Abstract:
A method of unambiguous range estimation is provided for use with a range imaging system that derives phase images from image pixels of a digital image. The method involves generating (a) a first phase image having one or more ambiguous phase intervals and (b) at least one additional phase image that is generated by shifting the phase intervals of the first phase image. Then at least one region of intersection between phase intervals in the two phase images is identified. Finally, the phase of at least one of the ambiguous phase intervals in the first phase image is adjusted based on values of the phase of the image pixels that belong to the region of intersection. As a result, the phase adjustment unwraps the phase ambiguity in the phase intervals of the first phase image.

Description:
FIELD OF THE INVENTION 
     The invention relates generally to the field of processing range images captured by a range imaging system, and in particular to the estimation and unwrapping of phase images captured by a scanner-less laser radar range imaging system. 
     BACKGROUND OF THE INVENTION 
     A scanner-less laser radar range imaging system is described in U.S. Pat. No. 4,935,616. The system described therein illuminates a scene object with an amplitude modulated laser source, wherein the amplitude modulation is in the range of 10MHz. The image capture portion of the system includes a micro-channel plate that is capable of modulating the optical signal reflected from the scene object. The phase shift of the intensity modulation reflected from the scene object can be calculated by capturing two images, one without modulating the optical signal, and another with the optical signal modulated by the micro-channel plate in phase with the same amplitude modulated frequency as used to modulate the laser source. Both images are registered spatially, and the difference between them is caused by the interference of the two modulating wave patterns, which produces a dc signal proportional to the phase shift. Once the phase shift has been established, range to the object can be computed. 
     Since the phase shift can only be determined modulo 27π the resulting range can only be found to within one wavelength of the modulation of the laser. To calculate the range at each point in the image, the correct integer number of phase cycles must be added to each phase measurement; that is, the phase must be “unwrapped”. It is therefore desirable to resolve the ambiguous (or wrapped) phase measurements to determine unambiguous (or unwrapped) phase. 
     The unambiguous phase, in turn, can be used to calculate unambiguous range. The aforementioned &#39;616 patent suggests modulating the laser and receiver with different frequencies in order to produce two range images with different modulating frequencies. This would yield range unambiguous to within one wavelength of the wave whose frequency is the greatest common factor of the frequencies of the laser and receiver, which is a lower frequency than either of the two modulating frequencies. Even though this may reduce ambiguity problems in many situations, they still exist albeit on a smaller scale. 
     There are two main types of methods for solving the phase ambiguity problem: branch-cut methods and weighted least-squares methods. 
     Branch-cut methods (such as those described in Goldstein, Zebker, and Werner, “Satellite radar interferometry: two-dimensional phase unwrapping”,  Radio Science , Vol. 23, pp. 713-720, 1998; and Prati, Giani, and Leurati, “SAR interferometry: A 2-D phase unwrapping technique based on phase and absolute values informations”,  Proc. Int. Geoscience  &amp;  Remote Sensing Symposium IGARSS  1990, Washington, D.C., pp. 2043-2046, 1990) use lines to connect phase inconsistencies (or residues). Branch-cut methods fail to perform adequately when the number of residues is high. They often resort to local averaging, which is undesirable because it can dampen high frequency information. Least-squares methods (such as those described in Ghiglia and Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods”,  Journal of the Optical Society of America,  Vol. 11, pp.107-117, 1994; and Pritt and Shipman, “Least-squares two dimensional phase unwrapping using FFT&#39;s”,  IEEE Transactions on Geoscience and Remote Sensing,  Vol. 32, pp. 706-708, 1994) determine a phase finction that minimizes the error in the gradient estimates. If there are areas in the ambiguous phase image with a high noise content, nearby areas with a low noise content can be distorted by a least-squares phase unwrapping algorithm. 
     SUMMARY OF THE INVENTION 
     An object of the invention is to provide a method of unambiguous range estimation without the drawbacks in the known branch-cut and weighted least squares methods. 
     It is a further object of this invention to unambiguate (or unwrap) the measured phase shifts using a morphological technique. 
     It is a further object of this invention to provide a method of unwrapping phase shifts obtained by a scanner-less laser radar range imaging system that produces an image bundle (at least three images corresponding to distinct phase offsets of the capture device and/or illumination source). 
