 3 α / 2 ) 3 cos ( α / 2 ) cos ( 3 α / 2 ) ] .
tan ( φ ) = N I 5 D I 5 = [-1    4    0    -4    1] I 5 [-1    -2    6    -2    -1] I 5 = I 1 4 I 2 + 4 I 4 I 5 I 1 + 2 I 2 6 I 3 + 2 I 4 + I 5 .
W 3 4 = { a , b , c , d } 3 4 = { ( a , b , c ) , ( a , b , d ) , ( a , c , d ) , ( b , c , d ) } .
W 2 4 = { a , b , c , d } 2 4 = { ( a , b ) , ( a , c ) , ( a , d ) , ( b , c ) , ( b , d ) , ( c , d ) } .
r = 0 n C n r = r = 0 n n ! ( n r ) ! r ! n = 2 n ,
Σ W 3 4 = Σ { a , b , c , d } 3 4 = { ( a + b + c ) , ( a + b + d ) , ( a + c + d ) , ( b + c + d ) } .
cos [ Σ { a , b , c , d } 3 4 ] = { cos ( a + b + c ) , cos ( a + b + d ) , cos ( a + c + d ) , cos ( b + c + d ) } .
σ + ( a , b , c , d ) = ( σ + a , σ + b , σ + c , σ + d ) .
σ + Σ { a , b , c , d } 3 4 = { ( σ + a + b + c ) , ( σ + a + b + d ) , ( σ + a + c + d ) , ( σ + b + c + d ) } .
Σ cos ( σ + Σ W 3 4 ) = cos ( σ + a + b + c ) + cos ( σ + a + b + d ) + cos ( σ + a + c + d ) + cos ( σ + b + c + d ) .
N D = Ω k = 1 M 1 { [ cos α k 1 ] [ sin α k 0 ] } = [ b 1 b 2 b 3 ... b M ] [ a 1 a 2 a 3 ... a M ] .
b 1 = sin [ Σ { α 1 , α 2 , α 3 ... , α M 1 } M 1 M 1 ] = sin ( σ ) ,
b 2 = ( 1 ) sin [ Σ { α 1 , α 2 , α 3 ... , α M 1 } M 2 M 1 ] = ( 1 ) k = 1 M 1 sin ( σ α k ) ,
b M = ( 1 ) M + 1 sin [ Σ { α 1 , α 2 , α 3 ... , α M 1 } M M M 1 ] = ( 1 ) M + 1 sin ( 0 ) = 0 ,
σ = k = 1 M 1 α k .
b k = ( 1 ) k + 1 sin [ Σ { α 1 , α 2 , α 3 ... , α M 1 } M k M 1 ]   ;  For  1 k M .
a 1 = cos [ Σ { α 1 , α 2 , α 3 ... , α M 1 } M 1 M 1 ] = cos ( σ ) ,
a 2 = ( 1 ) 2 cos [ Σ { α 1 , α 2 , α 3 ... , α M 1 } M 2 M 1 ] = ( 1 ) 2 k = 1 M 1 cos ( σ α k ) ,
a M = ( 1 ) M cos [ Σ { α 1 , α 2 , α 3 ... , α M 1 } M M M 1 ] = ( 1 ) M cos ( 0 ) = ( 1 ) M .
a k = ( 1 ) k cos [ Σ { α 1 , α 2 , α 3 ... , α M 1 } M k M 1 ]   ;  For  1 k M .
tan ( φ ) = N I M D I M = k = 1 M { ( 1 ) k + 1 sin [ Σ { α 1 , α 2 , ... α M 1 } M k M 1 ] }     I k k = 1 M { ( 1 ) k cos [ Σ { α 1 , α 2 , ... α M 1 } M k M 1 ] }     I k ,
tan ( φ ) = N I M D I M = k = 1 M { ( 1 ) k + 1 cos [ Σ { α 1 , α 2 , ... α M 1 } M k M 1 ] }     I k k = 1 M { ( 1 ) k + 1 sin [ Σ { α 1 , α 2 , ... α M 1 } M k M 1 ] }     I k .
