Abstract:
A wide-angle zoom lens with as few as two plastic elements codes the wavefront produced by the imaging system such that the imaging system is substantially invariant to aberrations related to misfocus. Signal processing is used to decode the wavefront to form the final image. A first type of zoom lens configuration uses as few as two lens elements. In this type, image processing may be modified to take into account the positioning of the lenses.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
   This patent application is a continuation-in-part of commonly-owned and U.S. patent application Ser. No. 09/910,675, now abandoned, filed on Jul. 20, 2001 and incorporated herein by reference. This patent application is also a continuation-in-part of commonly-owned and U.S. patent application Ser. No. 09/070,969, filed on May 1, 1998 now abandoned and incorporated herein by reference. U.S. patent application Ser. No. 09/070,969 is a continuation-in-part of U.S. patent application Ser. No. 08/823,894, filed Mar. 17, 1997, now U.S. Pat. No. 5,748,371, issued May 5, 1998 and incorporated herein by reference. U.S. patent application Ser. No. 08/823,894 is a continuation of U.S. patent application Ser. No. 08/384,257, filed Feb. 3. 1995, now abandoned. U.S. application Ser. No. 09/875,435, filed Jun. 6, 2001, now U.S. Pat. No. 6,525,302, and pending U.S. application Ser. No. 09/875,766, filed June 6, 2001, and Ser. No. 09/766,325, filed Jan. 19, 2001, are each incorporated herein by reference. 

