Passive interferometric array optimization based on redundant spacing calibration

Yuntao He; Yuesong Jiang; Li Liu; Changwei Wang

doi:10.1364/OE.17.021598

1. Introduction

In the past decades, passive synthetic aperture interferometric imaging has been widely implemented in surveillance and detection applications [1–6]. The development of this technique is driven by the desire to achieve relatively high resolution with relatively small collecting area. That is primarily because larger collecting area brings on cost and weight, and the latter consideration is especially important in air- and space-borne systems [7,8]. The range of operation frequency has almost covered all the frequencies from radio to visible [1–8]. Interferometric telescope array imaging is an attractive approach for high angular resolution imaging. However, sparse telescope arrays have limited (u, v)-plane coverage. The relative arrangement of the elementary telescopes (the so-called aperture configuration, or pupil configuration) is a key aspect in design of synthetic-aperture instruments.

There are two ways to form an image according to the imaging requirement. In radio astronomy observations, partial spatial frequency coverage is sufficient and the image is obtained by Fourier inversion of the asynchronously collected spatial spectrum. However, when there is no simple model for observation targets or when field confusion is a problem, dense (u, v)-plane coverage is needed, especially in instantaneous imaging systems. Many nonredundant array geometries with good instantaneous (u, v)-plane coverage have already been published [9–12]. Moffet has given a class of linear arrays with maximum resolution for a fixed number of elements by minimizing the number of redundant spacings in the array [9]. Golay presented two-dimensional arrays with nonredundant and compact autocorrelation functions with optimal or nearly optimal properties [10]. Cornwell has mentioned circular crystalline arrays for achieving uniform coverage in the (u, v)-plane [11], while Lannes showed a different way to optimize the circular array based on the criterion of the Field-to-Resolution Ratio (FRR) [12]. Besides, much research has been done for the array configuration optimization [13–22].

With the study development of passive interferometric array optimization, there are more factors considered. Phase calibration is one of the most important steps for the image reconstruction. It is noticed that controlling the phase of each antenna (or aperture) to an accuracy of less than one tenth of a wavelength can achieve high quality images [2–5,23–25].The redundant spacing calibration (RSC) derived from phase closure technique has been proved to be a very useful and effective approach for phase error correction in passive interferometric synthetic aperture imaging systems [6,23–26]. It has been used to produce high dynamic range images of the star earlier [2–4]. The advantage of RSC is that it can be applied at any electromagnetic wavelength, and it is model-independent and one-iteration solution [6,23–25]. While the drawback is that the array must be designed to meet the redundancy requirement. That means there should be n-3 redundant baselines at least for an n-element array [6,23]. Some linear arrays and “Y” shape arrays designed based on RSC have been proved effective and practical [6,25–27]. While in passive interferometric circular arrays, the placements of array elements based on RSC are different. And the arrays should be optimized to achieve maximum uniformity and minimum redundancy of spatial frequency.

In this paper, we discuss the phase correction principle based on RSC and constrains for optimizing passive interferometric circular arrays in section 2. The third section accounts for the optimization of circular arrays based on RSC by simulated annealing algorithm, and the placements are listed out for 8~16 elements circular arrays. An example of phase error correction to the optimized ten-element array based on RSC is investigated in section 3.4.

2. Principles of redundant spacing calibration

In a passive interferometric synthetic aperture imaging system, the Fourier transform of the brightness distribution of a self-luminous object can be determined by measuring the cross-correlation (visibility function) between the complex amplitudes of a series of spatial separated aperture elements. Any aberration from these points will lead to not only errors in phase measurement of the cross-correlations, but also image quality degradation ultimately.

For an n-aperture array, the number of baselines (spatial frequency samples) in this array is given by:

p = n (n - 1) / 2

Considering each baseline pair (i, j), the phase of measured cross-correlation is composed of two parts [24]:

m_{i j} = ϕ_{i j} + (e_{i} - e_{j})

Where

ϕ_{i j}

is the true baseline phase;

e_{i}

and

e_{i}

are the channel phase errors respectively which vary randomly and instantaneously. So

(e_{i} - e_{j})

is the phase error for baseline (i,j), and could not be known in advance. The input phase error due to the system modeling is removed by testing beforehand and not exists in Eq. (2) [24]. So all of these p equations can be combined in matrix form as:

