Patent Publication Number: US-8971625-B2

Title: Generating dolly zoom effect using light field image data

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     The present application claims priority as a continuation of U.S. Utility application Ser. No. 13/688,026 for “Extended Depth of Field and Variable Center of Perspective in Light field Processing”, filed on Nov. 28, 2012, now U.S. Pat. No. 8,811,769 the disclosure of which is incorporated herein by reference in its entirety. 
     U.S. Utility application Ser. No. 13/688,026 claims priority from U.S. Provisional Application Ser. No. 61/604,155 for “Compensating for Sensor Saturation and Microlens Modulation During Light field Image Processing”, filed on Feb. 28, 2012, the disclosure of which is incorporated herein by reference in its entirety. 
     U.S. Utility application Ser. No. 13/688,026 further claims priority from U.S. Provisional Application Ser. No. 61/604,175 for “Compensating for Variation in Microlens Position During Light field Image Processing”, filed on Feb. 28, 2012, the disclosure of which is incorporated herein by reference in its entirety. 
     U.S. Utility application Ser. No. 13/688,026 further claims priority from U.S. Provisional Application Ser. No. 61/604,195 for “Light field Processing and Analysis, Camera Control, and User Interfaces and Interaction on Light field Capture Devices”, filed on Feb. 28, 2012, the disclosure of which is incorporated herein by reference in its entirety. 
     U.S. Utility application Ser. No. 13/688,026 further claims priority from U.S. Provisional Application Ser. No. 61/655,790 for “Extending Light field Processing to Include Extended Depth of Field and Variable Center of Perspective”, filed on Jun. 5, 2012, the disclosure of which is incorporated herein by reference in its entirety. 
     The present application is related to U.S. Utility application Ser. No. 11/948,901 for “Interactive Refocusing of Electronic Images,”, filed Nov. 30, 2007, the disclosure of which is incorporated herein by reference in its entirety. 
     The present application is related to U.S. Utility application Ser. No. 12/703,367 for “Light field Camera Image, File and Configuration Data, and Method of Using, Storing and Communicating Same,”, filed Feb. 10, 2010, the disclosure of which is incorporated herein by reference in its entirety. 
     The present application is related to U.S. Utility application Ser. No. 13/027,946 for “3D Light Field Cameras, Images and Files, and Methods of Using, Operating, Processing and Viewing Same”, filed on Feb. 15, 2011, the disclosure of which is incorporated herein by reference in its entirety. 
     The present application is related to U.S. Utility application Ser. No. 13/155,882 for “Storage and Transmission of Pictures Including Multiple Frames,”, filed Jun. 8, 2011, the disclosure of which is incorporated herein by reference in its entirety. 
     The present application is related to U.S. Utility Application Serial 13/664,938 for “Light-field Camera Image, File and Configuration Data, and Method of Using, Storing and Communicating Same,”, filed Oct. 31, 2012, the disclosure of which is incorporated herein by reference in its entirety. 
    
    
     FIELD OF THE INVENTION 
     The present invention relates to systems and methods for processing and displaying light field image data. 
     SUMMARY 
     According to various embodiments, a dolly zoom effect is generated using light field image data. The dolly zoom effect simulates an in-camera technique wherein a camera moves toward or away from the subject in such a way that the subject is kept at the same size throughout the effect. The effect causes the relative size of foreground background elements to change while foreground elements such as the subject remain the same size. By varying a parameter while projecting the light field image, the size of each object in the projection image scales depending on its relative depth as compared with the depth of the target subject, thus simulating the dolly zoom effect without any need to physically move the camera. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The accompanying drawings illustrate several embodiments of the invention and, together with the description, serve to explain the principles of the invention according to the embodiments. One skilled in the art will recognize that the particular embodiments illustrated in the drawings are merely exemplary, and are not intended to limit the scope of the present invention. 
         FIG. 1  depicts a portion of a light field image. 
         FIG. 2  depicts transmission of light rays through a microlens to illuminate pixels in a digital sensor. 
         FIG. 3  depicts an arrangement of a light field capture device wherein a microlens array is positioned such that images of a main-lens aperture, as projected onto the digital sensor, do not overlap. 
