Abstract:
A software implemented system for algorithmic correction of systematic image distortions within fingerprint imaging systems. The system may implement a three dimensional geometric model of a fingerprint imaging system to discover where a configuration prescribed by a conceptual fingerprint imaging system and an actual configuration of a manufactured fingerprint imaging system differ. By describing this difference using the model, images captured by the manufactured fingerprint imaging system can be rectified to generate rectified images with relatively low amounts of distortion present. Rectification to remove distortion based on the model, without physically adjusting and/or correcting the manufactured fingerprint imaging system or its components, may enable the fingerprint imaging system to be manufactured with relatively lower tolerances without degrading a precision of the images generated by the system, potentially enabling enhanced precision of generated images without increasing various costs of the fingerprint imaging system (or its components) generating the images.

Description:
REFERENCE TO RELATED APPLICATIONS 
     This application claims priority to U.S. Provisional Application No. 60/713,765, filed Sep. 6, 2005, which is hereby incorporated by reference in its entirety. 
    
    
     FIELD OF THE INVENTION 
     The invention relates generally to software implemented, algorithmic correction of image distortions within fingerprint imaging systems. 
     BACKGROUND OF THE INVENTION 
     Conventional fingerprint imaging systems may implement distortion correction, also known as rectification, to correct for distortion that may typically include correction of radial distortions introduced by lenses. Such radial distortions may include barrel and pincushion distortion. Rectification also may correct for perspective distortions—those distortions which may be introduced into an imaging system due to the physical relationship between a principle point on the lens (e.g., the iris) and the scene being captured. Some fingerprint imaging systems may rectify radial and perspective correction in the system. For example, some conventional systems include hardware that may rectify radial and perspective distortion. However, other distortions may be present within a captured image due to curved surfaces in the light path. For instance, imaging through glass balls or from the reflection in a hyperbolic mirror may introduce distortions in a captured image. 
     Collectively, these types of image distortions may be referred to as systematic imaging distortions. More specifically, systematic image distortions may include those distortions caused by surfaces within a fingerprint imaging system at which light may be processed (e.g., lens surfaces, mirror surfaces, other refractive surfaces, other reflective surfaces, and/or other surfaces formed by optical elements). 
     Generally, conventional rectifying solutions share a focus on correcting images that have been captured with a simple optical configuration within a fingerprint imaging system—a system which may include a camera and a lens that takes images or photographs of a scene in three dimensions. Conventional solutions often fail to address more complex optical configurations that include optical elements beyond the lens. That is, the simple imaging systems may not compensate for images captured with multiple optical elements, or a more complex optical element, in the light path, such as windows, prisms, non-aligned lenses, balls, or other transparent or semi-transparent media or reflective surfaces. 
     Traditionally, attempts to minimize distortion in fingerprint imaging systems have been conducted during a design phase. Often, an iterative design process may be used in which building and analyzing prototype systems results in the final fingerprint imaging system design. The resulting design that meets the distortion goals may incorporate allowable tolerances on system components. Often, additional optical components may be required in the design to correct distortions to an acceptable level. 
     During manufacturing, fingerprint imaging systems may be built according to the final fingerprint imaging system design. Typically, components are assembled and during a quality assurance step, the components are adjusted to ensure that the distortions in each manufactured system do not exceed various tolerances. In order ensure that system distortions do not exceed these tolerances, optical components of a relatively high quality and expense are often used. For example, to adhere to the overall system tolerances, the components themselves may be required to be manufactured with high precision, thereby increasing a cost of manufacturing the corresponding optical component. 
     In some instances a calibration may be implemented to assure that the optical components are aligned with a predetermined precision (i.e., within predetermined alignment tolerances). Calibration typically involves adjusting component locations with respect to one another to correct distortion in images captured by the manufactured systems. This calibration step can be rather involved for higher precision fingerprint imaging system designs with relatively tight alignment tolerances. 