M_{p \times 1} = Φ_{p \times 1} + B_{p \times n} \times E_{n \times 1}

The subscripts in Eq. (3) stand for the matrix dimension, the same as follows (just for the capital letters). Here, M is the vector of p measurements; Φ is the vector of true baseline phases, and B is the phase error operator from all the phase errors vector E to the measured phase vector M. The above matrix equation can also be described as:

\begin{array}{l} M_{p \times 1} = I_{p \times p} \times Φ_{p \times 1} + B_{p \times n} \times E_{n \times 1} \\ = [I_{p \times p} ⋮ B_{p \times n}] \times [\begin{matrix} Φ_{p \times 1} \\ E_{n \times 1} \end{matrix}] = A_{p \times (p + n)} \times U_{(p + n) \times 1} \end{array}

Here I stands for a unit matrix. A is the coefficient matrix. From the Eq. (4), we know that there are $(p + n)$ unknown quantities while contrasting against plinear equations, which cannot lead to a unique solution.

Specially designed arrays based on the principle of RSC, have enough redundancy to permit calibration of the unknown antenna phase errors, leading to a unique solution [6,23–27]. In fact, each pair of elements would measure the same Fourier component (point of visibility function) of the scene in RSC technique when the two pairs of elements have the same baseline vector. If the complex visibilities are different, the difference can only be produced due to instrumental and atmospherical aberrations. Utilizing RSC, such phase errors can be distinguished without model-building or prior assumptions under condition of imaging a far-field incoherent radiation [23,24].

The redundancy in the baselines gives additional linear relationships which can be written as differences between baseline phases being equal to zero. Besides, there are still three disposable parameters (free parameters), which are two true phases of any non-redundant baselines and one aperture phase error [6,23]. Thanks to these additional relationships, a larger matrix relationship can be written for an n-element array with r independent redundant baselines:

[\begin{matrix} M_{p \times 1} \\ O_{(r + 3) \times 1} \end{matrix}] = [\begin{matrix} I_{p \times p} & B_{p \times n} \\ R_{(r + 3) \times p} & G_{(r + 3) \times n} \end{matrix}] \times [\begin{matrix} Φ_{p \times 1} \\ E_{n \times 1} \end{matrix}] = A_{(p + r + 3) \times (p + n)} \times U_{(p + n) \times 1}

The matrix O stands for a zero matrix, and R is matrix induced by the redundant baselines, while G denotes the matrix induced by the three disposable parameters whose values are all set to zero for convenience. So the new larger matrix A can be inverted to compute, from the measurements M, the phases Φand the errors E with enough independent redundancies.

Two spaces K and L are firstly defined:

K = s p a n {A_{1}, A_{2}, \dots, A_{p}}

Where

A_{(p + r + 3) \times (p + n)} = [A_{1}, A_{2}, \dots, A_{p}, A_{p + 1}, A_{p + 2}, \dots, A_{p + n}]

According to the RSC condition [24]:

K \cap L = {0}

The RSC promises that the two parts $Φ_{p \times 1}$ and $E_{n \times 1}$ are independent in the solution $U_{(p + n) \times 1}$ of Eq. (5), and that the coefficient matrix is a full rank one (that why we call this kind of arrays full rank arrays). From the RSC condition, we can get that:

r a n k (A_{(p + r + 3) \times (p + n)}) = r a n k ([\begin{matrix} I_{p \times p} & B_{p \times n} \\ R_{(r + 3) \times p} & G_{(r + 3) \times n} \end{matrix}]) = \min {p + n, p + r + 3}

In order to achieve a minimum-norm least-square solution from Eq. (5), we must ensure that:

\min {p + n, p + r + 3} \geq p + n

As the solution of Eq. (5) and phases of cross-correlations are just measured modulo $2 π$ , it may be sometime that we need to unwrap the measured phases of cross-correlations if any element in the inverse of matrix A is non-integer.