     The present invention is directed to overcoming one or more of the problems set forth above. Briefly summarized, according to one aspect of the present invention, a method of unambiguous range estimation is provided for use with a range imaging system that derives phase images from image pixels of a digital image. The method involves generating (a) a first phase image having one or more ambiguous phase intervals and (b) at least one additional phase image that is generated by shifting the phase intervals of the first phase image. Then at least one region of intersection between phase intervals in the two phase images is identified. Next, the phase of at least one of the ambiguous phase intervals in the first phase image is adjusted based on values of the phase of the image pixels that belong to the region of intersection. As a result, the phase adjustment unwraps the phase ambiguity in the phase intervals of the first phase image. 
     Instead of using a path-following technique or a technique based on a least-squares solution, this invention will resolve phase ambiguities by the use of a “region-moving” algorithm. By utilizing a model where ambiguous phase is determined by capturing images from at least three different phase shifts of either the illumination source (e.g., a laser) or the capture device (e.g., an intensifier) to obtain a wrapped phase image, a parameter in the model is then perturbed that spatially shifts regions of ambiguity in the wrapped phase image. 
     Using morphological image processing techniques, the spatial movements of the regions of ambiguity are followed. Appropriate phase offsets are determined by this region-moving technique, generating an unambiguous phase image. The advantage of this region-moving technique in relation to other phase-unwrapping techniques is that it determines unambiguous phase without distortions (i.e. given only the output of the region-moving technique, the input can be found exactly). 
     These and other aspects, objects, features and advantages of the present invention will be more clearly understood and appreciated from a review of the following detailed description of the preferred embodiments and appended claims, and by reference to the accompanying drawings. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a block diagram of a known range imaging system which can be used in the practice of the invention to capture a bundle of images. 
     FIG. 2 is a diagram of a method of computing an ambiguous range image from an image bundle captured by the range imaging system shown in FIG.  1 . 
     FIG. 3 is an illustration a shift in a region of ambiguity. 
     FIGS. 4A and 4B are illustrations of the effect of using a region-moving algorithm to determine appropriate phase offsets to remove ambiguities in the ambiguous range images captured in FIG.  2 . 
     FIG. 5 is an illustration of a region of intersection between the two phase images shown in FIGS. 4A and 4B. 
     FIG. 6 is an illustration of the steps involved in unwrapping an ambiguous range image in accordance with the invention. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Because range imaging devices employing illumination sources with laser illuminators, and capture devices employing image intensifiers and electronic sensors, are well known, the present description will be directed in particular to elements forming part of, or cooperating more directly with, apparatus in accordance with the present invention. Elements not specifically shown or described herein may be selected from those known in the art. Certain aspects of the embodiments to be described may be provided in software. Given the system as described in the following materials, all such software implementation needed for practice of the invention is conventional and within the ordinary skill in such arts. 
     In the following description, a preferred embodiment of the present invention will be described at least in part as a software program. Those skilled in the art will readily recognize that the equivalent of such software may also be constructed in hardware. Because phase unwrapping algorithms and methods are well known, the present description will be directed in particular to aspects of such algorithms and methods forming part of, or cooperating more directly with, the present invention. Other aspects of such algorithms and systems, and hardware and/or software for producing and otherwise processing the image signals involved therewith, not specifically shown or described herein may be selected from such systems, algorithms, components and elements known in the art. 
     Still further, as used herein, the computer program may be stored in a computer readable storage medium, which may comprise, for example; 
     magnetic storage media such as a magnetic disk (such as a floppy disk) or magnetic tape; optical storage media such as an optical disc, optical tape, or machine readable bar code; solid state electronic storage devices such as random access memory (RAM), or read only memory (OM); or any other physical device or medium employed to store a computer program. 
     Referring first to FIG. 1, an range imaging system  10  is shown as a laser radar that is used to illuminate a scene  12  and then to capture an image bundle comprising a minimum of three images of the scene  12 . An illuminator  14  emits a beam of electromagnetic radiation whose frequency is controlled by a modulator  16 . Typically the illuminator  14  is a laser device which includes an optical diffuser in order to effect a wide-field illumination. The modulator  16  provides an amplitude varying sinusoidal modulation. The modulated illumination source is modeled by: 
     
