Abstract:
Systems and methods for generating an optical image include forming an optical image with at least one optical element of an optical imager while modifying wavefront phase. Modifying the phase does not reduce an optical bandpass limited by an aperture of the optical imager. The systems and methods also include detecting the optical image over a range of spatial frequencies such that there are no zeros in an optical transfer function of the optical imager over detected spatial frequencies within the optical bandpass and over an extended depth of focus that is larger than a depth of focus occurring without modifying wavefront phase.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a divisional application that claims priority to commonly owned and U.S. patent application Ser. No. 10/758,740, filed on Jan. 16, 2004 now U.S. Pat. No. 7,436,595, which is a continuation of commonly owned U.S. patent application Ser. No. 09/070,969, now U.S. Pat. No. 7,218,448, filed on May 1, 1998, which is a continuation-in-part of U.S. patent application Ser. No. 08/823,894 filed on Mar. 17, 1997, now U.S. Pat. No. 5,748,371, which is a continuation of application Ser. No. 08/384,257, filed Feb. 3, 1995, now abandoned. All of the above-identified U.S. patents and patent applications, and U.S. Pat. No. 5,521,695, issued May 28, 1996 and entitled “Range Estimation Apparatus and Method,” are incorporated herein by reference. 
    
    
     U.S. GOVERNMENT RIGHTS 
     This invention was made with Government support awarded by the National Science Foundation and the Office of Naval Research. The Government has certain rights in this invention. 
    