For $r = n - 3$ : $A_{(p + r + 3) \times (p + n)}$ is a full rank square matrix. Then arrays of this kind are called as critical arrays, and as discussed in [23], the solution is given by:

U_{(p + n) \times 1} = A_{(p + n) \times (p + n)}^{- 1} \times M_{(p + n) \times 1}

For $r > n - 3$ : To get the unique minimum-norm least-square solution, some pre-processing of matrix A and M _{p × 1} is needed to make sure all the phases lie on the same Riemann sheet. A effectual method is to do singular value decomposition (SVD) to matrix A and the solution could be given by the decomposed matrices and measured phases [23]:

U_{(p + n) \times 1} = [\begin{matrix} Φ_{p \times 1} \\ E_{n \times 1} \end{matrix}] = S_{(p + n) \times (p + n)} V_{(p + n) \times (p + r + 3)}^{- 1} D_{(p + r + 3) \times (p + r + 3)}^{T} [\begin{matrix} M_{p \times 1} \\ O_{(r + 3) \times 1} \end{matrix}]

In Eq. (12), matrices S, V and D are given by the following equation:

A_{(p + n) \times (p + r + 3)} = S_{(p + n) \times (p + n)} V_{(p + n) \times (p + r + 3)} D_{(p + r + 3) \times (p + r + 3)}^{T}

Here $V_{(p + n) \times (p + r + 3)}$ is the singular value matrix of A, $S_{(p + n) \times (p + n)}$ and $D_{(p + r + 3) \times (p + r + 3)}$ are both unitary matrices.

3. Circular array Optimization based on RSC

3.1 RSC in circular arrays

Antenna/aperture arrays have been used widely in different applications including radar, sonar, biomedicine, communications, and imaging. In passive interferometric imaging systems, arrays may be linear, circular and spherical in element arrangement. A type of antenna arrays used widely is the circular array which has several advantages over other types such as two-dimensional field of view capability compared with linear array, achieving non-redundant uniform symmetric u-v coverage and accordingly almost the same imaging quality in all-azimuth. But one of the most important advantages of circular arrays is that, they eliminate well the phase errors resulted from the far field approximation in passive interferometric array, especially in near field imaging situation [28].

In order to using RSC in passive interferometric circular arrays, constrains need to be set on array configurations. From Fig. 1(a) , it can be concluded that the number of redundant baselines is consequentially even if redundant baselines exist. That is because, quadrilateral (1, 2, 3, 4) is a rectangular figure if baseline (2, 1) is redundant to baseline (3, 4) in a circle, which means baseline (2, 3) and (1, 4) is a redundant pair. Besides, baseline (1, 3) and baseline (2, 4) will consequentially cross the center of circle. But only one independent redundancy condition is available from one rectangular figure. Furthermore, if there are totally q baselines passing through the center of circle in an n-element circular array, as showed in Fig. 1(b), these elements form a closed figure and provide only q-1 independent redundancies from all $2 C_{q}^{2}$ redundant baselines. To apply RSC, it follows that from Eq. (12):

Fig. 1 The redundancy in a circular array. Point 1, 2, 3 and 4 are the elements’ position and O is the circle center.

Download Full Size | PDF

n - 3 \leq q - 1 \leq n / 2 - 1

This yields $n \leq 4$ . While $n > 4$ , arrays of this kind will not produce enough independent redundancies for a circular array. One way to overcome the difficulty is to set an extra aperture on the center of circle. This will produce q extra independent redundancies for an n-element circular array with an element in the center of circle and q baselines crossing the center of circle. Then Eq. (14) is rewritten as:

n - 3 \leq (q - 1) + q \leq (n / 2 - 1) + n / 2

And thus

n / 2 - 1 \leq q \leq n / 2

. So the minimum value of q is

q = {\begin{matrix} (n - 1) / 2 if  n is even \\ n / 2 - 1 if  n is odd \end{matrix}

The following table shows the corresponded q with circular arrays of 8~16 elements.

When we obtain the values of q and n, the constraints in circular array optimization based on RSC can be described as that, only n-q-1 elements’ coordinates need to be moved in calculation and the others are symmetrical to some of the n-q-1 elements.

3.2 Optimization strategy

As a key stage, optimization of the passive interferometric array configuration is completed in the engineering development of a passive interferometric imaging system. One challenge is to form an optimized objective (or measurement) function [15]. The objective function can be determined by the imaging plane, u-v plane or array configuration plane. The first step is choosing a criterion for forming appropriate objective function. Various criterions have been taken in literatures to choose an objective function for optimization of array configurations, such as redundancies [9], compactness [10], Field-to-Resolution Ratio (FRR) [12], point spread function (PSF) and modulation transfer function (MTF) [20,21]. Besides, Cornwell has developed an objective function to maximum uniformity of u-v coverage and minimum redundancy of spatial frequency. Results show perfect u-v coverage uniformity without redundancies for 3~12-element circular arrays [11].