         FIG. 4  depicts an example of projection and reconstruction to reduce a 4-D light field representation to a 2-D image. 
         FIG. 5  depicts an example of incorrect calibration. 
         FIG. 6  depicts an example of a pull-push image stack containing three levels, for implementing pull-push infilling to reduce or eliminate holes in the image resulting from some pixels having low influence values, according to one embodiment. 
         FIG. 7  depicts an example of exaggerated center-of-perspective movement, according to one embodiment. 
         FIG. 8A  depicts an example of an architecture for implementing the present invention in a light field capture device, according to one embodiment. 
         FIG. 8B  depicts an example of an architecture for implementing the present invention in a post-processing system communicatively coupled to a light field capture device, according to one embodiment. 
         FIG. 9  depicts an example of an architecture for a light field camera for implementing the present invention according to one embodiment. 
         FIG. 10  is a flow diagram depicting an example of a method for pulling 2-D image values up into array of images of gradually reduced dimensions, according to one embodiment. 
         FIG. 11  is a flow diagram depicting an example of a method for pushing reduced-dimension 2-D images down into higher-resolution images, according to one embodiment. 
         FIG. 12A  is a flow diagram depicting an example of a method for noise calibration, according to one embodiment. 
         FIG. 12B  is a flow diagram depicting an example of a method for normalizing image noise, according to one embodiment. 
         FIG. 13  is a flow diagram depicting an example of a method of grayscale image erosion, according to one embodiment. 
         FIG. 14  is a flow diagram depicting an example of a method of gridded calibration, according to one embodiment. 
         FIG. 15  is a flow diagram depicting an example of a method of interactive projection of 2-D images with varying center of perspective, according to one embodiment. 
         FIG. 16  depicts an example of gridded calibration in a hexagonal grid, according to one embodiment. 
         FIGS. 17 and 18  depict an example of extended depth-of-field projection according to one embodiment. 
         FIG. 19  depicts an example of sheared EDOF projection, according to one embodiment. 
         FIG. 20  depicts an example of a distribution of centers of perspective within a bounding aperture, according to one embodiment. 
     
    
    
     DETAILED DESCRIPTION 
     Definitions 
     For purposes of the description provided herein, the following definitions are used:
         aggregated irradiance: total irradiance over a period of time, e.g., on a sensor pixel while the shutter is open.   anterior nodal point: the nodal point on the scene side of a lens.   automatic white balance (AWB): the process of computing white-balance (WB) factors and estimating color of a scene&#39;s illumination.   Bayer pattern: a particular 2×2 pattern of different color filters above pixels on a digital sensor. The filter pattern is 50% green, 25% red and 25% blue.   center of perspective: relative to a scene being photographed, the center of perspective is the point (or locus of points) where light is being captured. Relative to the camera&#39;s sensor image, it is the point (or locus of points) from which light is being emitted to the sensor. For a pinhole camera, the pinhole is the center of perspective for both the scene and the sensor image. For a camera with a more complex main lens, the scene-relative center of perspective may be best approximated as either the anterior nodal point of the main lens, or the center of its entrance pupil, and the sensor-relative center of perspective may be best approximated as either the posterior nodal point of the main lens, or as the center of its exit pupil.   clamp: in the context of the described invention, to “clamp a signal to a value” means to select the smaller of the signal value and the clamp value.   chrominance: a mapping of color channel values to a lower (typically n−1) space.   CoP: abbreviation for center of perspective.   demosaicing: a process of computing and assigning values for all captured color channels to each pixel, especially when that pixel initially includes a value for only one color channel.   disk: a region in a light field image that is illuminated by light passing through a single microlens; may be circular or any other suitable shape.   entrance pupil: the image of the aperture of a lens, viewed from the side of the lens that faces the scene.   exit pupil: the image of the aperture of a lens, viewed from the side of the lens that faces the image.   exposure value (EV): a measure of net sensor sensitivity resulting from ISO, shutter speed, and f-stop.   flat-field image: a light field image of a scene with undifferentiated rays.   flat-field response contour: a continuous plot of the value that a hypothetical sensor pixel would take if centered at various locations on the surface of a sensor;   image: a two-dimensional array of pixel values, or pixels, each specifying a color.   ISO: a measure of the gain of a digital sensor.   lambda: a measure of distance perpendicular to the primary surface of the microlens array. One lambda corresponds to the perpendicular distance along which the diameter of the cone of light from a point in the scene changes by a value equal to the pitch of the microlens array.   