    
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     This invention relates to apparatus and methods for increasing the depth of field and decreasing the wavelength sensitivity of incoherent optical systems. This invention is particularly useful for increasing the useful range of passive ranging systems. The same techniques are applicable to passive acoustical and electromagnetic ranging systems. 
     2. Description of the Prior Art 
     Improving the depth of field of optical systems has long been a goal of those working with imaging systems. A need remains in the art for a simple imaging system, with one or only a few lenses, which none the less provides greatly expanded depth of field focusing. Depth of field refers to the depth in the scene being imaged. Depth of focus refers to the depth in the image recording system. 
     A drawback of simple optical systems is that the images formed with red light focus in a different plane from the images formed with blue or green light. There is only a narrow band of wavelengths in focus at one plane; the other wavelengths are out of focus. This is called chromatic aberration. Currently, extending the band of wavelengths that form an in-focus image is accomplished by using two or more lenses with different indices of refraction to form what is called an achromatic lens. If it were possible to extend the depth of field of the system, the regions would extended where each wavelength forms an in-focus image. If these regions can be made to overlap the system, after digital processing, can produce (for example) a high resolution image at the three different color bands of a television camera. The extended depth of focus system can, of course, be combined with an achromatic lens to provide even better performance. 
     There are several other aberrations that result in misfocus. Astigmatism, for example, occurs when vertical and horizontal lines focus mat different planes. Spherical aberration occurs when radial zones of the lens focus at different planes. Field curvature occurs when off-axis field points focus on a curved surface. And temperature dependent focus occurs when changes in ambient temperature affect the lens, shifting the best focus position. Each of these aberrations is traditionally compensated for by the use of additional lens elements. 
     The effects of these aberrations that causes a misfocus are reduced by the extended depth of imaging system. A larger depth of field gives the lens designer greater flexibility in balancing the aberrations. 
     The use of optical masks to improve image quality is also a popular field of exploration. For example, “Improvement in the OTF of a Defocussed Optical System Through the Use of Shaded Apertures”, by M. Mino and Y. Okano, Applied Optics, Vol. 10 No. 10, October 1971, discusses decreasing the amplitude transmittance gradually from the center of a pupil towards its rim to produce a slightly better image. “High Focal Depth By Apodization and Digital Restoration” by J. Ojeda-Castaneda et al, Applied Optics, Vol. 27 No. 12, June 1988, discusses the use of an iterative digital restoration algorithm to improve the optical transfer function of a previously apodized optical system. “Zone Plate for Arbitrarily High Focal Depth” by J. Ojeda-Castaneda et al, Applied Optics, Vol. 29 No. 7, March 1990, discusses use of a zone plate as an apodizer to increase focal depth. 
     All of these inventors, as well as all of the others in the field, are attempting to do the impossible: achieve the point spread function of a standard, in-focus optical system along with a large depth of field by purely optical means. When digital processing has been employed, it has been used to try to slightly clean up and sharpen an image after the fact. 
     SUMMARY OF THE INVENTION 
     The systems described herein give in-focus resolution over the entire region of the extended depth of focus. Thus they are especially useful for compensating for misfocus aberrations, astigmatism, field curvature, chromatic aberration, and temperature-dependent focus shifts. 
     An object of the present invention is to increase depth of field in an incoherent optical imaging system by adding a special purpose optical mask to the system that has been designed to make it possible for digital processing to produce an image with in-focus resolution over a large range of misfocus by digitally processing the resulting intermediate image. The mask causes the optical transfer function to remain essentially constant within some range away from the in-focus position. The digital processing undoes the optical transfer function modifying effects of the mask, resulting in the high resolution of an in-focus image over an increased depth of field. 
     A general incoherent optical system includes a lens for focusing light from an object into an intermediate image, and means for storing the image, such as film, a video camera, or a Charge Coupled Device (CCD) or the like. The depth of field of such an optical system is increased by inserting an optical mask between the object and the CCD. The mask modifies the optical transfer function of the system such that the optical transfer function is substantially insensitive to the distance between the object and the lens, over some range of distances. Depth of field post-processing is done on the stored image to restore the image by reversing the optical transfer alteration accomplished by the mask. For example, the post-processing means implements a filter which is the inverse of the alteration of the optical transfer function accomplished by the mask. 
     In general, the mask is located either at or near the aperture stop of the optical system or an image of the aperture stop. Generally, the mask is placed in a location of the optical system such that the resulting system can be approximated by a linear system. Placing the mask at the aperture stop or an image of the aperture stop may have this result. Preferably, the mask is a phase mask that alters the phase while maintaining the amplitude of the light. For example, the mask could be a cubic phase modulation mask. 
     The mask may be utilized in a wide field of view single lens optical system, or in combination with a self focusing fiber or lens, rather than a standard lens. 
     A mask for extending the depth of field of an optical system may be constructed by examining the ambiguity functions of several candidate mask functions to determine which particular mask function has an optical transfer function which is closest to constant over a range of object distances and manufacturing a mask having the mask function of that particular candidate. The function of the mask may be divided among two masks situated at different locations in the system. 
     A second object of the invention is to increase the useful range of passive ranging systems. To accomplish this object, the mask modifies the optical transfer function to be object distance insensitive as above, and also encodes distance information into the image by modifying the optical system such that the optical transfer function contains zeroes as a function of object range. Ranging post-processing means connected to the depth of field post-processing means decodes the distance information encoded into the image and from the distance information computes the range to various points within the object. For example, the mask could be a combined cubic phase modulation and linear phase modulation mask. 
     A third object of this invention is to extend the band of wavelengths (colors) that form an in-focus image. By extending the depth of field of the system, the regions are extended where each wavelength forms an in-focus image. These regions can be made to overlap and the system, after digital processing, can produce a high resolution image at the three different color bands. 
     A fourth object of this invention is to extend the depth of field of imaging system s which include elements whose optical properties vary with temperature, or elements which are particularly prone to chromatic aberration. 
     A fifth object of this invention is to extend the depth of field of imaging system s to minimize the effects of misfocus aberrations like spherical aberration, astigmatism, and field curvature. By extending the depth of field the misfocus aberration s can have overlapping regions of best focus. After digital processing, can produce images that minimize the effects of the misfocus aberrations. 
     A sixth object of this invention is to physically join the mask for extending depth of field with other optical elements, in order to increase the depth of field of the imaging system without adding another optical element. 
     Those having normal skill in the art will recognize the foregoing and other objects, features, advantages and applications of the present invention from the following more detailed description of the preferred embodiments as illustrated in the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  shows a standard prior art imaging system. 
         FIG. 2  shows an Extended Depth of Field (EDF) imaging system in accordance with the present invention. 
         FIG. 3  shows a mask profile for a Cubic-PM (C-PM) mask used in  FIG. 2 . 
         FIG. 4  shows the ambiguity function of the standard system of  FIG. 1 . 
         FIG. 5  shows a top view of the ambiguity function of  FIG. 4 . 
         FIG. 6  shows the OTF for the standard  FIG. 1  system with no misfocus. 
         FIG. 7  shows the OTF for the standard  FIG. 1  system with mild misfocus. 
         FIG. 8  shows the Optical Transfer Function for the standard  FIG. 1  system with large misfocus. 
         FIG. 9  shows the ambiguity function of the C-PM mask of  FIG. 3 . 
         FIG. 10  shows the OTF of the extended depth of field system of  FIG. 2 , with the C-PM mask of  FIG. 3 , with no misfocus and before digital processing. 
         FIG. 11  shows the OTF of the C-PM system of  FIG. 2  with no misfocus, after processing. 
         FIG. 12  shows the OTF of the C-PM system of  FIG. 2  with mild misfocus (before processing). 
         FIG. 13  shows the OTF of the C-PM system of  FIG. 2  with mild misfocus (after processing). 
         FIG. 14  shows the OTF of the C-PM system of  FIG. 2  with large misfocus (before processing). 
         FIG. 15  shows the OTF of the C-PM system of  FIG. 2  with large misfocus (after processing). 
         FIG. 16  shows a plot of the Full Width at Half Maximum (FWHM) of the point spread function (PSF) as misfocus increases, for the standard system of  FIG. 1  and the C-PM EDF system of  FIG. 2 . 
         FIG. 17  shows the PSF of the standard imaging system of  FIG. 1  with no misfocus. 
         FIG. 18  shows the PSF of the standard system of  FIG. 1  with mild misfocus. 
         FIG. 19  shows the PSF of the standard system of  FIG. 1  with large misfocus. 
         FIG. 20  shows the PSF of the C-PM system of  FIG. 2  with no misfocus, before digital processing. 
         FIG. 21  shows the PSF of the C-PM system of  FIG. 2  with no misfocus after processing. 
         FIG. 22  shows the PSF of the C-PM system of  FIG. 2  with small misfocus after processing. 
         FIG. 23  shows the PSF of the C-PM system of  FIG. 2  with large misfocus after processing. 
         FIG. 24  shows a spoke image from the standard system of  FIG. 1  with no misfocus. 
         FIG. 25  shows a spoke image from the standard system of  FIG. 1 , with mild misfocus. 
         FIG. 26  shows a spoke image from the standard  FIG. 1  system, with large misfocus. 
         FIG. 27  shows a spoke image from the  FIG. 2  C-PM system with no misfocus (before processing). 
         FIG. 28  shows a spoke image from the  FIG. 2  C-PM system with no misfocus (after processing). 
         FIG. 29  shows a spoke image from the  FIG. 2  C-PM system with mild misfocus (after processing). 
         FIG. 30  shows a spoke image from the  FIG. 2  C-PM system with large misfocus (after processing). 
         FIG. 31  shows an imaging system according to the present invention which combines extended depth of field capability with passive ranging. 
         FIG. 32  shows a phase mask for passive ranging. 
         FIG. 33  shows a phase mask for extended depth of field and passive ranging, for use in the device of  FIG. 31 . 
         FIG. 34  shows the point spread function of the  FIG. 31  embodiment with no misfocus. 
         FIG. 35  shows the point spread function of the  FIG. 31  embodiment with large positive misfocus. 
         FIG. 36  shows the point spread function of the  FIG. 31  embodiment with large negative misfocus. 
         FIG. 37  shows the point spread function of the  FIG. 31  embodiment with no extended depth of field capability and no misfocus. 
         FIG. 38  shows the optical transfer function of the  FIG. 31  embodiment with no extended depth of field capability and with large positive misfocus. 
         FIG. 39  shows the optical transfer function of the  FIG. 31  embodiment with no extended depth of field capability and with large negative misfocus. 
         FIG. 40  shows the optical transfer function of the extended depth of field passive ranging system of  FIG. 31  with a small amount of misfocus. 
         FIG. 41  shows the optical transfer function of a passive ranging system without extended depth of field capability and with a small amount of misfocus. 
         FIG. 42  shows an EDF imaging system similar to that of  FIG. 2 , with plastic optical elements used in place of the lens of  FIG. 2 . 
         FIG. 43  shows an EDF imaging system similar to that of  FIG. 2 , with an infrared lens used in place of the lens of  FIG. 2 . 
         FIG. 44  shows a color filter joined with the EDF mask of  FIG. 3 . 
         FIG. 45  shows a combined lens/EDF mask according to the present invention. 
         FIG. 46  shows a combined diffractive grating/EDF mask according to the present invention. 
         FIG. 47  shows and EDF optical system similar to that of  FIG. 2 , the lens having misfocus aberrations. 
         FIG. 48  shows an EDF optical system utilizing two masks in different locations in the system which combine to perform the EDF function, according to the present invention. 
         FIG. 49  shows an EDF imaging system similar to that of  FIG. 2 , with a self focusing fiber used in place of the lens of  FIG. 2 . 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
       FIG. 1  (prior art) shows a standard optical imaging system. Object  15  is imaged through lens  25  onto Charge Coupled Device (CCD)  30 . Such a system creates a sharp, in-focus image at CCD  30  only if object  15  is located at or very close to the in-focus object plane. If the distance from the back principal plane of lens  25  to CCD  30  is d i , and focal length of lens  25  is f, the distance, d 0 , from the front principal plane of lens  25  to object  15  must be chosen such that: 
                 1     d   o       +     1     d   i       -     1   f       =   0         
in order for the image at CCD  30  to be in-focus. The depth of field of an optical system is the distance the object can move away from the in-focus distance and still have the image be in focus. For a simple system like  FIG. 1 , the depth of field is very small.
 