     During operation, performance of a fingerprint imaging system may degrade over time. Generally, restoring the calibration of a fingerprint imaging system typically includes a physical adjustment of components within the system to correct for image distortions, and to ensure that the alignment of the optical components within the fingerprint imaging system falls within the predetermined alignment tolerances, which may be costly and/or time consuming. Other drawbacks in conventional fingerprint imaging systems exist. 
     SUMMARY 
     These and other drawbacks are addressed by various embodiments of the invention. 
     One aspect of the invention relates to a software implemented system and method for algorithmic correction of image distortions, such as systematic distortions, within fingerprint imaging systems. The system and method may implement a three dimensional geometric model of a fingerprint imaging system to discover where a configuration of a conceptual fingerprint imaging system, built according to a system design with no (or substantially no) imperfections, and an actual configuration of a manufactured fingerprint imaging system built according to the system design differ. The difference (or differences) between the conceptual fingerprint imaging system and the manufactured fingerprint imaging system may arise due to imperfections in the manufactured fingerprint imaging system, including imperfections in alignment and/or configuration of the optical elements, and imperfections in the optical elements themselves. By describing this difference using the geometric model, systematic distortion caused by the imperfections within the manufactured fingerprint imaging system may be corrected for, thereby enabling images captured by the manufactured fingerprint imaging system to be rectified in operational use to correct for the systematic distortion to generate rectified images with relatively low amounts of residual distortion present. Rectifying the images to remove systematic distortion based on the geometric model, without physically adjusting and/or correcting the manufactured fingerprint imaging system or its components, enables the manufactured fingerprint imaging system to be manufactured with relatively lower tolerances, without degrading the precision of the images generated by the system. This may enable an enhancement in the precision of generated images and/or a lower cost for comparable precision. 
     In a system design phase, a three dimensional geometric model of the conceptual fingerprint imaging system may be determined. This geometric model may describe the surfaces in the conceptual fingerprint imaging system that transmit, reflect, refract light, and/or otherwise process light. For example, an optical design tool such as Zemax may provide an ability to define such a geometric model as a series of surfaces. 
     In a manufacturing phase, a fingerprint imaging system may be manufactured according to the conceptual fingerprint imaging system. Once a fingerprint imaging system is built, a geometric model of the fingerprint imaging system may be determined based on one or more images of a predetermined target captured by the fingerprint imaging system. The target may include objects that may be measured with accuracy below the tolerance level desired in output images from the fingerprint imaging system. For instance, the target may include a precision glass target with dark circles provided at a periodic pitch on a reflective background. 
     In some embodiments of the invention, determining the geometric model of the fingerprint imaging system may include inputting an image of the target captured by the fingerprint imaging system, adjusting surface parameters that describe the surfaces within the fingerprint imaging system at which light may be processed, and returning the actual relative optical component locations within the manufactured fingerprint imaging system. The surface parameters may describe, for example, a location of a surface, a directional orientation of a surface, an index of refraction of an optical element that forms the surface, and/or other parameters. In some instances, one tool implemented to adjust the surface parameters may include a merit function that numerically compares predicted locations of the objects in the target to the observed locations of the objects in the captured image of the target. 
     According to various embodiments of the invention, once an image of the target is captured using the fingerprint imaging system, the merit function may be implemented to compare the conceptual fingerprint imaging system to the fingerprint imaging system. By running a geometric model determination method, surface parameters of the geometric model describing the conceptual fingerprint imaging system can be adjusted so that the relative location of the surfaces in the geometric model describe the surfaces as they are located and formed within the manufactured fingerprint imaging system. The surface parameters may be adjusted until the geometric model predicts an image that adequately coincides with image of the target captured by the manufactured fingerprint imaging system. That is, by adjusting the merit function&#39;s value toward a predetermined value (e.g., 0), the geometric model defining the relative three-dimensional location of the surfaces in the manufactured fingerprint imaging system may become an adequate predictor of how light within the manufactured fingerprint imaging system may be processed to generate an image. The surface parameters determined for the geometric model may then be stored for image rectification. 