In this work, in order to obtain maximum u-v coverage uniformity and redundancy condition for RSC, Cornwell’s objective function is transformed. As discussed earlier, some positions of elements are connected with other ones because baselines of elements crossed the circle center. If there are q baselines crossing the center of circle, the number of changeable positions will reduces by q. Besides, an extra aperture is set on the center of circle. So only (n-q-1) positions of elements are variable parameters in the optimization process, and the other q + 1 elements do not need to be considered.

In the end, the objective function is designed as follows [11]:

E (ρ_{1}, ρ_{2}, ..., ρ_{n}) = \sum_{i = 1}^{n} \sum_{j = 1}^{n} \log (d_{i j})

d_{i j} = {\begin{matrix} \sqrt{{(u_{i} - u_{j})}^{2} + {(v_{i} - v_{j})}^{2}} \sqrt{{(u_{i} - u_{j})}^{2} + {(v_{i} - v_{j})}^{2}} > ε \\ C_{c} \sqrt{{(u_{i} - u_{j})}^{2} + {(v_{i} - v_{j})}^{2}} \leq ε \end{matrix}

In (17) and (18), $ρ_{i}$ (i = 1, 2, …, n) is the antenna’s coordinate position; $(u_{i}, v_{i})$ and $(u_{j}, v_{j})$ are corresponded to u-v point i and j. εis a constant determined by the least sampling interval. And C_c is positive far less than ε which usually is assigned to be the smallest value, so that the optimization software can achieve (in this paper it is equal to 1.0E-100). The purpose is to punish redundant u-v coverage of the array during searching process so that redundant array configurations have less chance to be preserved.

3.3 Optimization for circular array based on RSC

The direct-search computation methods have been usually used for passive interferometric array optimization [20]. But they are just used when the derivatives of the objective function or constraints are not continuous and just give a good solution but not an optimal one. Other well-known optimization method is the simulated annealing algorithm (SAA) [11,29]. It is a global random optimization algorithm based on Monte Carlo method [30,31]. The basic process begins with an initial solution and user-controlled “temperature” which will affects the decision to accept or reject a solution when the new solution is not as good as the last one. Then loop calculations to current solution are done as “generating new solutions→calculating the objective function→acceptance or abandonment”. And the user-controlled “temperature” is lowed down gradually. The solution is just the optimized solution when the procedure stops. According to the above introduction, the optimization process of passive circular interferometric array based on RSC can be simply described as follows:

i. Initialization: set the n-r-1 element positions (the initial solution) uniformly in the circle, and initialize the optimization calculation parameters;
ii. Annealing loop: the objective function is calculated for the current solution by Eq. (17), and a random solution (the new n-r positions) in the neighborhood of the current solution is generated and the resulting change $Δ E$ in objective function is computed. If $Δ E > 0$ , then the modification is accepted. Otherwise, calculate $\exp (\frac{Δ E}{T})$ (here T is the current temperature) and compare it with a random number $E_{r a n d}$ uniformly distributed on the interval [0, 1]. If it is bigger, then accept the new solution, otherwise reject it and return to step ii. If the new solution is accepted, then reduce the temperature and go on to step iii.
iii. Termination test: judge the objective function (or temperature in SAA) if it is small enough or has finished the set loop times to avoid infinite loop. If not, return to step ii, or terminate the optimization.

By applying the above approach, optimized results are showed in the Table 2 based on RSC for circular arrays of 8~16 elements. Only n-1 positions of sub-apertures are listed and the last one is at the center of circle.

Table 2. Optimized Positions of 8 ~16 elements array

View Table | View all tables in this article

To illuminate the optimized results directly, two groups of comparisons are done. One group is the u-v coverage formed by the above optimized arrays against one formed by the uniformly distributed circular arrays, and the other group is u-v coverage formed by the above optimized arrays against the optimized arrays without constraints from RSC. Figure 2 shows the coverage for passive interferometric circular arrays with n = 8 and n = 16. The calculated objective function values based on Eq. (17) are shown in Table 2.