   BACKGROUND 
   Zoom lens designs are based on the property that the power of an optical system consisting of at least two lens groups can be varied by changing the distance between the groups. The lens capabilities depend on the number of moving groups in the system. This is discussed by W. J. Smith in “Modern Optical Engineering” McGraw-Hill, 1990. In any zoom system, at least two lens groups must be moved with respect to each other in order to have a variable focal length system and a fixed image plane position. 
   The complexity of a lens mechanical mount, or cam, is determined by the number of moving groups within the zoom lens. An example of a simple cam with two grooves is shown in W. J. Smith, FIG. 9.31, p. 276. 
   More moving optical groups may be required if other optical system characteristics are needed such as quality imaging over a range of object distances with large zoom power, or if the entrance and exit pupil locations need to be fixed. More elements within each group are often required to compensate for aberrations, as is the case with any traditional lens system. 
   Most of the modem miniature zoom lenses are composed of two groups of negative and positive powers. Such systems then have small size but a long back focal length, which is a serious drawback. For minimization purposes, these lens groups are further divided into subgroups that move independently to extend the zooming range and to attempt to minimize the overall size of the system. See, for example, U.S. Pat. No. 4,936,661 granted to E. I. Betensky, et al Jun. 26, 1990, U.S. Pat. No. 5,270,861 and U.S. Pat. No. 5,270,867 both granted to L. R. Estelle on Dec. 14, 1993. A two-element zoom system with negative and positive plastic elements is discussed in U.S. Pat. No. 5,473,473 granted to L. R. Estelle on Dec. 5, 1995. This is a 35 mm format lens with a speed of F/11 in the wide-angle position. 
   There is a continuing need for a small, compact, and inexpensive zoom lens system. 
   SUMMARY OF THE INVENTION 
   One feature herein is to provide a fast zoom lens imaging system with a reduced number of lens elements that provides high quality images over a large field of view, and at different zoom positions. Such a system enables simple and inexpensive fast wide-angle zoom lens with as few as two plastic elements. The cost of the zoom lens imaging system is directly reduced by minimizing the number of elements in the system and/or indirectly by reducing fabrication and assembly tolerances required to produce the system. 
   In one aspect, the number of elements in the zoom lens imaging system is reduced by coding the wavefront produced by the zoom lens system such that the imaging system is substantially invariant to aberrations related to misfocus. Such aberrations include, for example, chromatic aberration, spherical aberration, curvature of field, astigmatism, fabrication and assembly related misfocus, and temperature related misfocus. Image processing is used to decode the formed images and produce the final images. 
   In the prior art, such aberrations are not easily accommodated in a zoom lens with few lenses because of the large number of aberrations to be controlled and because of the changing parameters in the zoom imaging system. One feature of the zoom lens system herein shows how high quality images can be formed with a reduced number of lenses. 
   An extended depth of field zoom lens system according to one aspect includes a detector, a lens system between the object to be imaged and the detector comprising at least two lenses, and Wavefront Coding optics between the object and the detector. The Wavefront Coding optics are constructed and arranged to alter the optical transfer function of the zoom lens system in such a way that the altered optical transfer function is substantially less sensitive to focus related aberrations than was the unaltered optical transfer function. The Wavefront Coding optics affects the alteration to the optical transfer function substantially by affecting the phase of light transmitted by the optics. A post processing element processes the image captured by the detector, by reversing the alteration of the optical transfer function accomplished by the optics. 
   The Wavefront Coding optics may be integrally formed with at least one of the lenses. In one aspect, information regarding the location of the lenses in the lens system are provided to the post processing element. The processing applied by the post processing element is adjusted according to the lens information. More generally, information regarding the point spread function (PSF) of the lens system is provided to the post processing element and processing is modified according to the information. 
   In another aspect, the lens system comprises at least three lenses, and the lens system is constructed and arranged to have a constant working F/#. In this aspect, it is not necessary to provide the processing element with any information regarding PSF or lens position. 
   As a feature, the detector may be a charge coupled device (CCD). At least one of the lenses in the lens system may be made of optical plastic. The lens system may comprise two lenses in a positive/positive lens element configuration. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIGS. 1A and 1B  show one zoom imaging system, with two lens elements; one or more of the lenses perform Wavefront Coding. 
       FIGS. 2A and 2B  show one zoom imaging system, with three lens elements such that the working F/# is constant; one or more of the lenses perform Wavefront Coding. 
       FIG. 3  shows a cubic phase function that produces an extended depth of field. 
       FIGS. 4A and 4B  show ray traces for a two-element zoom lens. 
       FIGS. 5A-5D  show modulation transfer functions (MTFs) for an imaging system with no Wavefront Coding at wide angle and telephoto settings. 
       FIGS. 6A-6D  show through-focus MTFs at 10 lp/mm for a two element zoom system without Wavefront Coding for wide angle and telephoto settings. 
       FIGS. 7A-7D  show MTFs for an imaging system with Wavefront Coding at wide angle and telephoto settings, before processing. 
       FIGS. 8A-8D  show through-focus MTFs at 10 lp/mm for a two element zoom system with Wavefront Coding for wide angle and telephoto settings, before processing. 
       FIGS. 9A-9D  show the wide angle and telephoto MTFs of  FIGS. 6A-6D  after signal processing. 
       FIG. 10A  shows a spatial domain linear filter according to the present invention for processing the intermediate image in order to produce the final image. 
       FIG. 10B  shows the transfer function of the linear filter of FIG.  10 A. 
   

   DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
   By coding the image forming wavefront and performing image processing on the resulting images zoom lenses can be designed that are very fast (small F/#) with a reduced number of optical elements. These zoom lenses can also have a very wide field of view and the equivalent of a flat image plane. By coding the wavefront and using image processing the zoom system can have an increased the depth of field and depth of focus well as reduced system sensitivity to misfocus aberrations. The extension of the depth of focus also means that the zoom lens can be made insensitive to temperature changes. In a similar fashion, manufacturing and assembly tolerances can be relaxed so that the accuracy with which the optics and detector array must be placed is reduced. 
   There are two primary forms of zoom lens systems that use Wavefront Coding. The first form, shown in FIG.  1 A and  FIG. 1B , uses as few as two lens elements  302 ,  304 . By changing the distance between the two lens elements  302 ,  304  the value of the system focal length is varied, but the working F/# of the system also changes. With the working F/# varying, the PSFs and MTFs of the system can also vary. Image processing has access to lens position information so that the configuration of the optics is known to image processing. Image processing is optimized for groups of working F/#s, or equivalently for regions of system focal lengths, which are then automatically selected and used to process the resulting images as a function of zoom system configuration. 
   A second form of zoom system is shown in FIG.  2 A and  FIG. 2B , which uses a minimum of three lens elements, and which can maintain a constant working F/# with system focal length. When the working F/# is held constant, the PSFs and MTFs are also constant with zoom configuration. Since the PSFs and MTFs are not a function of the zoom system configuration, the digital processing (element  410 ) does not require information about the position of the optics. 
   More particularly,  FIG. 1A  shows a zoom imaging system  305  with two lens elements  302  and  304 , at least one of which has a modified surface to code the wavefront. Lens position information  307 A is used to select appropriate image processing  310  such that a final image  312  is formed.  FIG. 1B  shows the same zoom imaging system  305  in a different zoom position, which has a different lens position information  307 B sent to the image processing  310  to form the final image  312 . One reason image processing block  310  uses lens position information  307  in a two lens system  305  is illustrated by the ray angles near the detector  308  in  FIG. 1A  compared to the ray angles near the detector of FIG.  1 B. The rays enter the detector at different angles for the two lens configurations. When the ray angles are different for the two configurations the working F/#s, PSFs and MTFs for the two configurations  307 A,  307 B are also different. The processing applied by image processing block  310  accounts for these differences. 
     FIGS. 2A and 2B  show a zoom imaging system  405  with three lens elements  402 ,  404 , and  406  which are constructed and arranged such that the working F/# is constant as the system focal length is varied. One or more of the lens elements  402 ,  404 , and  406  have modified optics to perform Wavefront Coding. Image processing block  410  of system  405  does not necessarily utilize lens position information associated with positions of lens elements  402 ,  404 ,  406  because image processing applied by block  410  does not depend on knowledge of the configuration of lens elements  402 ,  404 , and  406  to obtain the final image. This is illustrated by the ray angles to the right of element  406  in  FIG. 2A  compared to the ray angles FIG.  2 B. The rays enter the detector at the same angles independent of the system focal length. Thus the working F/#, PSFs, and MTFs are not a function of the focal length of the system and image processing  410  does not need knowledge of the configuration of the optics. 
   To make such zoom lenses, one or more of the optical elements  302  and  304  of FIG.  1 A and  FIG. 1B , and  402 ,  404 , and  406  of FIG.  2 A and  FIG. 2B , are wavefront coded so that the resulting images  312 ,  412  are insensitive to focus related aberrations. In one aspect, a phase variation structure is applied to one or more of these optical elements. For example, the thickness of one or more of the lenses can be varied in such a manner as to apply the desired wavefront (phase) modifications. Other methods of modifying the wavefront include use of optical materials that have a spatially varying index of refraction and/or thickness, use of spatial light modulators, use of holograms, and/or use of micro mirror devices. 
     FIG. 3  shows one example of a wavefront coding suitable for application to lens  302 ,  304 ,  402 ,  404 , or  406 , illustrating thickness variations that encode the wavefront of light passing through the lens. Such lens modifications apply a wavefront phase function that produces an extended depth of field in the resulting images, after post processing by image processing  310 ,  410 . In one example, the phase function is a cubic phase function mathematically described as: 
    separable-cubic-phase( x,y )= K[x   3   +y   3 ] 
   where K is a constant. 
   In another example, the phase fuction is a non-separable conventional Wavefront Coding phase function, which in normalized coordinates is: 
         non   ⁢     -     ⁢   separable   ⁢     -     ⁢   cubic   ⁢     -     ⁢     phase   ⁡     (     p   ,   θ     )         =       ∑   i     ⁢       a   i     ⁢     p   bi     ⁢     cos   ⁡     (         w   i     ⁢   θ     +     ϕ   i       )               
 0≦p≦1, 0≦θ≦2pi.
 
Other alternative Wavefront Coding phase functions may be described as: 
         cubic   ⁢     -     ⁢   related   ⁢     -     ⁢     forms     ⁢           ⁢     (     x   ,   y     )       =       ∑   i     ⁢       a   i     ⁡     [       sign   ⁡     (   x   )       |   x   ⁢     |   bi     ⁢     +     sign   ⁡     (   y   )         |   y   ⁢     |   bi       ]             
 |x|≦1, |y|≦1
         where sign(x)=+1 for x≧0, sign(x)=−1 otherwise.
 