     In operation, a positional relationship between an observed pixel in an image captured by the fingerprint imaging system and a rectified pixel in a rectified image suitable for output and/or further processing may be determined using the geometric model&#39;s surface parameters. More specifically, the geometric model may enable ray tracing to be used to generate one or more rectified pixels based on one or more observed pixels in a captured image. In some instances, an interpolation method may be employed to determine the rectified pixels. For example, the relationship between rectified pixel locations and observed pixel locations can be encoded as one or several pre-calculated lookup tables which may then be used for real-time image reconstruction. 
     In some embodiments of the invention, one or both of the observed image and the rectified image may be analyzed to determine the likelihood that the system is still within the tolerances, for the alignment of optical components within the system as well as the components themselves, specified when the system was manufactured. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is an exemplary illustration of a system for determining a geometric model of a fingerprint imaging system, according to some embodiments of the invention. 
         FIG. 2  is an exemplary illustration of a fingerprint imaging system, in accordance with some embodiments of the invention. 
         FIG. 3  is an exemplary illustration of a method of determining a geometric model of a fingerprint imaging system, according to some embodiments of the invention. 
         FIG. 4  is an exemplary illustration of a captured image and a rectified image generated from the captured image, in accordance with various embodiments of the invention. 
     
    
    
     DETAILED DESCRIPTION 
       FIG. 1  is an exemplary illustration of a system  110  for determining a geometric model of a fingerprint imaging system  112 . The geometric model may be implemented, by system  110 , for example, to rectify images captured by fingerprint imaging system  112 . System  110  may include fingerprint imaging system  112 , processor  114 , and target  116 . Although processor  114  may be illustrated as a single component in  FIG. 1 , it may be appreciated that processor  114  may include a plurality of processors connected via an operative link. In some embodiments, the plurality of processors may be located centrally in a single location. In other embodiments, one or more of the plurality of processors may be located remotely from each other. The operative link between the plurality of processors may include a communications link, such as a wired, or wireless communications link, and may include a connection established over a network, or via a direct connection. Information may pass between fingerprint imaging system  112  and processor  114  via an operative link. The information may include control information, geometric model information, setting information, image information, or other information. 
       FIG. 2  is an exemplary illustration of fingerprint imaging system  112 , according to some embodiments of the invention. Fingerprint imaging system  112  may include a radiation source  210  and one or more optical elements. For example, the optical elements may include a prism  212 , a mirror  214 , and a sensor  216 . In other embodiments of the invention, the optical elements may include other types of elements, such as lenses, apertures, and/or other optical elements. Additionally, the various optical elements may be arranged in configurations different from the one illustrated in  FIG. 2  without departing from the scope of the invention. 
     In some embodiments of the invention, light emitted by radiation source  210  may be incident on a platen  218  of prism  212  at an angle greater than the critical angle of prism  212  such that the light may be internally reflected from platen  218 . At platen  218 , a pattern of an object in contact with platen  218 , such as the fingerprints of an individual holding at least a portion of his/her hand in contact with platen  218 , may be imparted to the light. The light reflected from platen  218  may pass out of prism  212  at a prism face  220  and become incident on mirror  214 . The light from prism  212  may be reflected by mirror  214  through a principle point  222  of fingerprint imaging system  112  and onto sensor  216 . Sensor  216  may include an electronic image sensor, such as a CMOS chip, a CCD chip, or another image sensor. The light incident on sensor  216  may form an image of the object in contact with platen  218 . Sensor  216  may capture the image formed by the light. 
     It should be appreciated that the embodiment of fingerprint imaging system  112  shown in  FIG. 2  is provided for illustrative purposes, and that the invention contemplates the implementation of any arrangement of optical components capable of imaging the fingerprints of an individual. For example, fingerprint imaging system  112  may include a finger print imaging system as described in the co-pending U.S. patent application Ser. No. 11/030,327, which is incorporated herein by reference. 