       
           L ( t )=μ L +η sin(2 πλt )  (Eq. 1)  
       
     
     where μ L  is the mean illumination, η is the modulus of the illumination source, and λ is the modulation frequency applied to the illuminator  14 . The modulation frequency is sufficiently high (e.g., 10 mHz) to attain sufficiently accurate range estimates. The output beam  18  is directed toward the scene  12  and a reflected beam  20  is directed back toward a receiving section  22 . As is well known, the reflected beam  20  is a delayed version of the transmitted output beam  18 , with the amount of phase delay being equivalent to the distance of the scene  12  from the image capture system. The reflected beam  20  is sensed by a photocathode  24 , which converts its amplitude variations into a modulated electron stream that strikes an image intensifier  26 . The output of the image intensifier  26  is modeled by: 
     
       
           M ( t )=μ M +γ sin(2 πλt )  (Eq. 2)  
       
     
     where μ M  is the mean intensification, γ is the modulus of the intensification and λ is the modulation frequency applied to the intensifier  26 . The purpose of the image intensifier is not so much to intensify the image, but rather to act as a modulating shutter. Upon entering the image intensifier  26 , the electron stream first strikes a thin conductive sheet  28  connected to the modulator  16 , where the electron stream is modulated by a modulating signal from the modulator  16 . The modulated electron stream is then amplified through secondary emission by a microchannel plate  30 . The intensified electron stream bombards a phosphor screen  32 , which converts the energy into a visible light image. The intensified light image signal is captured by a capture mechanism  34 , such as a charge-coupled device (CCD). The captured image signal is applied to a range processor  36  to determine the phase offset at each point in the scene. The phase term ω of an object at a range ρ meters is given by:              ω   =         2      ρ                 λ     c        mod                 2                 π             (     Eq   .              3     )                                
     where c is the velocity of light in a vacuum. The reflected light at this point is modeled by: 
     
       
           R ( t )=μ L +κ sin(2 πλt +ω)  (Eq. 4)  
       
     
     where κ is the modulus of illumination reflected from the object. The pixel response P at this point is an integration of the reflected light and the effect of the intensification:              P   =         ∫   0     2      π              R        (   t   )            M        (   t   )                          t         =       2                   μ   L          μ   M        π     +     κ                 π                 γ                   cos        (   ω   )                     (     Eq   .              5     )                                
     In the range imaging system disclosed in the aforementioned &#39;616 patent, a reference image is captured during which time the micro-channel plate  30  is not modulated, but rather kept at a mean response. In that case, equation (5) fundamentally is unchanged, though M(t) is now simply a constant μ M . The range is estimated for each pixel by recovering the phase term as a function of the value of the pixel in the reference image and the phase image. There are several reasons why this approach is not robust. Included in this is the fact that the analysis depends upon continuous values. The range estimation is based upon the portion of the phase image relative to the reference image. For digital systems the relative quantization of the phase image to the reference image decreases as the response of the reference image decreases. The system is also somewhat noise sensitive. 
     A robust approach which overcomes the limitations of the method proposed in the &#39;616 patent is described in copending application Serial No. 09/342,370, entitled “method and Apparatus for Scannerless Range Image Capture Using Photographic Film” and filed in the names of Lawrence Allen Ray and Timothy P. Mathers, which is assigned to the assignee of this application and which is incorporated herein by reference. Instead of collecting a phase image and a reference image, the improved approach collects at least three phase images (referred to as an image bundle). In the previous approach, the micro-channel plate  30  and the laser illuminator  12  were phase locked. The improved approach shifts the phase of the micro-channel plate  30  relative to the phase of the illuminator  12 , and each of the phase images has a distinct phase offset. For this purpose, the range processor  36  is suitably connected to control the phase offset of the modulator  16 , as well as the average illumination level and such other capture functions as may be necessary. If the image intensifier  26  (or laser illuminator  14 ) is phase shifted by θ i , the pixel response from equation (5) becomes: 
     