For an odd integer b, these related forms trace out “cubic like” profiles of increasing slopes near the end of the aperture. For b with values between the odd integers, the related forms trace out other “cubic like” profiles that lie between the ones generated when b is an odd integer.
       

   The above phase functions are for example useful in controlling misfocus and for minimizing optical power in high spatial frequencies. Minimizing the optical power at high spatial frequencies is often called antialiasing. When using a digital detector such as a CCD or CMOS device to capture an image, optical power that is beyond the spatial frequency limit of the detector masquerades or “aliases” as low spatial frequency power. For example, say that the normalized spatial frequency limit of a digital detector is 0.5. If the in-focus MTF from the conventional system with no Wavefront Coding produces a considerable amount of optical power beyond this spatial frequency limit, then aliasing artifacts can degrade the resulting images. By adding misfocus to the system without Wavefront Coding, the amount of high spatial frequency optical power can be decreased, reducing aliasing. With Wavefront Coding, as in  FIGS. 1 and 2 , the amount of optical power that can be aliased also decreases. In comparison to the prior art, the amount of aliasing in a wavefront coded system does not increase with a change of focus. 
     FIGS. 4A and 4B  show ray traces for a two-element zoom lens  602  with Wavefront Coding in two configurations. Lens system  602  is in the zoom lens form used in FIG.  1 .  FIG. 4A  shows ray traces for the wide angle configuration (top plot) and the telephoto configuration (bottom plot) for standard imaging of objects at infinity.  FIG. 4B  shows ray traces for the wide angle configuration (top plot) and the telephoto configuration (bottom plot) in a macro mode for objects at 200 mm. 
   A two element zoom lens system has a total of three combinations of lens elements that can be used. These combinations are:
         1. Positive/positive   2. Positive/negative   3. Negative/positive       

   Traditional two element zoom systems nearly always employ either the positive/negative or negative/positive lens element configurations. This is because the use of positive and negative lens element combinations allows the lens designer to minimize the aberration of petzval curvature that otherwise would drastically limit the field of view of the traditional zoom system. Designs that employ the positive/positive lens element combination can have the shortest overall length, compared to designs that use negative lens elements, but also implicitly have the largest amount of petzval curvature. In traditional designs this petzval curvature is large enough to preclude the practical use of the positive/positive arrangement for traditional two element zoom systems. 
   In many zoom lens designs, minimum overall length and wide field of view are both desired. By using Wavefront Coding methods described above, the two element zoom lens design can use the positive/positive lens element combination in order to minimize the overall length of the zoom lens system while correcting the aberration of petzval curvature and other focus related aberrations by coding the wavefront and image processing the resulting images. Use of Wavefront Coding thus enables the design of a shorter zoom lens than is possible with traditional design methods. FIG.  4 A and  FIG. 4B  show a positive/positive zoom system  602 . 
   One embodiment of the positive/positive two-element zoom system  602  is specified below. This zoom system has been designed to image in a standard mode with objects at infinity, and in a macro mode with objects near 200 mm. The zoom system also works well with objects at intermediate positions. The full field of view of lens system  602  continuously varies from about 23° to 52°. This system is designed to be used with a digital detector with 5.6 micron square pixels and a Bayer color filter array. This detector also has lenslet array. In order to ensure maximum light collection by the lenslet array, the maximum chief ray angles for each of these configurations have been designed to be under 11°. Those skilled in the art of optical design will realize that this or similar lens systems can be used with a variety of other digital detector formats as well. All dimensions below are given in mm and indices of refraction and dispersions (V) are for the d line of the spectrum. Surface number 1 is the front of the first lens element. The mechanical layout of preferred embodiment is: 
                                                                 SURFACE   RADIUS   THICKNESS   INDEX   V                                1   ASPHERE   0.482   1.530   55.8       2   ASPHERE   (A)       3   ASPHERE   2.855   1.530   55.8       4   ASPHERE   (B)       Image                    
Surface #2 is the stop. Surface #2 also contains the Wavefront Coding surface. The thickness of surfaces 2 and 4 vary with zoom configuration. The lens material is the optical plastic zeonex.
 