     In some embodiments of the invention, fingerprint imaging system  112  may be built according to a fingerprint imaging system design for a conceptual fingerprint imaging system. As with most manufactured systems, the specifications of fingerprint imaging system  112 , including the location and orientation of optical elements  212 ,  214  and  216 , as well as optical elements  212 ,  214 , and  216  themselves, may be manufactured and assembled within predefined tolerances of the specifications of the conceptual fingerprint imaging system. However, the manufacturing of fingerprint imaging system  112  according to the specifications may not be perfect. Deviations of fingerprint imaging system  112  from the conceptual fingerprint imaging system, even deviations within the predefined tolerances, may lead to distortion in the image of the object that is captured by sensor  216 . 
     Returning to  FIG. 1 , processor  114  may include a model determination module  118  and a rectification module, among other modules. It should be appreciated that the representation of modules  114  and  116  are provided for illustrative purposes, and that each module may include one or more components that perform the functionalities assigned to modules  114  and  116 , as well as other functions. Modules  114  and  116  may include components implemented as hardware, software, firmware, a combination of hardware, software, and/or firmware, as well as in other mediums. 
     In some embodiments of the invention, model determination module  118  may operate to determine a geometric model of fingerprint imaging system  112 . The geometric model may account for the deviations between the specifications of fingerprint imaging system  112  and the conceptual fingerprint imaging system. In some instances, model determination module  118  may implement a geometric model determining method such as the one described in further detail below to determine the geometric model. 
     According to various embodiments of the invention, rectification module  120  may operate to rectify images captured by fingerprint imaging system  112  in accordance with the geometric model. Rectification module  120  may use the geometric model of fingerprint imaging system  112  determined by model determination module  118  to rectify the images captured. In some embodiments, rectification module  120  may implement the geometric model in a bilinear interpolation to rectify captured images. 
       FIG. 3  illustrates a method  310  of determining a geometric model of a fingerprint imaging system, in accordance with some embodiments of the invention. At a step  312 , a conceptual fingerprint imaging system may be designed. For example, designing the conceptual fingerprint imaging system may include determining the location and orientation of one or more optical elements within the conceptual fingerprint imaging system to capture an image. In some embodiments, the one or more optical elements may include a prism, a mirror, an imaging surface, a lens, a beam splitter, a film, or other optical elements. The location and orientation of the one or more optical elements may be determined to capture a certain type (or types) of images, such as, for instance, hand or finger prints. The conceptual fingerprint imaging system may be designed using optical design software such as Zemax that includes ray tracing and other optical design capabilities. In one embodiment, the conceptual fingerprint imaging system may include the conceptual fingerprint imaging system for fingerprint imaging system  112  of  FIG. 1 . 
     At a step  314 , one or more surfaces at which light is processed (e.g., reflected, refracted, captured, etc.) within the conceptual fingerprint imaging system may be determined. The surfaces may include the surfaces of the optical elements at which light may be processed within the fingerprint imaging system. For example, some embodiments in which the conceptual fingerprint imaging system includes the conceptual fingerprint imaging system for fingerprint imaging system  112  of  FIG. 1 , the surfaces may include platen  218 , prism face  220 , mirror  214 , and sensor  216 . 
     At a step  316 , the location and orientation of the surfaces within the conceptual fingerprint imaging system may be defined in terms of one or more parameters, such as, for example, surface parameters. The surface parameters may describe, for example, a location of a surface, a directional orientation of a surface, a shape of a surface, an index of refraction of an optical element that forms a surface, and/or other parameters. In combination, the surface parameters may form an initial geometric model that may describe the conceptual fingerprint imaging system. Or, in other words, the initial geometric model would describe the manufactured fingerprint imaging system, if that fingerprint imaging system were manufactured to be substantially identical to the conceptual fingerprint imaging system, and included virtually no imperfections in the configuration of the one or more optical elements and/or the optical elements themselves. 