Fig. 2 u-v coverage of 8-element arrays (in first row) and 16-element arrays (in second row). (a) and (d) is with optimized arrays based on RSC; (b) and (e) is with optimized arrays without constraint from RSC; (c) and (f) is with uniformly distributed arrays.

Download Full Size | PDF

From Table 3 we can get that the objective function values for passive interferometric circular array optimized based on RSC is lower than that without RSC constraint for about 14.5%, and only 7.2% smaller than that of uniformly distributed arrays for 16-element array.

Table 3. Objective function values with different array configuration of 8 and 16 elements

View Table | View all tables in this article

It is easy to understand because the u-v coverage is improved after the array configurations are optimized, but it could corrupt the u-v coverage when adding redundant baseline in array optimization considering the phase calibration. These optimized results in Table 2 are very useful for passive interferometric circular arrays, for they give attentions to both the u-v coverage optimization and special requirements for RSC.

3.4 The RSC matrix for optimized ten-element array

In order to demonstrate the above method with optimized results, the ten-element circular array is chosen to carry out the RSC. Figure 3 shows the position of ten elements on the circle and the relevant u-v coverage.

Fig. 3 Optimized circular array configuration based on RSC with ten elements and its u-v coverage figure.

Download Full Size | PDF

From Fig. 3 and Table 1 , we know that there are $q = 4$ baselines crossed the circle center, so there are $r = 2 q - 1 = 7$ redundant baselines. For $n = 10$ , so $p = 45$ .According to Eq. (5), the corresponded matrix equation can be written as follows:

[\begin{matrix} M_{45 \times 1} \\ O_{10 \times 1} \end{matrix}] = [\begin{matrix} I_{45 \times 45} & B_{45 \times 10} \\ R_{10 \times 45} & G_{10 \times 10} \end{matrix}] \times [\begin{matrix} Φ_{45 \times 1} \\ E_{10 \times 1} \end{matrix}] = A_{55 \times 55} \times U_{55 \times 1}

Where

Table 1. The corresponded q with arrays of 8~16 elements

View Table | View all tables in this article

[\begin{matrix} M_{45 \times 1} \\ O_{10 \times 1} \end{matrix}] = {[\underset{45}{\underset{︸}{m_{1, 2}^{(1)} m_{1, 3}^{(2)} \dots m_{1, 10}^{(9)} m_{2, 3}^{(10)} \dots m_{2, 10}^{(18)} \dots m_{9, 10}^{(45)}}} \underset{10}{\underset{︸}{0 \dots 0}}]}^{T}

The superscript of $m_{i, j}$ will be used later. And we define a collection H:

H = {(i, j) : i = 1, 2, ..., 10; j = 1, 2, ..., 10; i < j}

So the elements in matrix $B_{45 \times 10}$ could be written as:

b_{i, j} = {\begin{matrix} 1 for (i, j) \in H \\ - 1 for (i, j) \in H \\ 0 otherwise \end{matrix}

Here $R_{10 \times 45}$ and $G_{10 \times 10}$ are both sparse matrices. For convenience, the redundant baseline pairs in Fig. 3 are set to be an ordered collectionℜ:

\begin{array}{l} ℜ = {ℜ_{1} \dots ℜ_{7}} \\ = {{\begin{cases} (1, 2) \\ (5, 6) \end{cases}}, {\begin{cases} (2, 3) \\ (6, 7) \end{cases}}, {\begin{cases} (3, 4) \\ (7, 8) \end{cases}}, {\begin{cases} (1, 10) \\ (5, 10) \end{cases}}, {\begin{cases} (2, 10) \\ (6, 10) \end{cases}}, {\begin{cases} (3, 10) \\ (7, 10) \end{cases}}, {\begin{cases} (4, 10) \\ (8, 10) \end{cases}}} \end{array}

So the nonzero elements of matrix $R_{10 \times 45}$ induced by the redundant baseline in the Fig. 3 are given by:

r_{k, i} = r_{k, j} = 1 if {\begin{cases} (k_{1}, k_{2}) \\ (k_{3}, k_{4}) \end{cases}} = ℜ_{k} for m_{k_{1}, k_{2}}^{(i)} and m_{k_{3}, k_{4}}^{(j)} (k \leq 7)

The two disposable parameters in $Φ_{10 \times 1}$ are appointed to the first two elements so that $r_{8, 1} = r_{9, 2} = 1$ .