       FIG. 2  shows the interaction and operation of a multi-component extended depth of field system in accordance with the invention. Object  15  is imaged through optical mask  20  and lens  25  onto Charge Coupled Device (CCD) system  30 , and image post-processing is performed by digital processing system  35 . Those skilled in the art will appreciate that any image recording and retrieval device could be used in place of CCD system  30 . 
     Mask  20  is composed of an optical material, such as glass or plastic film, having variations in opaqueness, thickness, or index of refraction. Mask  20  preferably is a phase mask, affecting only the phase of the light transmitted and not its amplitude. This results in a high efficiency optical system. However, mask  20  may also be an amplitude mask or a combination of the two. Mask  20  is designed to alter an incoherent optical system in such a way that the system response to a point object, or the Point Spread Function (PSF), is relatively insensitive to the distance of the point from the lens  25 , over a predetermined range of object distances. Thus, the Optical Transfer Function (OTF) is also relatively insensitive to object distance over this range. The resulting PSF is not itself a point. But, so long as the OTF does not contain any zeroes, image post processing may be used to correct the PSF and OTF such that the resulting PSF is nearly identical to the in-focus response of a standard optical system over the entire predetermined range of object distances. 
     The object of mask  20  is to modify the optical system in such a way that the OTF of the  FIG. 2  system is unaffected by the misfocus distance over a particular range of object distances. In addition, the OTF should not contain zeroes, so that the effects of the mask (other than the increased depth of field) can be removed in post-processing. 
     A useful method of describing the optical mask function P(x) (P(x) is described in conjunction with  FIGS. 3-30  below) is the ambiguity function method. It happens that the OTF equation for an optical system can be put in a form similar to the well known ambiguity function A(u,v). The ambiguity function is used in radar applications and has been extensively studied. The use and interpretation of the ambiguity function for radar systems are completely different from the OTF, but the similarity in the form of the equations helps in working with the OTF. The ambiguity function is given by:
 
