Abstract:
The invention concerns an imaging assembly comprising a synthetic aperture instrument including a plurality of pupils separated by sampling detection means. The sampling frequency values in column (u) and in line (v) are less than the values corresponding to the Shannon criterion, and are such that zones of response modulation transfer function resulting from sampling are inserted in zones of null modulation transfer function of the central section of the modulation transfer function, so that said zones do not form an intersection and, in a central section of smaller dimension ( 22   0 ) than the central section ( 20   0.0 ) of the modulation transfer function, the set of zones representing the central section of the modulation transfer function is present. The acquired image spectrum is reconfigured so as to reconstruct said image.

Description:
BACKGROUND OF THE INVENTION 
   The present invention relates to an imaging assembly with a synthetic aperture optical instrument. 
   Using imaging optical systems with synthetic apertures is envisaged in the aerospace industry in particular. These systems consist in a combination of subsystems, each of small size, that achieves practically the same result as a large optical system. A system made up of small subsystems is easier to produce than a large optical system and causes less problems to launch by means of a satellite. An imaging system of this type has a telescope function, for example. 
   The signals obtained by each of the instruments are generally combined by Fizeau or Michelson interferometry. 
     FIG. 1  represents by way of illustration one pupil configuration of a three-telescope interferometer. 
   The main optical characteristic of a synthetic aperture optical device is determined by the diameter of the pupils  10 ,  12 ,  14  and their respective positions. In this example, the pupils all have the same diameter D′ and their centers are disposed in accordance with an equilateral triangle of side length B′. 
   It is known that synthetic aperture devices of this kind may have a modulation transfer function (MTF) whose support is discontinuous if the distance between the pupils is sufficiently large compared to their diameter, i.e. this function features cancellation ranges.  FIG. 2  represents the modulation transfer function of the instrument represented in  FIG. 1 . 
   The modulation transfer function of an optical device is the response of the instrument to the diverse input spatial frequencies. In the  FIG. 2  diagram, the column spatial frequencies are represented on the abscissa axis u and the row frequencies are represented on the ordinate axis v. The support of the modulation transfer function of the instrument represented in  FIG. 1  therefore features seven circular regions, each having a diameter 2D (where D=D′/λ, λ being the wavelength): a “central” region LL for low column and line frequencies (L), and six peripheral circular regions: LH, HH, H′H, L′H, H′H′ and HH′. The notation LH means that the region relates to low row frequencies (L) and high column frequencies (H). Similarly, the notation HH, H′H, H′H′ and HH′ signifies high row and column frequencies. 
   The centers of the peripheral circular regions are on a circle of diameter 2B (where B=B′/λ) centerd on the origin. 
   The line joining the centers of the circles HH and H′H′ is at an angle of 60° to the abscissa axis and, similarly, the line joining the centers of the circles H′H and HH′ is at an angle of 120° to the abscissa axis. 
   In practice, the images are sampled with a spatial sampling frequency that may be different for the rows and the columns. To prevent aliasing, i.e. loss of information, it is necessary to comply with Shannon&#39;s theorem, i.e. the column sampling frequency must be greater than or equal to twice the maximum frequency of the spectrum to be reproduced, i.e. twice the distance I ( FIG. 2 ) such that I=B+D, and, the row sampling frequency must be equal to twice the distance μ ( FIG. 2 ), where μ=√3B/2+D. 
   Accordingly, when the column sampling frequency is 2B+2D and the row sampling frequency is √3B/2+D, an image is obtained for which the support of the spectrum is of the type represented in  FIG. 3  with a central spectrum section comprising the seven circular regions LL, L′H, LH, HH, H′H′, H′H and HH′ and delimited in  FIG. 3  by a rectangle  20   0,0 . The spectrum also comprises a set of replicas identical to the central section offset on the abscissa axis by an integer number of column sampling frequencies and on the ordinate axis by an integer number of row sampling frequencies. 
   Accordingly, as may be seen in  FIG. 3 , the central rectangle  20   0,0  of the usable portion of the spectrum is replicated to form paving with slabs identical to the central slab  20   0,0  constituting replicas  20   1,0 ,  20   0,1 ,  20   1,1 ,  20   0,−1 ,  20   −1,1 , etc. 
   Until now, it has been considered that the sampling frequency could not fall below values corresponding to those of  FIG. 3  since, for lower sampling frequencies, the MTF regions of the replicas overlap the MTF regions of the central section  20   0,0 , which would lead to aliasing, i.e. to degrading of the information. 
   The number of pixels (picture elements) necessary for sampling the image is, of course, a direct function of the sampling frequency. As a result of this, the higher the sampling frequency, the higher the number of pixels that is necessary. 
   SUMMARY OF THE INVENTION 
   The invention allows a significantly lower sampling frequency to be used without aliasing. In other words, at constant field, the invention reduces the number of pixels necessary to reproduce images and to increase the quality (signal to noise ratio) of the image. 