     In some embodiments, the surface parameters may include: three 3-dimensional points that may define flat sensor  216 , the three points may include two points that define an edge of sensor  216  as a vector and a third point on sensor  216  that does not lie on this edge; a single 3-dimensional point that defines the principal point of the imaging system; one 3-dimensional point on the surface of mirror  214  and one 3-dimensional direction vector that defines a normal to the surface of mirror  214 ; the distances from each of three corners on sensor  216  to prism face  220  along the optical path; the intersection vector between the prism face  220  and the platen  218 ; and the angle between the prism face  220  and the platen  218 . These 28 parameters may define the spatial relationship between platen  218 , prism face  220 , mirror  214 , principal point  222 , and sensor  216  in three dimensional space. It should be appreciated that in other embodiments alternative parameters may be used to completely define the relationships of the surfaces (or the optical components), and that the parameters above are recited for illustrative purposes. 
     At a step  318 , a target may be imaged by a fingerprint imaging system built according to the design of the conceptual fingerprint imaging system. The target may be located at a predetermined imaging location with respect to the fingerprint imaging system. The target may include graphics in which one or more distinguishing points may be emphasized graphically. The distinguishing points may include lines, squares, points, circles, rectangles, or any other shape or mark that may be geometrically analyzed to according to the algorithm described herein. The locations and/or distances between the distinguishing points may be measured and recorded with some degree of precision prior to step  318 , as a reference. It should be appreciated that in order measure and/or record the locations and/or distances between the distinguishing points, the same position on each distinguishing point may be identified (e.g., the center, a common corner, etc.). 
     In some embodiments of the invention, the target may include target  116  of system  110 . Target  116  may include a precision target that includes distinctive objects located in a periodic manner on a reflective background. For example, target  116  may include a chrome on glass target that has chrome dots with a diameter of 1.5 mm provided at a 3.0 mm pitch. Target  116  may be mated with platen  218  using an index matching fluid. Such a target may be commercially available from various sources including Applied Optics. 
     At a step  320 , an evaluation function may be determined. The evaluation function yields values expressing differences between predictions related to a calculated location (or locations) of the image of the target based on the conceptual fingerprint imaging system and a measured location (or locations) of the image of the target captured by the actual fingerprint imaging system. The predictions may include predictions of the location of the images of the distinguishing points, predictions of the distances between the images of the distinguishing points, or other predictions. These predictions may be made via a ray tracing capability of the optical design software used to design conceptual fingerprint imaging system. More specifically, ray tracing may be used to predict what the locations of the images of the distinguishing points, or the distances of the images of the distinguishing points, would be by tracing light backwards through the conceptual fingerprint imaging system, from the surface in the conceptual fingerprint imaging system wherein the image of the target would be captured, through the optical elements in the fingerprint imaging system, to the surface in the conceptual fingerprint imaging system where the target would be positioned. The optical design software may leverage surface parameters to trace the light through the conceptual fingerprint imaging system in order to make these predictions. Thus, the surface parameters may be parameters of the evaluation functions. 
     For example, an evaluation function F, may describe a difference between a predicted distance between two particular distinguishing points, points i and j, for example, and a distance between a captured image of the distinguishing point i and a captured image of the distinguishing point j, as a function of the points i and j, in which the surface parameters (p) are implemented as parameters of F. This evaluation function F may be described as
 
 d   i,j   =F ( i, j|p ).  (1)
 
     Another way of conceptualizing the evaluation function may be as a plurality of evaluation functions, one for each pair of distinguishing points in the image of the target, that determines the difference between a predicted distance between two or more distinguishing points in the captured image, and the measured distance between the corresponding distinguishing points in the captured image, as a function of the surface parameters (p). This set of evaluation functions may be represented as F 1,2 (p), F 1,3 (p), . . . , F 1,n (p), F 2,1 (p), F 2,3 (p), . . . , F 2,n (p), . . . , F n,n (p) for distinguishing points i=1 to n. Note that F i,i (p) is 0, because the distance between any distinguishing point and itself is 0. It should be appreciated that alternate evaluation functions that describe the performance of the fingerprint imaging system may be implemented. 