 A ( u,v )=∫{circumflex over ( P )}( x+u/ 2) {circumflex over (P)} *( x−u/ 2) e   j2πxv   dx  
 
where * denotes complex conjugate and where the mask function P(x) is in normalized coordinates:
 
                   P   ^     ⁡     (   x   )       =       P   ^     ⁡     (     x   ⁢           ⁢     D     2   ⁢   π         )         ,     
     ⁢         P   ^     ⁡     (   x   )       =       0   ⁢        x          &gt;   π             
with D being the length of the one-dimensional mask. The above assumes two dimensional rectangularly separable masks for simplicity. Such systems theoretically can be completely described by a one dimensional mask.
 
     As is known to those skilled in the art, given a general optical mask function P(x), one can calculate the response of the incoherent OTF to any value of misfocus ψ by the equation:
 
 H ( u ,ψ)=∫( {circumflex over (P)} ( x+u/ 2) e   j(x+u/2)     2     ψ )( {circumflex over (P)} *( x−u/ 2) e   −j(x−u/2)     2     ψ ) dx  
 
The independent spatial parameter x and spatial frequency parameter u are unitless because the equation has been normalized.
 
ψ is a normalized misfocus parameter dependent on the size of lens  25  and the focus state:
 
             ψ   =         L   2       4   ⁢   πλ       ⁢     (       1   f     -     1     d   0       -     1     d   i         )             
Where L is the length of the lens, λ is the wavelength of the light, f is the focal length of lens  25 , d 0  is the distance from the front principal plane to the object  15 , and d i  is the distance from the rear principal plane to the image plane, located at CCD  30 . Given fixed optical system parameters, misfocus ψ is monotonically related to object distance d 0 .
 
     It can be shown that the OTF and the ambiguity function are related as:
 
 H ( u ,ψ)= A ( u,u ψ/π)
 
Therefore, the OTF is given by a radial slice through the ambiguity function A(u,v) that pertains to the optical mask function {circumflex over (P)}(x). This radial line has a slope of ψ/π. The process of finding the OTF from the ambiguity function is shown in  FIGS. 4-8 . The power and utility of the relationship between the OTF and the ambiguity function lie in the fact that a single two dimensional function, A(u,v), which depends uniquely on the optical mask function {circumflex over (P)}(x), can represent the OTF for all values of misfocus. Without this tool, it would be necessary to calculate a different OTF function for each value of misfocus, making it difficult to determine whether the OTF is essentially constant over a range of object distances.
 
     A general form of the one family of phase masks is Cubic phase Modulation (Cubic-PM). The general form is:
 
 P ( x,y )=exp( j (α x   3   +βy   3   +γx   2   y+δxy   2 )), | x|≦π, |y|≦π 
 
     Choice of the constants, α, β, γ, and δ allow phase functions that are rectangularly separable (with γ=δ=0) to systems whose modulation transfer functions (MTF&#39;s) are circularly symmetric (α=β=α 0 , γ=δ==−3α 0 ). For simplicity we will use the symmetric rectangularly separable form, which is given by:
 
 P ( x,y )=exp( j α( x   3   +y   3 )), | x|≦π, |y|≦π 
 
     Since this form is rectangularly separable, for most analysis only its one dimensional component must be considered:
 
 {circumflex over (P)} ( x )=exp( jαx   3 ), | x|≦π 
 
where α is a parameter used to adjust the depth of field increase.
 