   The rotationally symmetric aspheric surface height as a function of spatial position, or radius, is given: 
       Z   =         C   ·     r   2         1   +       1   -       (     K   +   1     )     ⁢       C   2     ·     r   2                 +     D   ·     r   4       +     E   ·     r   6       +     F   ·     r   8       +     G   ·     r   10       +     H   ·     r   12             
 
Certain constants that define the rotationally symmetric surfaces are given as:
 
   
     
       
             
             
             
             
           
         
             
                 
             
           
           
             
               Surface 1 
               C = 0.233386 
               D = −0.031277 
               F = −0.128988 
             
             
                 
               K = 3.656 
               E = 0.080978 
               G = 0.087080 
             
             
                 
                 
                 
               H = −0.010498 
             
             
               Surface 2 
               C = 0.002507 
               D = 0.029598 
               F = 0.103280 
             
             
                 
               K = 0.0 
               E = −0.089061 
               G = 0.0 
             
             
                 
                 
                 
               H = 0.0 
             
             
               Surface 3 
               C = −0.085283 
               D = −0.012930 
               F = 0.011175 
             
             
                 
               K = 53.030 
               E = −0.014721 
               G = 0.004873 
             
             
                 
                 
                 
               H = 5.699E−04 
             
             
               Surface 4 
               C = −0.459841 
               D = 0.006828 
               F = −2.809E−04 
             
             
                 
               K = −0.344 
               E = −3.565E−04 
               G = 7.026E−05 
             
             
                 
                 
                 
               H = −5.739E−06 
             
             
                 
             
           
        
       
     
   
   Surface 2 contains the stop as well as the Wavefront Coding surface. The Wavefront Coding surface is used in addition to the rotationally symmetric surface 2 defined above. The Wavefront Coding surface form may be defined as:
 
 S ( x,y )=β 1 [sign( x )| x|   α     1   +sign( y )| y|   α     1   ]+β 2 [sign( x )| x|   α     2   +sign( y )| y|   α     2   ]
         where 
         x   =       x     un   -   normalized         |     x   max     |         ,           ⁢     y   =       y     un   -   normalized         |     y   max     |             
 
and where sign(x)=+1 for x≧0, and sign(x)=−1 otherwise, The parameters β 1 , and β 2  control the contribution of each term and α 1  and α 2  control the maximum slope of each term. The values of α and β are:
   β 1 =26.666, α 1 ,=3.006   β 2 =69.519, α 2 =9.613       

   The distance between the two lenses (A) of system  602  is a function of the focal length the zoom system. The distance from the second lens to the image detector (B), also known as the back focal length, is a function of the focal length and object position. In the standard imaging mode, with the object at infinity, the system distances, lengths, and working F/#s are: 
   Standard imaging, object at infinity 
   
     
       
             
             
             
             
             
           
             
             
             
             
             
           
         
             
                 
             
             
                 
                 
               Back focal 
                 
                 
             
             
                 
               Lens spacing 
               length 
               Overall 
               Working 
             
             
               Focal Length 
               (A) 
               (B) 
               Length 
               F/# 
             
             
                 
             
           
           
             
                 
             
           
        
         
             
               3.864 
               0.725 
               2.794 
               6.857 
               2.8 
             
             
               6.136 
               4.226 
               1.549 
               9.113 
               4.3 
             
             
               9.454 
               6.315 
               0.100 
               9.753 
               6.2 
             
             
                 
             
           
        
       
     
   
   When used in macro mode, the object position can be as close as 200 mm. Back focal length (B) varies with object distance. Lens spacing (A) is the same in standard and macro imaging. In the macro imaging mode, with the object at 200 mm, the system distances, lengths, and working F/#s are: 
   Macro imaging, object at 200 mm 
   
     
       
             
             
             
             
             
           
             
             
             
             
             
           
         
             
                 
             