       
           P   i =2μ L μ M π+κπγ cos(ω+θ i )  (Eq. 6)  
       
     
     It is desired to extract the phase term ω from the expression. 
     However, this term is not directly accessible from a single image. In equation (6) there are three unknown values: the mean term μ L μ M , the moduli term κγ and the phase term ω. As a result, mathematically only three samples (from three images) are required to retrieve an estimate of the phase term caused by the distance to an object in the scene. Therefore, a set of three images captured with unique phase shifts is sufficient to determine ω. For simplicity, the phase shifts are given by θ i =2πi/3; i=0,1,2. In the following description, an image bundle shall be understood to constitute a collection of images which are of the same scene, but with each image having a distinct phase offset obtained from the modulation applied to the micro-channel plate. It should also be understood that an analogous analysis may be performed by phase shifting the illuminator  14  instead of the micro-channel plate  30 . 
     FIG. 2 shows stages in the computation of an ambiguous phase image from the image bundle. If images are captured with n≧3 distinct phase offsets of the intensifier (or laser or a combination of both) these images form an image bundle  40 . Applying Equation (6) to each image in the image bundle and expanding the cosine term (i.e., P i =2μ L μ M π+κπγ(cos(ω)cos(θ i )−sin(ω)sin(θ i ))) results in the following system of linear equations in n unknowns at each point:                (           P   1               P   2             ⋮             P   n           )     =       (         1         cos                   θ   1               -   sin                     θ   1               1         cos                   θ   2               -   sin                     θ   2               ⋮       ⋮       ⋮           1         cos                   θ   n               -   sin                     θ   n             )          (           Λ   1               Λ   2               Λ   3           )               (     Eq   .              7     )                                
     where Λ=2μ L μ M π, Λ 2 =κπγ cos ω, and Λ 3 =κπγ sin ω. If only three images are used, this linear system of equations is well-determined and its solution lends itself to a wide variety of numerical algorithms. In order to minimize rounding errors, the algorithm used is the LU decomposition with pivoting (which is described in Golub and Van Loan,  Matrix Computations , Johns Hopkins University Press, 3 rd  ed., 1996). Many other methods of solution exist; e.g. decomposition-substitution methods, computing an inverse directly, Cramer&#39;s rule, etc.; however, all of these methods perform an equivalent task. If more than three images are used, the linear system is over-determined and its solution lends itself to a wide variety of least squares techniques. For example, if an image bundle comprising more than three images is captured, then the estimates of range can be enhanced by a least squares analysis using a singular value decomposition (see, e.g., W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling,  Numerical Recipes  ( the Art of Scientific Computing ), Cambridge University Press, Cambridge, 1986). In particular, a singular value decomposition is chosen (see Golub and Van Loan) because of its robustness against rounding errors. Many other methods of solution exist; e.g. QR decomposition, Givens transformation, Lanczos transformation, modified Gram-Schmidt, transformation to a quadratic programming problem, etc.; however, all of these methods perform an equivalent task. 
     This system of equations is solved by one of the aforementioned methods of solution  42  to yield the vector Λ=[Λ 1 , Λ 2 , Λ 3 ] τ . Since this calculation is carried out at every (x,y) location in the image bundle, Λ is really a vector image  44  containing a three element vector at every point. The phase term ω is computed in block  46  at each point using the four-quadrant aretangent calculation: 
     
       
         ω=tan −1 (Λ 3 , Λ 2 )  (Eq. 8)  
       