       FIG. 3  shows the mask implementing this rectangularly separable cubic phase function. When α=0, the mask function is the standard rectangular function given by no mask or by a transparent mask. As the absolute value of α increases, the depth of field increases. The image contrast before post-processing also decreases as a increases. This is because as α increases, the ambiguity function broadens, so that it is less sensitive to misfocus. But, since the total volume of the ambiguity function stays constant, the ambiguity function flattens out as it widens. 
     For large enough α, the OTF of a system using a cubic PM mask can be approximated by: 
                 H   ⁡     (     u   ,   ψ     )       ≈         π     3   ⁢          α   ⁢           ⁢   u                ⁢     ⅇ       -   j     ⁢       α   ⁢           ⁢     u   3       4             ,     u   ≠   0                     H   ⁡     (     u   ,   ψ     )       ≈   2     ,     u   =   0           
Appendix A gives the mathematics necessary to arrive at the above OTF function.
 
     Thus, the cubic-PM mask is an example of a mask which modifies the optical system to have a near-constant OTF over a range of object distances. The particular range for which the OTF does not vary much is dependent of α. The range (and thus the depth of field) increases with α. However, the amount that depth of field can be increased is practically limited by the fact that contrast decreases as a increases, and eventually contrast will go below the system noise. 
       FIGS. 4 through 30  compare and contrast the performance of the standard imaging system of  FIG. 1  and a preferred embodiment of the extended depth of field system of  FIG. 2 , which utilizes the C-PM mask of  FIG. 3 . 
     In the following description, the systems of  FIG. 1  and  FIG. 2  are examined using three methods. First, the magnitude of the OTFs of the two systems are examined for various values of misfocus. The magnitude of the OTF of a system does not completely describe the quality of the final image. Comparison of the ideal OTF (the standard system of  FIG. 1  when in focus) with the OTF under other circumstances gives a qualitative feel for how good the system is. 
     Second, the PSFs of the two systems are compared. The full width at half maximum amplitude of the PSFs gives a quantitative value for comparing the two systems. Third, images of a spoke picture formed by the two systems are compared. The spoke picture is easily recognizable and contains a large range of spatial frequencies. This comparison is quite accurate, although it is qualitative. 
       FIG. 4  shows the ambiguity function of the standard optical system of  FIG. 1 . Most of the power is concentrated along the v=0 axis, making the system very sensitive to misfocus.  FIG. 5  is the top view of  FIG. 4 . Large values of the ambiguity function are represented by dark shades in this figure. The horizontal axis extends from −2π to 2π. As discussed above, the projection of a radial line drawn through the ambiguity function with slope ψ/π determines the OTF for misfocus ψ. This radial line is projected onto the spatial frequency u axis. For example, the dotted line on  FIG. 5  was drawn with a slope of 1/(2π). This line corresponds to the OTF of the standard system of  FIG. 1  for a misfocus value of ψ=½. The magnitude of this OTF is shown in  FIG. 7 . 
       FIG. 6  shows the magnitude of the OTF of the standard system of  FIG. 1  with no misfocus. This plot corresponds to the radial line drawn horizontally along the horizontal u axis in  FIG. 5 . 
       FIG. 7  shows the magnitude of the OTF for a relatively mild misfocus value of ½. This OTF corresponds to the dotted line in  FIG. 5 . Even for a misfocus of ½, this OTF is dramatically different from the OTF of the in-focus system, shown in  FIG. 6 . 
       FIG. 8  shows the magnitude of the OTF for a rather large misfocus value of ψ=3. It bears very little resemblance to the in-focus OTF of  FIG. 6 . 
       FIG. 9  shows the ambiguity function of the extended depth of field system of  FIG. 2  utilizing the C-PM mask of  FIG. 3  (the C-PM system). This ambiguity function is relatively flat, so that changes in misfocus produce little change in the system OTF. α, as previously defined, is set equal to three for this particular system, designated “the C-PM system” herein. 
       FIG. 10  shows the magnitude of the OTF of the C-PM system of  FIG. 2  before digital filtering is done. This OTF does not look much like the ideal OTF of  FIG. 6 . However, the OTF of the entire C-PM EDF system (which includes filtering) shown in  FIG. 11  is quite similar to  FIG. 6 . The high frequency ripples do not affect output image quality much, and can be reduced in size by increasing α. 
       FIG. 12  shows the magnitude of the OTF of the C-PM system of  FIG. 2  with mild misfocus (ψ=½), before filtering. Again, this OTF doesn&#39;t look like  FIG. 6 . It does, however look like  FIG. 10 , the OTF for no misfocus. Thus, the same filter produces the final OTF shown in  FIG. 13 , which does resemble  FIG. 6 . 
       FIG. 14  shows the magnitude of the OTF of THE C-PM system of  FIG. 2  with large misfocus (ψ=3), before filtering.  FIG. 15  shows the magnitude of the OTF of the entire C-PM system. Notice that it is the fact that the OTFs before processing in all three cases (no misfocus, mild misfocus, and large misfocus) are almost the same that allows the same post-processing, or filter, to restore the OTF to near ideal. 
     Note that while the OTF of the  FIG. 2  C-PM system is nearly constant for the three values of misfocus, it does not resemble the ideal OTF of  FIG. 10 . Thus, it is desirable that the effect of the  FIG. 3  mask (other than the increased depth of field) be removed by post-processing before a sharp image is obtained. The effect of the mask may be removed in a variety of ways. In the preferred embodiment, the function implemented by post-processor (preferably a digital signal processing algorithm in a special purpose electronic chip, but also possible with a digital computer or an electronic or optical analog processor) is the inverse of the OTF (approximated as the function H(u), which is constant over ψ). Thus, the post-processor  35  must, in general, implement the function: 
     