   The invention exploits the fact that the central region  20   0,0  has large portions with a null modulation transfer function. 
   The method conforming to the invention consists in:
         conferring on the column and row sampling frequencies values lower than the sampling frequencies that are determined by the Shannon criterion, these lower sampling frequencies producing replicas occupying the null transfer function portions of the central section of the spectrum, without intersecting with the non-null transfer function regions of the central section, the set of different useful regions nearest the origin then forming a new central section smaller than the original central section, and   in order to reconstruct the image, reconfiguring the small central section according to the original central section.       

   Thus row and column sampling frequencies are chosen whose values are such that replicas overlap this central section, but do not intersect with the (circular) non-null modulation transfer function regions (i.e. the non-null MTF regions of the replicas fill null value regions of the central section of the MTF), and there are selected in the central section filled in this way those of the regions that are nearest the frequency origin and which may be used to reconstruct the modulation transfer frequency, and this new central section, which is smaller than the original central section, is reconfigured so that it corresponds, for the purpose of image reconstruction, to the original central section. 
   In the case of an onboard instrument, the image is preferably reconstructed on the ground. 
   It can be shown that in the case of the interferometer system with three pupils represented in  FIG. 1 , the row sampling period may be increased by a factor of approximately 1.62 and the column sampling period may be increased by a factor of approximately 1.67. Thus, at constant field, the total number of pixels necessary for sampling the image is reduced by a factor of 2.7. 
   Accordingly, the invention generally provides an imager assembly comprising a synthetic aperture instrument comprising a plurality of separate pupils with sampling detection means, characterized in that:
         the values of the column and row sampling frequencies are less than the values conforming to the Shannon criterion and are such that modulation transfer function regions of replicas resulting from sampling are inserted into null modulation transfer function regions of the central section of the modulation transfer function so that these regions do not intersect and a central section smaller than the central section of the modulation transfer function contains all the regions representing the central section of the modulation transfer function, and   it comprises means for reconfiguring the spectrum of the image acquired in order to reconstruct it, said means being used to position the regions of the smaller central section so that they correspond to the positions of the regions of the original central section.       

   In one embodiment the values of the sampling frequencies are such that in at least the row direction or the column direction the regions of the smaller central section of the modulation transfer function are tangential. 
   The instrument may comprise three pupils of diameter D′ whose centers are disposed in accordance with an equilateral triangle of side length B′ such that 2D′=√3B′/4, the column sampling frequency being equal to 3B/2 and the row sampling frequency being equal to 6D, where B=B′/λ and D=D′/λ, λ being a wavelength. 
   A variant of the instrument comprises four pupils each of which has a diameter D′ and whose centers are disposed in accordance with a square of diagonal length B′ such that: 2D′=B′/2, the column and row sampling frequencies being 6D, where D=D′/λ, λ being a wavelength. 
   The assembly is of the interferometer type, in particular of the telescope type, for example. 
   The invention also provides a method of determining the optical sampling frequency of an imaging assembly comprising a synthetic aperture type instrument comprising a plurality of pupils, such as an interferometric telescope, which method is characterized in that said sampling frequency is chosen with a value lower than that imposed by the Shannon criterion and such that replicas adjoining the central section of the modulation transfer function overlap the central section so that the non-null transfer function regions of the replicas are inserted into null transfer function regions of the central section without intersecting with a non-null transfer function region and so that a smaller section inside the central section contains all the regions useful for reconstructing the modulation transfer function. 
   In one embodiment the row and column sampling frequency is chosen so that in at least the row direction or the column direction adjacent regions of the smaller section are tangential. 
   The invention further provides a method of reconstructing images in an imaging assembly for which the sampling frequency corresponds to the method defined above, in which method the modulation transfer function is re-established from regions in the smaller section. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     Other features and advantages of the invention will become apparent in the course of the description of certain embodiments of the invention given with reference to the appended drawings, in which: 
       FIG. 1 , already described, represents an interferometer with three pupils, 
       FIG. 2 , already described, represents the support of the modulation transfer function before sampling of the interferometer represented in  FIG. 1 , 
       FIG. 3 , also described already, represents, for the interferometer represented in  FIG. 1 , the support of the spectrum after sampling conforming to the Shannon criterion, 
       FIG. 4  represents, for the interferometer represented in  FIG. 1 , the support of the spectrum resulting from a sampling frequency determined using the method according to the invention, 
       FIGS. 5   a,    5   b  and  5   c  are diagrams illustrating a process for recovering the original spectrum with a view to reconstructing the image after using the method represented in  FIG. 4 , and 
       FIGS. 6 ,  7 ,  8 ,  9 ,  10   a,    10   b  and  10   c  are figures analogous to  FIGS. 1 ,  2 ,  3 ,  4 ,  5   a,    5   b  and  5   c,  respectively, for an interferometer with four pupils represented in  FIG. 6 . 