             
                 
               Lens 
               Back focal 
                 
                 
             
             
               Focal 
               spacing 
               length 
               Overall 
               Working 
             
             
               Length 
               (A) 
               (B) 
               Length 
               F/# 
             
             
                 
             
           
           
             
                 
             
           
        
         
             
               3.864 
               0.725 
               2.870 
               6.930 
               2.8 
             
             
               6.136 
               4.226 
               1.770 
               9.332 
               4.3 
             
             
               9.454 
               6.315 
               0.391 
               10.044 
               6.0 
             
             
                 
             
           
        
       
     
   
   The performance of wavefront coded zoom lens system  602 , as specified above, is described and compared to a zoom system not using Wavefront Coding in  FIGS. 5 through 10 .  FIGS. 5 and 6  describe the MTF characteristics of the zoom system without Wavefront Coding.  FIGS. 7 and 8  describe the MTF performance of the zoom system with Wavefront Coding but before image processing  410 .  FIG. 9  describes the MTF performance of the zoom system  602  after image processing  410 .  FIG. 10  describes the digital filters used in image processing  410 . 
   The MTFs of the zoom system without Wavefront Coding are described in FIG.  5 . The zoom system without Wavefront Coding is as described above but with the Wavefront Coding parameters β 1 =β 2 =0.  FIGS. 5A and 5B  describe the system in standard imaging mode with the object at infinity at the shortest focal length or widest imaging angle and at the longest focal length or narrowest imaging angle or telephoto respectively.  FIGS. 5C and 5D  are similar to  FIGS. 5A and 5B  with the system in macro imaging mode and the object being at 200 mm.  FIG. 5C  describes wide angle imaging while  FIG. 5D  describes telephoto imaging. The Wavefront Coding design method consists of minimizing, through traditional design methods, the non-focus related aberrations, such as coma, lateral color, and distortion. Focus related aberrations are controlled both through traditional design techniques and through Wavefront Coding via the optics and image processing. 
   With the positive/positive lens element configuration of zoom system  602 , the largest monochromatic aberrations are related to field curvature. The effects of field curvature are clearly seen in the off-axis MTFs of the  FIGS. 5A-5C . In these figures the full-field MTFs have lower responses than the on-axis MTFs. The full-field MTFs also have zeros caused by misfocus as a function of field angle (or field curvature) within the spatial frequency limit of the Bayer detector of 44 lp/mm. This two element zoom system without Wavefront Coding would image well only at small field angles or with a very small sized detector. 
     FIG. 6  describes the MTFs of the zoom system without Wavefront Coding at a spatial frequency of 10 lp/mm over a −0.2 mm to +0.2 mm deviation from the best focused image plane, or the through focus MTFs at 10 lp/mm. These curves again clearly show the limiting nature of field curvature on the zoom system without Wavefront Coding.  FIGS. 6A-6D  are arranged as in  FIG. 5  with  FIGS. 6A and 6B  describing imaging with the object at infinity at wide angle and telephoto positions respectively.  FIGS. 6C and 6D  describe similar in a macro mode with the object at 200 mm. In  FIGS. 6A and 6C  the peak of the full field MTF is seen to be around −0.2 mm from best focus while the peak of the on-axis MTF is about +0.1 mm from best focus. Best focus has been adjusted to balance the effects field curvature so that the 0.7 field MTF is at best focus.  FIGS. 6B and 6D  show similar but less dramatic effects of field curvature due to the smaller field angles of the telephoto configurations. From  FIG. 6  there is no one focus position with the system without Wavefront Coding where all field angles are well focused. 