       
         
           
             
               
                 
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       FIGS. 16-23  show the Point Spread Functions (PSFs) for the standard system of  FIG. 1  and the C-PM system of  FIG. 2  for varying amounts of misfocus.  FIG. 16  shows a plot of normalized Full Width at Half Maximum amplitude (FWHM) of the point spread functions versus misfocus for the two systems. The FWHM barely changes for the  FIG. 2  C-PM system, but rises rapidly for the  FIG. 1  standard system. 
       FIGS. 17 ,  18 , and  19  show the PSFs associated with the  FIG. 1  standard system for misfocus values of 0, 0.5, and 3, (no misfocus, mild misfocus, and large misfocus) respectively. The PSF changes dramatically even for mild misfocus, and is entirely unacceptable for large misfocus. 
       FIG. 20  shows the PSF for the  FIG. 2  C-PM system with no misfocus, before filtering (post-processing). It does not look at all like the ideal PSF of  FIG. 17 , but again, the PSF after filtering, shown in  FIG. 21  does. The PSFs of the  FIG. 2  C-PM system for mild misfocus is shown in  FIG. 22 , and the PSF for the  FIG. 2  C-PM system with large misfocus is shown in  FIG. 23 . All three PSFs from the entire system are nearly indistinguishable from each other and from  FIG. 17 . 
       FIG. 24  shows an image of a spoke picture formed by the  FIG. 1  standard system with no misfocus.  FIG. 25  shows an image of the same picture formed by the  FIG. 1  standard system with mild misfocus. You can still discern the spokes, but the high frequency central portion of the picture is lost.  FIG. 26  shows the  FIG. 1  standard system image formed with large misfocus. Almost no information is carried by the image. 
       FIG. 27  is the image of the spoke picture formed by the  FIG. 2  C-PM system, before digital processing. The image formed after processing is shown in  FIG. 28 . The images formed by the complete  FIG. 2  system with mild and large misfocus are shown in  FIGS. 29 and 30 , respectively. Again, they are almost indistinguishable from each other, and from the ideal image of  FIG. 24 . 
       FIG. 31  shows an optical system according to the present invention for extended depth of field passive ranging. Passive ranging using an optical mask is described in U.S. Pat. No. 5,521,695 entitled “Range Estimation Apparatus and Method” by the present inventors, herein incorporated by reference. U.S. Pat. No. 5,521,695 discusses systems containing range dependent null space, which is substantially similar to the range dependent zeroes discussed below. 
     In  FIG. 31 , general lens system  40  has front principal plane (or focal plane)  42  and back principal plane  43 . Generally, optical mask  60  is placed at or near one of the principal planes, but mask  60  may also be placed at the image of one of the principal planes, as shown in  FIG. 31 . This allows beam splitter  45  to generate a clear image  50  of the object (not shown). Lens  55  projects an image of back focal plane  43  onto mask  60 . Mask  60  is a combined extended depth of field and passive ranging mask. CCD  65  samples the image from mask  60 . Digital filter  70  is a fixed digital filter matched to the extended depth of field component of mask  60 . Filter  70  returns the PSF of the image to a point as described above. Range estimator  75  estimates the range to various points on the object (not shown) by estimating the period of the range-dependant nulls or zeroes. 
     Briefly, passive ranging is accomplished by modifying the incoherent optical system of  FIG. 2  in such a way that range dependent zeroes are present in the Optical Transfer Function (OTF). Note that the OTF of the EDF system discussed above could not contain zeroes, because the zeroes cannot be removed by post filtering to restore the image. In  FIG. 31 , however, zeroes are added to encode the wavefront with range information. To find the range associated with small specific blocks of the image, the period of zeroes within a block is related to the range to the object imaged within the block. U.S. Pat. No. 5,521,695 primarily discusses amplitude masks, but phase masks can also produce an OTF with zeroes as a function of object range, and without loss of optical energy. Current passive ranging systems can only operate over a very limited object depth, beyond which it becomes impossible to locate the zeroes, because the OTF main lobe is narrowed, and the ranging zeroes get lost in the OTF lobe zeroes. Extending the depth of field of a passive ranging system makes such a system much more useful. 
     Consider a general mask  60  for passive ranging described mathematically as: 
                 P   ⁡     (   x   )       =       ∑     S   =   0       S   -   1       ⁢         μ   S     ⁡     (     x   -   sT     )       ⁢     ⅇ     j   ⁢           ⁢       w   s     ⁡     (     x   -   sT     )                 ,          x        ≤     π   /   S                       μ   s     ⁡     (   x   )       =       0   ⁢           ⁢   for   ⁢           ⁢        x          &gt;     π   s             
This mask is composed of S phase modulated elements μ s (x) of length T, where S·T=2π. Phase modulation of each segment is given by the exponential terms. If the above mask is a phase mask then the segments μ s (x), s=0, 1, . . . , s−1, satisfy |μ s  (x)|=1. A simple example of this type of mask is shown in  FIG. 32 . This is a two segment (S=2) phase mask where ω 0 =−π/2, ω 1 =π/2.
 