   

   DETAILED DESCRIPTION OF THE INVENTION 
   Refer first to  FIG. 4 . 
   In this example, corresponding to the interferometer with three pupils represented in  FIG. 1 , a column sampling frequency is chosen with the value 3B/2 and a row sampling frequency is chosen with the value 6D. 
   In this case, as may be seen in  FIG. 4 , the (replica) section  20 ′ 1,0 , for which the center of the central region LL has the abscissa  1  and the ordinate  0 , overlaps the central section  20   0,0 . Accordingly, the regions H′H′, L′H and H′H of this replica  20 ′ 1,0  are inside the central section  20   0,0 . In particular, it may be seen that the region L′H 1,0  is between the regions LL 0,0  and LH 0,0 . The subscripts used for the regions correspond to the co-ordinates of the center of the central region LL of the corresponding replica. 
   Similarly, the region LH −1,0  is between the regions L′H 0,0  and LL 0,0 . It may also be seen that the region H′H′ 0,1  is between the regions LH −1,0  and H′H 0,0  and is tangential to them. Similarly, the region HH′ 0,1  is between the regions HH 0,0  and L′H 1,0  and is tangential to them. Symmetrically, the region H′H 0,−1  is between the regions LH −1,0  and H′H′ 0,0  and is tangential to them; finally, the region HH 0,1  is between the regions L′H 1,0  and HH′ 0,0  and is tangential to them. 
   It can therefore be seen that a central section  22   0  that is smaller than the central section  20   0,0  contains the seven modulation transfer function regions: LL, L′H, LH, HH′, H′H, HH and H′H′; the region LL comes from central section  0 , 0 , the regions H′H′ and HH′ come from the replica  0 , 1 , the regions H′H and HH come from the replica  0 ,− 1 , the region L′H comes from the replica  1 , 0 , and the region LH comes from the replica − 1 , 0 . 
   Accordingly, although row and column sampling frequencies have been chosen that are lower than those conforming to the Shannon criterion, no aliasing occurs, because all seven regions of the modulation transfer function are contained in a central section and do not overlap. 
   The procedure for reconstructing the image is then as represented in  FIGS. 5   a,    5   b  and  5   c,  that is to say, after obtaining the smaller central section  22   0  ( FIGS. 4 and 5   a ), the regions LL, LH, L′H, HH, H′H, HH′ and H′H′ are rearranged so that they are disposed in the manner represented in the section  20   0  of  FIG. 3 , i.e. as in  FIG. 5   c.  To this end, HH and HH′ are interchanged, H′H′ and H′H are interchanged, and LH and L′H′ are interchanged. This achieves the disposition shown in  FIG. 5   b.  It is then sufficient to move the central region LL away from the six peripheral regions H′H, HH, L′H, LH, HH′ and H′H′ to obtain the configuration represented in  FIG. 5   c,  i.e. the starting configuration of the central section of the spectrum. 
   It has been found that, using this method, although the null transfer function portions contain energy coming from the noise present at all frequencies during sampling, the signal obtained is of the same quality, i.e. there is no loss of information, and of substantially the same amplitude as in the situation where sampling conforms to the Shannon criterion. 
   If 2D′=√3B′/4, the ratio between the column sampling frequency conforming to the Shannon criterion ( FIG. 3 ) and the sampling frequency determined using the method of the invention ( FIG. 4 ) has the value (8+√3)/6≈1.62. Thus the period between the pixels may be increased in this ratio. 
   The relationship indicated above between D′ and B′ constitutes the maximum value of D′ allowing use of the method conforming to the invention. This is because, for higher values of D′, it is no longer possible to insert replicas in the central region. 
   For the rows, the same ratio is 5/3 (again assuming that 2D′=√3B′/4). Thus the period between the rows of pixels may be increased by a factor of approximately 1.67. 
   The total number of pixels may therefore be reduced by a factor of 1.62×1.67=2.7 for a constant field. 
   Given that with this kind of sampling, and for a given field, the number of pixels is reduced, the size of the pixels may therefore be increased. In the case of a push-broom observation system, for which only one row of detectors is used, the integration time may be increased. Under these conditions, in the case of a push-broom system, the flux collected by each pixel is multiplied by 1.62×1.67 2  (1.67 2  is the result of the fact that the integration is effected along the columns). Thus the flux collected by each pixel is multiplied by a factor of approximately 4.52 compared to sampling conforming to the Shannon criterion. In the most favorable situation (in which the noise is independent of the area of the detector), the improvement in the signal to noise ratio is 4.52, while in the unfavourable situation (when the noise depends on the area of the detector), the signal to noise ratio improvement is √{square root over (4.52)}, i.e. 2.13. 