light field image: an image that contains a representation of light field data captured at the sensor.   luminance: a 1-component reduction of color that corresponds to perceived brightness or intensity.   microlens: a small lens, typically one in an array of similar microlenses.   MLA: abbreviation for microlens array.   modulation image: an image that is computed from a flat-field image by normalizing based on average values (per color channel).   nodal point: the center of a radially symmetric thin lens. For a lens that cannot be treated as thin, one of two points that together act as thin-lens centers, in that any ray that enters one point exits the other along a parallel path.   normalized pixel value: a sensor pixel value that has been adjusted to a range where 0.0 corresponds to black (no light) and 1.0 corresponds to saturation.   posterior nodal point: the nodal point on the image side of a lens.   quantization: a process of approximating a continuous value with one of a fixed set of pre-determined values. Quantization error increases as the separations between pre-determined values increases.   representative ray: a single ray that represents all the rays that reach a pixel.   saturated pixel: a pixel that has been driven by aggregated irradiance to its maximum representation.   sensor saturation: a sensor that has one or more saturated pixels.   uniform radiance: same radiance at all angles and at all spatial locations.   vignetting: a phenomenon, related to modulation, in which an image&#39;s brightness or saturation is reduced at the periphery as compared to the image center.       

     In addition, for ease of nomenclature, the term “camera” is used herein to refer to an image capture device or other data acquisition device. Such a data acquisition device can be any device or system for acquiring, recording, measuring, estimating, determining and/or computing data representative of a scene, including but not limited to two-dimensional image data, three-dimensional image data, and/or light field data. Such a data acquisition device may include optics, sensors, and image processing electronics for acquiring data representative of a scene, using techniques that are well known in the art. One skilled in the art will recognize that many types of data acquisition devices can be used in connection with the present invention, and that the invention is not limited to cameras. Thus, the use of the term “camera” herein is intended to be illustrative and exemplary, but should not be considered to limit the scope of the invention. Specifically, any use of such term herein should be considered to refer to any suitable device for acquiring image data. 
     In the following description, several techniques and methods for processing light field images are described. One skilled in the art will recognize that these various techniques and methods can be performed singly and/or in any suitable combination with one another. 
     Architecture 
     In at least one embodiment, the system and method described herein can be implemented in connection with light field images captured by light field capture devices including but not limited to those described in Ng et al., Light field photography with a hand-held plenoptic capture device, Technical Report CSTR 2005-02, Stanford Computer Science. Referring now to  FIG. 8A , there is shown a block diagram depicting an architecture for implementing the present invention in a light field capture device such as a camera  800 . Referring now also to  FIG. 8B , there is shown a block diagram depicting an architecture for implementing the present invention in a post-processing system communicatively coupled to a light field capture device such as a camera  800 , according to one embodiment. One skilled in the art will recognize that the particular configurations shown in  FIGS. 8A and 8B  are merely exemplary, and that other architectures are possible for camera  800 . One skilled in the art will further recognize that several of the components shown in the configurations of  FIGS. 8A and 8B  are optional, and may be omitted or reconfigured. 
     In at least one embodiment, camera  800  may be a light field camera that includes light field image data acquisition device  809  having optics  801 , image sensor  803  (including a plurality of individual sensors for capturing pixels), and microlens array  802 . Optics  801  may include, for example, aperture  812  for allowing a selectable amount of light into camera  800 , and main lens  813  for focusing light toward microlens array  802 . In at least one embodiment, microlens array  802  may be disposed and/or incorporated in the optical path of camera  800  (between main lens  813  and sensor  803 ) so as to facilitate acquisition, capture, sampling of, recording, and/or obtaining light field image data via sensor  803 . Referring now also to  FIG. 9 , there is shown an example of an architecture for a light field camera  800  for implementing the present invention according to one embodiment. The Figure is not shown to scale.  FIG. 9  shows, in conceptual form, the relationship between aperture  812 , main lens  813 , microlens array  802 , and sensor  803 , as such components interact to capture light field data for subject  901 . 