While $G_{10 \times 10}$ denotes the matrix induced by a disposable parameter in $E_{10 \times 1}$ whose value is equal to zero. And the only one nonzero element is $g_{10, 10} = 1$ with the disposable parameter $e_{10} = 0$ .

Up to now, the coefficient matrix $A_{55 \times 55}$ of the Eq. (19) is presented. As declared in the first part of this section, matrix $A_{55 \times 55}$ is full rank and n-3 = 2q-1, so the inverse of the coefficient matrix can be calculated and the solution can be given as follows:

[\begin{matrix} Φ_{45 \times 1} \\ E_{10 \times 1} \end{matrix}] = A_{55 \times 55}^{- 1} \times [\begin{matrix} M_{45 \times 1} \\ O_{10 \times 1} \end{matrix}]

4. Conclusion

The redundant spacing calibration (RSC) approach applied in passive interferometric circular arrays is presented and the constraint in array optimization is analyzed. During the optimization, only some positions of the elements in the circle must be changed and the rest elements are on the symmetrical positions about the circle center, so that right numbers of redundant baselines are generated. That is very important in the optimization of passive interferometric circular arrays where RSC approach is applied. Simulated annealing algorithm is used to optimize circular arrays based on RSC. Particular attentions are paid to the objective function and punishments are implemented when extra redundancies exist. The optimized arrays with 8~16 elements are listed. The comparisons of u-v coverage and the objective function values with 8 and 16 element arrays in different arrangements show that a small sacrifice of u-v coverage can get RSC arrays in return. Considering the special merits of circular arrays, the optimized circular array based on RSC in this paper could be adopted in the platform of the space-based or air-based devices.

Acknowledgements

This work was supported in part by the National High-tech Research and Development Program (863) under contact No.2007AA12Z114 and by the Innovation Foundation of BUAA for PhD Graduates.

References and links

1. W. T. Rhodes and J. W. Goodman, “Interferometric technique for recording and restoring images degraded by unknown aberrations,” J. Opt. Soc. Am. 63(6), 647–657 ( 1973). [CrossRef]

2. J. E. Noordam and A. G. de Bruyn, “High dynamic range mapping of strong radio sources with application to 3C84,” Nature 299(5884), 597–600 ( 1982). [CrossRef]

3. J. E. Baldwin, C. A. Haniff, C. D. Mackay, and P. J. Warner, “Closure phase in high-resolution optical imaging,” Nature 320(6063), 595–597 ( 1986). [CrossRef]

4. T. J. Cornwell, “The applications of closure phase to astronomical imaging,” Science 245(4915), 263–269 ( 1989). [CrossRef] [PubMed]

5. A. Lannes, “Remarkable algebraic structures of phase closure imaging and their algorithmic implications in aperture synthesis,” J. Opt. Soc. Am. A 7(3), 500–512 ( 1990). [CrossRef]

6. P. M. Blanchard, A. H. Greenaway, A. R. Harvey, and K. Webster, “Coherent Optical Beam Forming with Passive Millimeter-Wave Arrays,” J. Lightwave Technol. 17(3), 418–425 ( 1999). [CrossRef]

7. C. S. Ruf, C. T. Swift, A. B. Tanner, et al.., “Interferometric synthetic aperture microwave radiometry for the remote sensing of the earth,” IEEE Trans. On GRS 26, 597–611 ( 1988).

8. M. Peichl, H. Suess, M. Suess, and S. Kern, “Microwave imaging of the brightness temperature distribution of extended areas in the near and far field using two-dimensional aperture synthesis with high spatial resolution,” Radio Sci. 33(3), 781–801 ( 1998). [CrossRef]

9. A. T. Moffet, “Minimum-Redundancy Linear Arrays,” IEEE Trans. Antenn. Propag. 16(2), 172–175 ( 1968). [CrossRef]

10. M. J. E. Golay, “Point Arrays Having Compact, Nonredundant Autocorrelations,” J. Opt. Soc. Am. 61(2), 272–273 ( 1971). [CrossRef]