N D = Ω k = 1 M 1 { cos ( α k / 2 ) [ 1 1 ] sin ( α k / 2 ) [ 1    1 ]   } = [ b 1 b 2 b 3 ... b M ] [ a 1 a 2 a 3 ... a M ] .
b 1 = sin [ σ / 2 Σ { α 1 , α 2 , α 3 ... , α M 1 } 0 M 1 ] = b M = sin ( σ / 2 ) ,
a 1 = ( 1 ) cos [ σ / 2 Σ { α 1 , α 2 , α 3 ... , α M 1 } 0 M 1 ] = a M = cos ( σ / 2 ) ,
b ( m + 1 ) / 2 = ( 1 ) ( m + 3 ) / 2 sin [ σ / 2 Σ { α 1 , α 2 , α 3 ... , α M 1 } ( M 1 ) / 2 M 1 ] = 0.
b k = ( 1 ) k + 1 sin [ σ / 2 Σ { α 1 , α 2 , α 3 ... , α M 1 } k 1 M 1 ]   ;  For  1 k M ,
a k = ( 1 ) k cos [ σ / 2 Σ { α 1 , α 2 , α 3 ... , α M 1 } k 1 M 1 ]   ;  For  1 k M .
tan ( φ ) = N I M D I M = k = 1 M { ( 1 ) k + 1 sin [ σ / 2 Σ { α 1 , α 2 , α 3 ... , α M 1 } k 1 M 1 ] }       I k k = 1 M { ( 1 ) k cos [ σ / 2 Σ { α 1 , α 2 , α 3 ... , α M 1 } k 1 M 1 ] }       I k .
tan ( φ ) = N I M D I M = k = 1 M { ( 1 ) k + 1 cos [ σ / 2 Σ { α 1 , α 2 , α 3 ... , α M 1 } k 1 M 1 ] }       I k k = 1 M { ( 1 ) k + 1 sin [ σ / 2 Σ { α 1 , α 2 , α 3 ... , α M 1 } k 1 M 1 ] }       I k .
P ( x ) = k = 1 5 ( a k + i b k ) x k 1 = ( x 1 ) ( x + 1 ) [ x + cos ( α ) + i sin ( α ) ] [ x cos ( α ) + i sin ( α ) ] .
N D = { 2 sin ( α ) [ 0 1 0 1 0 ] [ 1 0 2 0 1 ] } { cos ( β / 2 ) [ 1 1 ] sin ( β / 2 ) [ 1 1 ] } { sin ( β / 2 ) [ 1 1 ] cos ( β / 2 ) [ 1 1 ] } .
H ( ω ) = ( 2 ) 6 sin ( ω )     [ sin ( ω ) sin ( α ) ] [ sin ( ω ) sin ( β ) ] .
N D = [ 1 0 +  4 sin( α ) sin( β 0  4 sin( α ) sin( β 0 1 ] [ 0 2 sin( β ) 2 sin( α 0 4 sin( α + 4sin( β 0 2 sin( β ) 2 sin( α 0 ] .
tan ( φ ) = N I 7 D     I 7 = ( I 1 I 7 ) + [ 3 + 4 sin ( α ) sin ( β ) ] ( I 3 I 5 ) 2 [ sin ( α ) + sin ( β ) ] ( I 2 2 I 4 + I 6 ) .
tan ( φ ) = ( I 1 I 7 ) + 3 ( I 3 I 5 ) 2 sin ( α ) ( I 2 2 I 4 + I 6 ) .
tan ( φ ) = I 1 3 I 3 + 3 I 5 I 7 2 I 2 4 I 4 + 2 I 6 .
tan ( φ ) = I 1 5 I 3 + 5 I 5 I 7 3 I 2 6 I 4 + 3 I 6 .
tan ( φ ) = ( I 1 I 7 ) + [ 3 + 4 sin 2 ( α ) ] ( I 3 I 5 ) 4 sin ( α ) ( I 2 2 I 4 + I 6 ) = ( I 1 I 7 ) [ 5 2 cos ( 2 α ) ] ( I 3 I 5 ) 4 sin ( α ) ( I 2 2 I 4 + I 6 ) ,
tan ( φ ) = I 1 7 I 3 + 7 I 5 I 7 4 ( I 2 2 I 4 + I 6 ) .
N D = lim α 1 0 Ω k = 1 4 { [ cos α 1 ] [ sin α 0 ] } [ cos ( α 1 ) 1 ] [ sin ( α 1 ) 0 ] .
H ( ω ) = ( 2 ) 5 exp ( 2 α i ) sin ( ω / 2 ) sin 4 [ ( ω α ) / 2 ] .
b k = ( 1 ) k + 1 sin [ Σ { α , α , α , α } 5 k 4 ] = ( 1 ) k + 1 ( 4 5 k ) sin [ ( 5 k ) α ]   ;  For  1 k 5 ,
a k = ( 1 ) k cos [ Σ { α , α , α , α } 5 k 4 ] = ( 1 ) k ( 4 5 k ) cos [ ( 5 k ) α ]