       FIG. 32  shows an example of a phase passive ranging mask  80 , which can be used as mask  60  of  FIG. 31 . This mask is called a Linear Phase Modulation (LPM) mask because each of the segments modulates phase linearly. Mask  80  comprises two wedges or prisms  81  and  82  with reversed orientation. Without optional filter  85 , the formed image is the sum of the left and right components. Optional filter  85  comprises two halves  86  and  87 , one under each wedge. Half  86  is orthogonal to half  87 , in the sense that light which passes through one half will not pass through the other. For example, the filters could be different colors (such as red and green, green and blue, or blue and red), or could be polarized in perpendicular directions. The purpose of filter  85  is to allow single-lens stereograms to be produced. A stereogram is composed of two images that overlap, with the distance between the same point in each image being determined by the object range to that point. 
       FIG. 33  shows the optical mask function of a combined LPM passive ranging mask and Cubic-PM mask  60  of  FIG. 31  which is suitable for passive ranging over a large depth of field. This mask is described by:
 
 P ( x )=μ( x ) e   j+x     3     e   jω     0     x +μ( x −π) e   j+(x-π)   3   e   jω     1     (x-π) ,
         where μ(x)=1 for 0≦x≦π,
           0 otherwise
 
By using two segments for the LPM component of mask  60 , two lobes of the PSF will be produced.
   
               