11. T. J. Cornwell, “A novel principle for optimization of the instantaneous Fourier plane coverage of correlation arrays,” IEEE Trans. Antenn. Propag. 36(8), 1165–1167 ( 1988). [CrossRef]

12. A. Lannes, E. Anterrieu, L. Koechlin, et al.., “Concept of field-to-resolution ratio in aperture synthesis,” in Space Optics 1994: Earth Observation and Astronomy, M. G. Cerutti-Maori ed. Proc. SPIE 2209, 402–412 ( 1994). [CrossRef]

13. M. Faucherre, F. Merkle, and F. Vakili, “Beam combination in aperture synthesis from space: Field of view limitations and (u, v) plane coverage optimization” in New Technologies for Astronomy, J. P. Swings, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1130, 138–145 (1989).

14. J. P. Fitch, and T. W. Lawrence, “Placement of multiple apertures for imaging telescopes,” in Amplitude and Intensity Spatial Interferometry,Jim B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1237, 61–69 (1990).

15. L. M. Mugnier, G. Rousset, and F. Cassaing, “Aperture configuration optimality criterion for phased arrays of optical telescopes,” J. Opt. Soc. Am. A 13(12), 2367–2374 ( 1996). [CrossRef]

16. M. A. Holdaway, Helfer, Tamara T. “Interferometric Array Design,” in Synthesis Imaging in Radio Astronomy II, Proc. ASP Conference 180, 537–563 (1999).

17. L. Kogan, “Optimizing a large array configuration to minimize the sidelobes,” IEEE Trans. Antenn. Propag. 48(7), 1075–1078 ( 2000). [CrossRef]

18. O. Guyon and F. Roddier, “Aperture rotation synthesis: Optimization of the (u, v)-plane coverage for a rotating phased array of telescopes,” The Astronomical Society of the Pacific 113(779), 98–104 ( 2001). [CrossRef]

19. Y. Su, R. D. Nan, B. Peng, N. Roddis, and J. F. Zhou, “Optimization of interferometric array configurations by “sieving” u - v points,” Astron. Astrophys. 414(1), 389–397 ( 2004). [CrossRef]

20. I. Tcherniavski and M. Kahrizi, “Optimization of the optical sparse array configuration,” Opt. Eng. 44(10), 103201 ( 2005). [CrossRef]

21. I. Tcherniavski and M. Kahrizi, “Optimization of design and operating parameters of a space-based optical-electronic system with a distributed aperture,” Appl. Opt. 47(33), 6159–6176 ( 2008). [CrossRef] [PubMed]

22. P. Armand, J. Benoist, E. Bousquet, L. Delage, S. Olivier, and F. Reynaud, “Optimization of a one dimensional hypertelescope for a direct imaging in astronomy,” Eur. J. Oper. Res. 195(2), 519–527 ( 2009). [CrossRef]

23. A. H. Greenaway, “Self-calibrating dilute-aperture optics”, in Digital image synthesis and inverse optics Proc. SPIE 1351, 738–748 ( 1990). [CrossRef]

24. A. Lannes and E. Anterrieu, “Redundant spacing calibration: phase restoration methods,” J. Opt. Soc. Am. A 16(12), 2866–2879 ( 1999). [CrossRef]

25. Y. Li, B. Braker, F. Schlottau, D. Gu, M. Colice, and K. H. Wagner, “Broadband RF imaging and spectrum analysis using spatial-spectral hole-burning in an inhomogeneously broadened absorber,” in Photonic Applications in Nonlinear Optics, Nanophotonics, and Microwave Photonics Proc. SPIE 5971, 597122 ( 2005). [CrossRef]

26. R. J. Eastwood, A. M. Johnson, and A. H. Greenaway, “Calculation and correction of piston phase aberration in synthesis imaging,” J. Opt. Soc. Am. A 26(1), 195–205 ( 2009). [CrossRef]

27. P. M. Blanchard, A. H. Greenaway, R. N. Anderton, and R. Appleby, “Phase calibration of arrays at optical and millimeter wavelengths,” J. Opt. Soc. Am. A 13(7), 1593–1600 ( 1996). [CrossRef]

28. A. Bandyopadhyay, A. Stepanov, B. Schulkin, M. D. Federici, A. Sengupta, D. Gary, J. F. Federici, R. Barat, Z.-H. Michalopoulou, and D. Zimdars, “Terahertz interferometric and synthetic aperture imaging,” J. Opt. Soc. Am. A 23(5), 1168–1178 ( 2006). [CrossRef]