     In at least one embodiment, light field camera  800  may also include a user interface  805  for allowing a user to provide input for controlling the operation of camera  800  for capturing, acquiring, storing, and/or processing image data. 
     In at least one embodiment, light field camera  800  may also include control circuitry  810  for facilitating acquisition, sampling, recording, and/or obtaining light field image data. For example, control circuitry  810  may manage and/or control (automatically or in response to user input) the acquisition timing, rate of acquisition, sampling, capturing, recording, and/or obtaining of light field image data. 
     In at least one embodiment, camera  800  may include memory  811  for storing image data, such as output by image sensor  803 . Such memory  811  can include external and/or internal memory. In at least one embodiment, memory  811  can be provided at a separate device and/or location from camera  800 . 
     For example, camera  800  may store raw light field image data, as output by sensor  803 , and/or a representation thereof, such as a compressed image data file. In addition, as described in related U.S. Utility application Ser. No. 12/703,367 for “Light field Camera Image, File and Configuration Data, and Method of Using, Storing and Communicating Same,”, filed Feb. 10, 2010, memory  811  can also store data representing the characteristics, parameters, and/or configurations (collectively “configuration data”) of device  809 . 
     In at least one embodiment, captured image data is provided to post-processing circuitry  804 . Such circuitry  804  may be disposed in or integrated into light field image data acquisition device  809 , as shown in  FIG. 8A , or it may be in a separate component external to light field image data acquisition device  809 , as shown in  FIG. 8B . Such separate component may be local or remote with respect to light field image data acquisition device  809 . Any suitable wired or wireless protocol can be used for transmitting image data  821  to circuitry  804 ; for example camera  800  can transmit image data  821  and/or other data via the Internet, a cellular data network, a WiFi network, a BlueTooth communication protocol, and/or any other suitable means. 
     Overview 
     Light field images often include a plurality of projections (which may be circular or of other shapes) of aperture  812  of camera  800 , each projection taken from a different vantage point on the camera&#39;s focal plane. The light field image may be captured on sensor  803 . The interposition of microlens array  802  between main lens  813  and sensor  803  causes images of aperture  812  to be formed on sensor  803 , each microlens in array  802  projecting a small image of main-lens aperture  812  onto sensor  803 . These aperture-shaped projections are referred to herein as disks, although they need not be circular in shape. 
     Light field images include four dimensions of information describing light rays impinging on the focal plane of camera  800  (or other capture device). Two spatial dimensions (herein referred to as x and y) are represented by the disks themselves. For example, the spatial resolution of a light field image with 120,000 disks, arranged in a Cartesian pattern 400 wide and 300 high, is 400×300. Two angular dimensions (herein referred to as u and v) are represented as the pixels within an individual disk. For example, the angular resolution of a light field image with 100 pixels within each disk, arranged as a 10×10 Cartesian pattern, is 10×10. This light field image has a 4-D (x, y, u, v) resolution of (400, 300, 10, 10). Referring now to  FIG. 1 , there is shown an example of a 2-disk by 2-disk portion of such a light field image, including depictions of disks  102  and individual pixels  203 ; for illustrative purposes, each disk  102  is ten pixels  203  across. 
     Many light rays in the light field within a light field camera contribute to the illumination of a single pixel  203 . Referring now to  FIG. 2 , there is shown an example of transmission of light rays  202 , including representative rays  202 A,  202 D, through microlens  201 B of array  802 , to illuminate sensor pixels  203 A,  203 B in sensor  803 . 