     
     The resulting collection of phase values at each point forms the ambiguous phase image  48 . 
     Once phase has been determined, range r can be calculated by:              r   =     ω                   c     4      π                 λ                 (     Eq   .              9     )                                
     Equations (1)-(9) describe the method of estimating range using an image bundle with at least three images (i.e., n=3) corresponding to distinct phase offsets of the intensifier and/or laser. However, since co is wrapped into a principal phase interval (−ππ], the estimated range is ambiguous to within one wavelength. 
     The principal object of the invention is to resolve phase ambiguities; i.e., to determine what integral multiples of 2π should be added to ω at each pixel location to yield an unwrapped phase image. The phase unwrapping method described in this invention spatially moves regions of ambiguity in the phase image by changing the principal phase interval from (−π, π] to (−π+α, π+α] where |a|&lt;π/2. FIG. 3 shows the shift in the principal phase interval. The original phase interval, (−π, π], is shown by an interval indicator  50 , and the shifted interval (−π+α, π+α] is shown by an interval indicator  52 . This shifting can be accomplished by letting the phase ω α {circumflex over (ω)}=α and modifying the value of Λ 2  and Λ 3  in Equation (7) to become Λ 2 =κπλ cos({circumflex over (ω)}+α) and Λ 3 =κπγ sin({circumflex over (ω)}+α). Expanding the sine and cosine terms and manipulating algebraically yields: 
     
       
         Λ 3 −Λ 2  tan α=sin {circumflex over (ω)}(cos α+sin α tan α)  (Eq. 10)  
       
     
     
       
         Λ 2 +Λ 2  tan α=cos {circumflex over (ω)}(cos α+sin α tan α)  (Eq. 11)  
       
     
     Solving the equations in (10) and (11) for {circumflex over (ω)} yields: 
     
       
         {circumflex over (ω)}=tan −1 (Λ 3 −Λ 2  tan α,Λ 2 +Λ 3  tan α)  (Eq. 12)  
       