     The PSF of the imaging system of  FIG. 31 , using a mask  60  having the  FIG. 33  characteristics, with misfocus ω=0 (no misfocus), is shown in  FIG. 34 . This system will be called the EDF/PR system, for extended depth of field/passive ranging. The PSF has two peaks because of the two segments of mask  60 .  FIG. 35  shows the PSF of the EDF/PR system with Ψ=10. The fact that Ψ positive indicates that the object is on the far side of the in-focus plane from the lens. The two peaks of the PSF have moved closer together. Thus, it can be seen that the misfocus (or distance from in-focus plane) is related to the distance between the peaks of the PSF. The actual processing done by digital range estimator  75  is, of course, considerably more complicated, since an entire scene is received by estimator  75 , and not just the image of a point source. This processing is described in detail in U.S. Pat. No. 5,521,695. 
       FIG. 36  shows the PSF of the EDF/PR system with ω=−10. The fact that ψ is negative indicates that the object is nearer to the lens than is the in-focus plane The two peaks of the PSF have moved farther apart. This allows estimator  75  to determine not only how far the object is from the in focus plane, but which direction. 
     It is important to note that while the distance between the peaks of the PSF varies with distance, the peaks themselves remain narrow and sharp because of the EDF portion of mask  60  combined with the operation of digital filter  70 . 
       FIG. 37  shows the PSF of a system with an LPM mask  80  of  FIG. 31 , without the EDF portion, and with no misfocus. Since there is no misfocus,  FIG. 37  is very similar to  FIG. 34 .  FIG. 38  shows the PSF of mask  80  without EDF and with large positive misfocus (ψ=10). The peaks have moved together, as in  FIG. 35 . It would be very difficult, however, for any amount of digital processing to determine range from this PSF because the peaks are so broadened.  FIG. 39  shows the PSF of mask  80  with no EDF and large negative misfocus (ψ=−10). The peaks have moved apart, but it would be difficult to determine by how much because of the large amount of misfocus. 
     That is,  FIG. 39  shows the PSF of the LPM system without extended depth of field capability and with large negative misfocus (ψ=−10). The peaks have moved further apart, but again it would be very difficult to determine the location of the peaks. 
       FIG. 40  shows the optical transfer function of the combined EDF and LPM system shown in  FIG. 31 , with a small amount of misfocus (ψ=1). The envelope of the OTF is essentially the triangle of the perfect system (shown in  FIG. 6 ). The function added to the OTF by the ranging portion of the mask of  FIG. 33  includes range dependent zeroes, or minima. The digital processing looks for these zeroes to determine the range to different points in the object. 
       FIG. 41  shows the optical transfer function of the  FIG. 31  embodiment with no extended depth of field capability and small misfocus (ψ=1). The envelope has moved from being the ideal triangle (shown in  FIG. 6 ) to having a narrowed central lobe with side lobes. It is still possible to distinguish the range dependent zeroes, but it is becoming more difficult, because of the low value of the envelope between the main lobe and the side lobes. As the misfocus increases, the main lobe narrows and the envelope has low values over a larger area. The range-dependant minima and zeroes tend to blend in with the envelope zeroes to the extent that digital processing  70 ,  75  cannot reliably distinguish them. 
       FIG. 42  shows an optical system  100 , similar to the imaging system of  FIG. 2 , but utilizing plastic optical elements  106  and  108  in place of lens  25 . Optical elements  106 ,  108  are affixed using spacers  102 ,  104 , which are intended to retain elements  106 ,  108  at a fixed location in the optical system, with a fixed spacing between elements  106 ,  108 . All optical elements, and especially plastic elements, are subject to changes in geometry as well as changes in index of refraction with variations in temperature. For example, PMMA, a popular plastic for optical elements, has an index of refraction that changes with temperature 60 times faster than that of glass. In addition, spacers  102  and  104  will change in dimension with temperature, growing slightly longer as temperature increases. This causes elements  106 ,  108  to move apart as temperature increases. 
     Thus, changes in temperature result in changes in the performance of optical systems like system  100 . In particular, the image plane of an optical system like system  100  will move with temperature. EDF mask  20 , combined with digital processing  35 , increases the depth of field of system  100 , reducing the impact of this temperature effect. In  FIG. 42 , mask  20  is located between elements  102  and  104 , but mask  20  may also be located elsewhere in the optical system. 
     EDF mask  20  (combined with processing  35 ) also reduces the impact of chromatic aberrations caused by elements  106 ,  108 . Plastic optical elements are especially prone to chromatic aberrations due to the limited number of different plastics that have good optical properties. Common methods of reducing chromatic aberrations, such as combining two elements having different indices of refraction, are usually not available. Thus, the increase of depth of field provided by the EDF elements  20 ,  35 , is particularly important in systems including plastic elements. 
       FIG. 43  shows an infrared lens  112  used in place of lens  25  in the imaging system of  FIG. 2 . Dotted line  114  shows the dimensions of lens  112  at an increased temperature. Infrared materials such as Germanium are especially prone to thermal effects such as changes in dimension and changes in index of refraction with changes in temperature. The change in index of refraction with temperature is 230 times that of glass. EDF filter  20  and processing  35  increase the depth of field of optical system  110 , reducing the impact of these thermal effects. 
     Like plastic optical elements, infrared optical elements are more prone to chromatic aberration than glass elements. It is especially difficult to reduce chromatic aberration in infrared elements, due to the limited number of infrared materials available. Common methods of reducing chromatic aberrations, such as combining two elements having different indices of refraction, are usually not available. Thus, the increase in depth of field provided by the EDF elements is particularly important in infrared systems. 
       FIG. 44  shows a color filter  118  joined with EDF mask  20 . In some optical systems it is desirable to process or image only one wavelength of light, e.g. red light. In other systems a grey filter may be used. In systems utilizing a color filter, EDF mask  120  may be affixed to the color filter or formed integrally with the color filter of a single material, to form a single element. 
       FIG. 45  shows a combined lens/EDF mask  124  (the EDF mask is not to scale). This element could replace lens  25  and mask  20  of the imaging system of  FIG. 2 , for example. In this particular example, the mask and the lens are formed integrally. A first surface  126  implements the focusing function, and a second surface  128  also implements the EDF mask function. Those skilled in the art will appreciate that these two functions could be accomplished with a variety of mask shapes. 
       FIG. 46  shows a combined diffractive grating/EDF mask  130 . Grating  134  could be added to EDF mask  132  via an embossing process, for example. Grating  134  may comprise a modulated grating, e.g. to compensate for chromatic aberration, or it might comprise a diffractive optical element functioning as a lens or as an antialiasing filter. 
       FIG. 47  shows an EDF optical system similar to that of  FIG. 2 , wherein lens  142  exhibits misfocus aberrations. Misfocus aberrations include astigmatism, which occurs when vertical and horizontal lines focus in different planes, spherical aberration, which occurs when radial zones of the lens focus at different planes, and field curvature, which occurs when off-axis field points focus on a curved surface. Mask  20 , in conjunction with post processing  35  extend the depth of field of the optical system, which reduces the effect of these misfocus aberrations. 
       FIG. 48  shows an optical system  150  utilizing two masks  152 ,  156  in different locations in the system, which combine to perform the EDF mask function of mask  20 . This might be useful to implement vertical variations in mask  152  and horizontal variations in mask  156 , for example. In the particular example of  FIG. 48 , masks  152 ,  156  are arrayed on either side of lens  154 . This assembly could replace lens  25  and mask  20  in the imaging system of  FIG. 2 , for example. 
       FIG. 49  shows an optical imaging system like that of  FIG. 2 , with lens  25  replaced by a self focusing element  148 . Element  148  focuses light not by changes in the thickness of the optical material across the cross section of the element (such as the shape of a lens), but rather by changes in the index of refraction of the material across the cross section of the element. 
     While the exemplary preferred embodiments of the present invention are described herein with particularity, those having normal skill in the art will recognize various changes, modifications, additions and applications other than those specifically mentioned herein without departing from the spirit of this invention.