     FIG. 7  shows the MTFs from the two element zoom system  602  with Wavefront Coding, but before image processing  410 , according to the present invention.  FIGS. 7A and 7B  represent MTFs with the object at infinity at wide angle and telephoto configurations respectively.  FIGS. 7C and 7D  represent the MTFs with the object at 200 mm at wide angle and telephoto configurations respectively. From the MTFs of  FIGS. 7A-7D  notice that there is very little change in MTFs with field angle. All MTFs for each configuration are essentially identical, especially compared to the MTFs from the system without Wavefront Coding shown in FIG.  5 . Notice also that the MTFs of  FIG. 7  does not match the diffraction limited MTFs. The wavefront coded MTFs are lower than the diffraction limited MTFs but higher than the off-axis MTFs from the system without Wavefront Coding in FIG.  5 . Image processing  410  is used to essentially transform the MTFs shown in  FIG. 7  to any desired MTF. Typically image processing  410  is used to form MTFs that lay between the unprocessed wavefront coded MTFs and the diffraction limited MTFs. 
     FIGS. 8A-8D  describes the through focus MTFs at 10 lp/mm of the zoom system  602  with Wavefront Coding, but without image processing  410 . The arrangement of  FIGS. 8A-8D  is similar to that of  FIGS. 7A-7D . Notice that the response of the through focus MTFs are much more independent of focus shift than the system without Wavefront Coding shown in FIG.  6 . From  FIG. 8A  there is a large region, at least +/−0.2 mm, where the image plane can be positioned and still have essentially identical performance. By not having separated peaks of the through focus MTFs as a function of field angle, the Wavefront Coding MTFs are seen to not suffer from effects of field curvature. By also having a large region over which the image plane can be positioned and still image clearly, the wavefront coded system is seen to also have a large depth of focus. The depth of focus is seen to be the least for  FIG. 8C  as the response curves as a function of field angle vary the most for this configuration (wide angle, object at 200 mm). 
     FIGS. 9A-9D  describes the MTFs for zoom system  602  with Wavefront Coding and with image processing  410 .  FIGS. 9A and 9B  describe the MTFs with the object at infinity imaging in wide angle and telephoto configurations respectively.  FIGS. 9C and 9D  describe the MTFs when the object is at 200 mm and in wide angle and telephoto configurations respectively. The MTFs of  FIG. 9  include the MTFs due to the optics and the MTFs due to the 5.6 micron square pixel Bayer detector. The diffraction limited MTFs shown in  FIG. 9  are those of  FIG. 7  with the addition of the detector MTFs. Each figure shows the diffraction limited MTF, the MTFs before image processing  410 , and the MTFs after image processing  410 . The MTFs after image processing, or filtering, extend to the spatial frequency limit of the digital detector or 44 lp/mm. The MTFs after filtering for  FIGS. 9A-9D  lay between the MTFs before filtering and the diffraction limited MTFs. The corresponding PSFs after filtering, not shown, are spatially very compact. Only one digital filter is applied to each configuration of the zoom system. For example when imaging with a wide angle and object at infinity ( FIG. 9A ) a single digital filter is applied to all images. When the optics are changed to image in telephoto mode with the object at infinity ( FIG. 9B ) another digital filter is applied to all images resulting from this configuration. 
     FIG. 10  describes one dimension of the two dimensional digital filter used to form the MTFs after filtering in FIG.  9 . The two dimensional filter can be implemented as a rectangularly separable digital filter.  FIG. 10A  describes one dimension of a rectangularly separable filter.  FIG. 10B  shows the transfer function of the spatial domain filter of FIG.  10 A. 
   For zoom system  602 , image processing  410  uses the digital filter from  FIG. 10A  in order to form the final images  412 . Computationally efficient rectangularly separable digital filtering may be used for implementations where the total number of multiplications and additions is minimized. General two dimensional linear filtering can also be used for processing flexibility. The operation of rectangularly separable filtering is to first filter each row (or column) independently with a one dimensional row (or column) filter. The filtered rows (or columns) form an intermediate image. Columns (or rows) of the intermediate image are then independently filtered with the column (or row) filter. This forms the final image. 
   The actual filter values as shown in  FIGS. 10A and 10B  are typically chosen to produce MTFs that match some desired MTF performance as well as produce PSFs that also match some desired spatial performance. MTF criteria after filtering typically include a minimum MTF values for groups of spatial frequencies. PSF criteria after filtering typically include a spatially compact shape with a maximum size for image artifacts. The actual digital filters can be calculated through least squares methods or through nonlinear computer optimization.