     
     Since {circumflex over (ω)}ε(−π,π], the phase shift ω α ={circumflex over (ω)}+αε(−π+α,π+α]. The ambiguity regions in the image of phase shifts (o. (which will be denoted Ω α , for the remainder of this disclosure) have been moved spatially from those in the image of phase shifts ω. 
     The phase unwrapping method described to this point will be seen to operate upon two perturbed phase images, the aforementioned first phase image Ω α  perturbed by the phase shift a and a second phase image Ω β  perturbed by a phase shift β. The second phase image Ω β  has regions of ambiguity that are spatially moved by changing the principal phase interval from (−π, π] to (−π+β, π+β], where α&lt;β. FIG. 3 also shows this shift in the principal phase interval; the shifted interval (−π+β, π+β] is shown by an interval indicator  54 . However, in its most general sense, the phase unwrapping method described in this invention can be understood to operate upon two phase images, one phase image with unperturbed ambiguous phase, i.e, the aforementioned first phase image Ω α  or second phase image Ω β  with either α=0 or β=0. In both the general case with one perturbed phase image and the specific case with two (or more) perturbed phase images, the invention is practiced similarly, that is, by finding (as will be explained) regions of intersection between the two phase images. (Also note that when α=0, then ω α ={circumflex over (ω)} for the “first” phase image, which accordingly is the “original” phase image derived from Equation (8).) 
     FIG. 4A shows the boundaries between regions in the first phase image Ω α  ( 56 ) whose principal phase interval is (−π+α, π+α], and FIG. 4B shows the boundaries between regions in the second phase image Ω β  ( 58 ), whose principal phase interval is (−π+β, π+β]. Tracking the movement of the ambiguity regions from image Ω α  to image Ω β , where α&lt;β without loss of generality, yields the appropriate offsets to unwrap the phase image. The first step S 100  in this process, as shown in FIG. 6, is therefore one of generating first and second phase images Ω α , Ω β  from an ambiguous original phase image by respectively shifting the ambiguous phase intervals by first and second phase shifts α, β, wherein the two phase shifts are bounded by −π/2&lt;α&lt;β&lt;π/2. The next step S 102  is to identify the ambiguity regions in image Ω α  and Ω β . Region boundaries exist where the phase shifts wrap; therefore any of a variety of edge-detection techniques can be used to find where the differences in phase approach 2π. It may also be necessary to employ a morphological technique to bridge boundary pixels together. Once all regions have been identified, each image can be represented as the union of mutually exclusive regions:                Ω   α     =         ⋃     i   =   1       m          Ω   α     (   i   )                 (     Eq   .              13     )                 Ω   β     =         ⋃     i   =   1       n          Ω   β     (   i   )                 (     Eq   .              14     )                                
     where Ω κ   (i)  is the i th  region of Ω κ  (any ordering), and m and n are the number of regions in Ω α  and Ω β , respectively. 
     If α and β differ significantly, then for every i=1,2 , . . . m, Ω α   (i)  will overlap at least one region of Ω β  which overlaps at least two regions of Ωα. Likewise, for every j=1,2, . . . n, Ω β   (j)  will overlap at least one region of Ω α  which overlaps at least two regions of Ω β . Referring again to FIG. 6, and without loss of generality, a first region Ω α(l)  is chosen in step S 104  to contain unwrapped phase values. Other regions of Ω α  can be unwrapped by the following process. First, in step S 106  a first region Ω β   (κ)  of the second phase image Ω β  that overlaps both the first region Ω α   (l)  and another second region in Ω α , say Ω α   (p)  of the first phase image is identified. (FIG. 4 illustrates the regions Ω α   (l) , Ω β   (k) , and Ω α   (p) ; FIG. 5 shows the region Ω β  with a region of intersection between Ω β   (k)  and Ω α   (p) ). If the pixels in Ω β  belonging to a region of intersection  60  (see FIG. 5) between Ω β   (k)  and Ω α   (p)  take on values in (π+α, π+β], which values are within an interval  62   a  of the plot in FIG. 3, then in step S 108  the pixels in Ω α  belonging to Ω α (p) should be offset by 2π. If the pixels in Op belonging to the region of intersection  60  between Ω β   (k)  and Ω α   (p)  take on values in (−π+β, π+α], which values are within an interval  62   b  of the plot in FIG. 3, then (also in step S 108 ) the pixels in Ω α  belonging to Ω α   (p)  should be offset by −2π. If any other regions of Ω β  overlap both Ω α   (l)  and another region in Ω α , say Ω α   (q) , then an offset for Ω α   (l)  is determined in the same way. 
     Once the process has been completed for Ω α   (l) , Ω α   (l)  is excluded from the set of regions making up Ω α , and the entire process is repeated for the other regions of Ω α , starting with the bordering regions of Ω α   (l) . However, new offsets that are determined must be added to the offsets of the current iteration&#39;s starting region of Ω α . Once all regions of Ω α  have been examined in turn, the offsets accumulated for each region are added to the phase shift values of Ω α , producing an unambiguous (unwrapped) phase image. 
     In summary, a technique of estimating unambiguous phase shifts using a laser radar range imaging system has been developed. After capturing at least three images corresponding to phase offsets of the modulated laser source or receiver, the phase shift of the intensity modulation reflected from an object can be calculated. This phase shift can be used to calculate range to the object; however, phase shifts are ambiguous (they can only be determined modulo 2π). An additional parameter is introduced into the phase calculation step. The perturbation of this parameter combined with a morphological image processing technique identifies regions where an additive offset should be applied in order to remove phase ambiguities. 
     The invention has been described with reference to a preferred embodiment. However, it will be appreciated that variations and modifications can be effected by a person of ordinary skill in the art without departing from the scope of the invention. 
     Parts List 
       10  range imaging system 
       12  scene 
       14  illuminator 
       16  modulator 
       18  output beam 
       20  reflected beam 
       22  receiving section 
       24  photocathode 
       26  image intensifier 
       28  conductive sheet 
       30  microchannel plate 
       32  phosphor screen 
       34  capture mechanism 
       36  range processor 
       40  image bundle 
       42  method of solution 
       44  vector image 
       46  four-quadrant arctangent 
       48  ambiguous phase image 
       50  original phase interval 
       52  first shifted phase interval 
       54  second shifted phase interval 
       56  first phase image 
       58  second phase image 
       60  region of intersection 
       62   a  interval indicator 
       62   b  interval indicator 
     S 100 -S 108  steps