   One example of an application of the invention to the situation where the interferometer comprises four pupils is described next with reference to  FIGS. 6 ,  7 ,  8 ,  9  and  10   a,    10   b,    10   c.    
     FIG. 6  shows an interferometer of this kind. It comprises four circular pupils  30 ,  32 ,  34  and  36  of diameter D′ whose centers are disposed in a square of diagonal length B′; in this example, B′=4D′, which is the minimum value of D′ allowing use of the method conforming to the invention. 
     FIG. 7  shows the support of the modulation transfer function of an interferometer of this kind. This modulation transfer function comprises nine circular regions all having a diameter 2D (D=D′/λ), i.e. B/2 (B=B′/λ). In addition to the central region LL, the center of which is at the origin of the column and row frequencies, this MTF comprises two regions LH and L′H whose centers are on the abscissa axis. The center of the region LH is at the abscissa B and that of the region L′H is at the abscissa −B. Furthermore, two regions HL and H′L have their centers on the ordinate axis. The ordinate of the center of the region HL is +B and the ordinate of the region H′L is −B. 
   The centers of the regions HL, LH, H′L and L′H therefore form a square and at the center of each side of this square is one of the centers of the four other circular regions HH, HH′, H′H′ and H′H. As in the  FIG. 2  diagram, LH signifies a low row frequency and a high column frequency and HL signifies a high row frequency and a low column frequency. 
   If the sampling frequency conforms to the Shannon criterion with a minimum value 2B+2D in the columns and the rows, there are obtained for the MTF (see  FIG. 8 ) a square central section  40   0,0  and replicas  40   1,0 ,  40   1,−1 ,  40   1,1 ,  40   0,1 , etc. of that square section, the subscripts having the same meaning as in the  FIG. 3  diagram. 
   In this case, when the method conforming to the invention is applied, the same column and row sampling frequency may be chosen, and this sampling frequency may have a value such that, in the smaller central section, all the circular regions are tangential. 
   In the example represented in  FIG. 9 , the column and row sampling frequency is 6D. This results in a smaller central section  40 ′ 0,0  comprising a central region LL 0,0  and eight other peripheral circular regions HH, H′H, H′L, LH, L′H, H′H′, HL and HH′ coming from the central section and replicas. As in the  FIG. 4  diagram, each circular region in the  FIG. 9  diagram is assigned subscripts corresponding to the co-ordinates of the center of the region LL of the corresponding replica. 
   The smaller central section  40 ′ 0,0  comprises, in addition to the region LL 0,0 , the regions H′H 0,0 , HH 0,0 , H′H′ 0,0  and HH′ 0,0 . Furthermore, the region HL 0,−1  is between the regions H′H′ 0,0  and HH′ 0,0  and is tangential to them. This region is also tangential to the central region LL 0,0 . The region L′H 1,0  is between the regions HH 0,0  and HH′ 0,0  and is tangential to them and to the region LL 0,0 . Similarly, the region H′L 0,1  is between the regions H′H 0,0  and HH 0,0  and is tangential to them and to the central region LL 0,0 . Finally, the region LH −1,0  is between the regions H′H 0,0  and H′H′ 0,0  and is tangential to them and to the central region LL 0,0 . 
   To reconstruct the image, it is necessary to reconfigure the smaller central section  40 ′ 0,0  so that the nine regions are distributed like the regions of the central section  40   0,0  represented in  FIG. 8 . To this end, starting from the section  40 ′ 0,0  regions are interchanged so that the region HL is on the positive ordinate side and the region H′L on the negative ordinate side, and the regions LH and L′H are likewise interchanged. This yields the  FIG. 10   b  configuration. It is then sufficient to position the regions HL, LH, L′H and H′L to obtain the configuration represented in  FIG. 10   c,  corresponding to the central section  40   0,0  represented in  FIG. 8 . 
   The value (6D) of the sampling frequency is equal to 5/3 times the value of the sampling frequency conforming to the Shannon criterion. Thus the period between pixels may be increased by a factor of approximately 1.67 and, at constant field, the total number of pixels necessary for sampling maybe reduced by 1.67 2 =2.79. Given that the size of the pixels may be increased and that the integration time may likewise also be increased in the case of a push-broom system, the flux collected by each pixel is multiplied by a factor of 1.67 2 ×1.67=4.66. The improvement in the signal to noise ratio is from 4.66 to 2.16. 
   Of course, the invention is not limited to a number of pupils equal to three or four. It applies regardless of the number of pupils of the interferometer or, more generally, of the synthetic aperture instrument. However, regardless of the embodiment, the ratio between the diameter of each pupil and the distance between the pupils must have a maximum value that can be determined easily so that the null transfer function regions are sufficiently extensive to enable insertion of the non-null replica transfer function supports into the central section of that MTF.