29. C. S. Ruf, “Numerical annealing of low-redundant linear arrays,” IEEE Trans. Antenn. Propag. 41(1), 85–90 ( 1993). [CrossRef]

30. S. Kirkpatrick, C. D. Gelatt Jr, and M. P. Vecchi, “Optimization by Simulated Annealing,” Science 220(4598), 671–680 ( 1983). [CrossRef] [PubMed]

31. P. J. M. Laarhoven, and E. H. L. Aarts, Simulated annealing: theory and applications. (D. Reidel Publishing Company, Dordrecht, (1987).

n	8	9	10	11	12	13	14	15	16
Positions of sub-apertures (rad)	4.0787	3.1233	3.8008	0.3441	4.6004	2.9168	0.4511	5.0921	5.6896
	0.0257	0.7494	6.2641	5.3371	6.0198	0.8042	4.7821	4.1710	2.9994
	2.2575	5.4699	5.5348	1.6029	0.8927	5.4844	4.0712	5.5581	0.8842
	1.6137	1.5524	1.3820	5.9819	2.3040	1.3069	6.2201	6.0189	4.8410
	5.3991	4.6940	1.8957	4.0882	0.3118	0.2377	2.6406	0.1539	1.3389
	3.1673	2.3283	4.5236	0.9466	1.8767	1.8520	2.1243	0.6530	2.1724
	0.9371	3.8910	2.3932	2.8404	3.4534	4.9936	1.2971	4.6751	0.4624
		6.2649	3.1225	4.7445	5.4455	3.3793	5.2659	1.5335	3.2932
			0.6592	2.1955	4.0343	4.4485	5.7821	3.7946	3.6040
				3.4856	2.8782	2.3428	3.0785	3.2955	5.3140
					1.4588	3.9458	0.9296	2.8773	4.4805
						6.0584	1.6405	2.4165	1.6994
							3.5927	1.0294	4.0258
								1.9505	6.1410
									2.5480

Element number	Optimized array based on RSC	Optimized array without RSC	Uniformly distributed array
8	218.4	374.3	268.7
16	5173.9	6050.3	5573.1

n	8	9	10	11	12	13	14	15	16
Positions of sub-apertures (rad)	4.0787	3.1233	3.8008	0.3441	4.6004	2.9168	0.4511	5.0921	5.6896
	0.0257	0.7494	6.2641	5.3371	6.0198	0.8042	4.7821	4.1710	2.9994
	2.2575	5.4699	5.5348	1.6029	0.8927	5.4844	4.0712	5.5581	0.8842
	1.6137	1.5524	1.3820	5.9819	2.3040	1.3069	6.2201	6.0189	4.8410
	5.3991	4.6940	1.8957	4.0882	0.3118	0.2377	2.6406	0.1539	1.3389
	3.1673	2.3283	4.5236	0.9466	1.8767	1.8520	2.1243	0.6530	2.1724
	0.9371	3.8910	2.3932	2.8404	3.4534	4.9936	1.2971	4.6751	0.4624
		6.2649	3.1225	4.7445	5.4455	3.3793	5.2659	1.5335	3.2932
			0.6592	2.1955	4.0343	4.4485	5.7821	3.7946	3.6040
				3.4856	2.8782	2.3428	3.0785	3.2955	5.3140
					1.4588	3.9458	0.9296	2.8773	4.4805
						6.0584	1.6405	2.4165	1.6994
							3.5927	1.0294	4.0258
								1.9505	6.1410
									2.5480

Element number	Optimized array based on RSC	Optimized array without RSC	Uniformly distributed array
8	218.4	374.3	268.7
16	5173.9	6050.3	5573.1

Passive interferometric array optimization based on redundant spacing calibration

Abstract

1. Introduction

2. Principles of redundant spacing calibration

3. Circular array Optimization based on RSC

3.1 RSC in circular arrays

3.2 Optimization strategy

3.3 Optimization for circular array based on RSC

3.4 The RSC matrix for optimized ten-element array

4. Conclusion

Acknowledgements

References and links

Cited By

Figures (3)

Tables (3)

Equations (26)

Optics Express