     At a step  322 , the actual image of the target captured by the fingerprint imaging system may be measured. Measuring the image of the target may include determining the locations of the images of the distinguishing points and/or measuring the distances between the distinguishing points within the image. In some embodiments of the invention in which the target includes circular dots provided at a predetermined pitch, the images may be distorted in the image captured by the fingerprint imaging system. In such embodiments, circular dots may be extracted from the target by implementing an algorithm that assumes that the circles are ellipses, and determines the centers of the ellipses. For example, an algorithm may be implemented that uses a super-resolution approach to identifying the edges of the ellipses to enable a determination of the centers of the ellipses within an acceptable tolerance. For each pair of ellipses, a distance between the centers of the ellipses may be determined. For instance, if 100 dots were identified in the captured image, there are 100*9/2=4,950 dot pairs for which distances may be determined. 
     Beginning at a step  324 , a data-fitting algorithm may be implemented to minimize the evaluation function(s) by adjusting the surface parameters to more accurately represent the actual configuration of the fingerprint imaging system that may include imperfections in the optical elements and/or their location and/or orientation within the system. The data-fitting algorithm may include a known iterative non-linear data-fitting algorithm, such as a Levenberg-Marquardt algorithm, a Gauss Newton algorithm, or another iterative or non-iterative data-fitting algorithm for linear or non-linear systems. For example, a Levenberg-Marquardt algorithm may be implemented to minimize a metric function (S) where the metric function is the sum of the evaluation functions. This sum may be represented mathematically as 
     
       
         
           
             
               
                 
                   
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     At the step  324 , the value of the metric function may be determined for the current surface parameters. The value of the metric function may represent a “goodness of a fit” between the current surface parameters and the actual surface parameters of the fingerprint imaging system. In other words, the metric function may quantify the accuracy with which the current surface parameters describe the manner in which light is processed within the fingerprint imaging system. In other embodiments, other functions account for differences between the predicted positions of, or distances between, the distinguishing points of the target and the values measured in the image of the target. For example, an average of the evaluation functions may be implemented as the metric function. Other metric functions and evaluation functions may be implemented. 
     The premise of the Levenberg-Marquardt algorithm and other data-fitting algorithms may include collecting the overall observed error and adjusting the surface parameters by a certain amount. As may be appreciated by one of ordinary skill in the art, in this algorithm, the amount to adjust the parameters by may be calculated from the gradient of the merit function with respect the each individual surface parameter. Thus, larger slopes may elicit larger movements in the surface parameters. 
     In some embodiments of the invention, calculating the gradient of a function that involves many calculations, such as implementing a ray tracing program to predict locations of the images of the distinguishing points as described above, especially when deriving the derivatives directly, may prove to be computationally intractable. In such embodiments, to calculate such gradients, an automatic differentiation software package named ADOL may be implemented. This program is accessible from various sources, such as, at www.math.tu-dresden.de/˜adol-c/ and may include a built in a DLL interface. Basically, ADOL may track a sequence of mathematical calculations, one building upon the other and, when the evaluation of a function is complete, a record of the sequence of calculations may be used (at a given input point) the evaluate the gradient at that point. Using ADOL, the function for which the gradient is being taken may be multi-variate and very complex, and yet this approach may accurately identify the gradient at the given input point. Thus, to find the gradient of the metric function for a set of parameter values, a call may be made to calculate the gradient within an ADOL library after the metric function is evaluated at the set of evaluation functions. 
     At a step  326 , the metric function value may be evaluated to determine if the current surface parameters adequately define the configuration of the fingerprint imaging system. In some embodiments, the metric function value may be compared to a threshold value at step  326 . If the metric function value is greater than the threshold value then method  310  may proceed to a step  328 . At the step  328 , the surface parameters may be adjusted in accordance with the implemented curve-fitting algorithm. From step  328 , method  310  may loop back for another iteration through steps  24  and  26 . If it is determined at step  326  that the metric function value of the current surface parameters adequately define the configuration of the fingerprint imaging system (e.g., the metric function value is less than the threshold value), then method  310  may proceed to a step  330 . At step  330 , the current surface parameters are adopted as the surface parameters of the fingerprint imaging system, and are implemented in a geometric model of the actual fingerprint imaging system. The geometric model may then be implemented to rectify images captured by the fingerprint imaging system. 
       FIGS. 4A and 4B  are exemplary illustrations of a captured image  410 , captured by a fingerprint imaging system that includes systematic distortion introduced by imperfections in the fingerprint imaging system, and a rectified image  412 , generated by rectifying captured image  410  based on a geometric model of the fingerprint imaging system to remove the systematic distortion. The systematic distortions that may be present within compound lens systems (e.g., barrel distortion, pincushion distortion, etc.) may also be corrected by implementing the method described above. For example, the geometric model used to generate rectified image  412  from captured image  410  may be determined according to method  310 . 
     In some embodiments of the invention, based on the geometric model, ray tracing software, such as the optical design software discussed above, may be implemented to trace ray paths through the fingerprint imaging system from the imaging location within the fingerprint system where the target (or other objects to be imaged) are placed to the surface within the fingerprint imaging system where the image of the target may be captured. Based on these ray paths, positional relationships between observed positions of one or more distinguishing points in an image of an object captured by the fingerprint imaging system and the positions within the image that the one or more distinguishing points may have been located if substantially no distortion were introduced by imperfections in the fingerprint imaging system can be ascertained with suitable accuracy. Using these positional relationships, rectified pixels in a rectified image may be generated from observed pixels in a captured image. The rectified image may be suitable for output and/or further processing using the geometric model. In short, the geometric model may enable ray tracing to be used to generate one or more rectified pixels based on one or more observed pixels in a captured image. In some instances, once the initial ray tracing and determination of the positional relationships, as described above, have been accomplished, the positional relationships may be used to employ an interpolation method to determine the rectified pixels in images captured by the fingerprint imaging system thereafter. For example, the positional relationships between rectified pixel locations and observed pixel locations can be encoded as one or more pre-calculated lookup tables which may then be used for real-time image reconstruction. 
     According to various embodiments of the invention, one or both of the observed image and the rectified image may be analyzed to determine the likelihood that the system is still producing images with less than a predetermined amount of distortion. For example, in some instances, fingerprint imaging system  112  may be implemented and at various times after the initial determination of the geometric model, image target  116  may be captured in order to determine a current metric function value. By way of illustration, at a step  332 , shown in  FIG. 3 , a user may initiate a re-determination of the geometric model that may cause an image of target  116  to be captured at step  318  of method  310 . From step  318 , method  310  may be followed, as described above, to adjust the surface parameters within the geometric model, if need. 
     It may be appreciated that although the invention has been described in terms of algorithmic correction of image distortions within fingerprint imaging systems, that this disclosure is not intended as limiting. Accordingly, the software implemented system and method contemplated by this disclosure may be implemented to correct image distortions within other optical imaging systems, thereby reducing a cost of providing imaging systems capable of generating images with a predetermined precision. 
     It can thus be appreciated that embodiments of the invention have now been fully and effectively accomplished. The foregoing embodiments have been provided to illustrate the structural and functional principles of the present invention, and are not intended to be limiting. To the contrary, the present invention is intended to encompass all modifications, alterations and substitutions within the spirit and scope of the appended claims.