     In the example of  FIG. 2 , solid rays  202 A,  202 B,  202 C illuminate sensor pixel  203 A, while dashed rays  202 D,  202 E,  202 F illuminate sensor pixel  203 B. The value at each sensor pixel  203  is determined by the sum of the irradiance of all rays  202  that illuminate it. For illustrative and descriptive purposes, however, it may be useful to identify a single geometric ray  202  with each sensor pixel  203 . That ray  202  may be chosen to be representative of all the rays  202  that illuminate that sensor pixel  203 , and is therefore referred to herein as a representative ray  202 . Such representative rays  202  may be chosen as those that pass through the center of a particular microlens  201 , and that illuminate the center of a particular sensor pixel  203 . In the example of  FIG. 2 , rays  202 A and  202 D are depicted as representative rays; both rays  202 A,  202 D pass through the center of microlens  201 B, with ray  202 A representing all rays  202  that illuminate sensor pixel  203 A and ray  202 D representing all rays  202  that illuminate sensor pixel  203 B. 
     There may be a one-to-one relationship between sensor pixels  203  and their representative rays  202 . This relationship may be enforced by arranging the (apparent) size and position of main-lens aperture  812 , relative to microlens array  802 , such that images of aperture  812 , as projected onto sensor  803 , do not overlap. Referring now to  FIG. 3 , there is shown an example of an arrangement of a light field capture device, such as camera  800 , wherein microlens array  802  is positioned such that images of a main-lens aperture  812 , as projected onto sensor  803 , do not overlap. All rays  202  depicted in  FIG. 3  are representative rays  202 , as they all pass through the center of one of microlenses  201  to the center of a pixel  203  of sensor  803 . 
     In at least one embodiment, the 4-D light field representation may be reduced to a 2-D image through a process of projection and reconstruction. Referring now to  FIG. 4 , there is shown an example of such a process. A virtual surface of projection  401  may be introduced, and the intersection of each representative ray  202  with surface  401  is computed. Surface  401  may be planar or non-planar. If planar, it may be parallel to microlens array  802  and sensor  803 , or it may not be parallel. In general, surface  401  may be positioned at any arbitrary location with respect to microlens array  802  and sensor  803 . The color of each representative ray  202  may be taken to be equal to the color of its corresponding pixel. In at least one embodiment, pixels  203  of sensor  803  may include filters arranged in a regular pattern, such as a Bayer pattern, and converted to full-color pixels. Such conversion can take place prior to projection, so that projected rays  202  can be reconstructed without differentiation. Alternatively, separate reconstruction can be performed for each color channel. 
     The color of an image pixel  402  on projection surface  401  may be computed by summing the colors of representative rays  202  that intersect projection surface  401  within the domain of that image pixel  402 . The domain may be within the boundary of the image pixel  402 , or may extend beyond the boundary of the image pixel  402 . The summation may be weighted, such that different representative rays  202  contribute different fractions to the sum. Ray weights may be assigned, for example, as a function of the location of the intersection between ray  202  and surface  401 , relative to the center of a particular pixel  402 . Any suitable weighting algorithm can be used, including for example a bilinear weighting algorithm, a bicubic weighting algorithm and/or a Gaussian weighting algorithm. 
     Artifacts Due to Incorrect Calibration 
     In at least one embodiment, the system of the present invention corrects artifacts resulting from incorrect calibration of sensor  803 . Existing light field cameras can experience artifacts resulting from incorrect calibration of sensors  803 . Each pixel  203  on sensor  803  is illuminated by actual rays  202  of light that pass through microlens array  802 . However, representative rays  202 , as described above, are not actual rays of light, but are instead mathematical rays that are defined based on the geometric relationship of microlens array  802  and sensor  803 . If representative rays  202  are to accurately represent the light that reaches a sensor pixel  203 , the geometric relationship between microlens array  802  and pixels  203  on sensor  803  must be known to a sufficient degree of accuracy. If this relationship may vary from one sensor  803  to another, then calibration of each sensor  803  may serve to compensate for such variation. If the actual geometric relationship between microlens array  802  and sensor  803  differs from the (known) relationship indicated by calibration, images created by projecting the light field image may contain unwanted artifacts. 
     Referring now to  FIG. 5 , there is shown an example of incorrect calibration. Microlenses  201  and sensor pixels  203  are depicted in their calibrated geometric relationship—that is, as they are believed to be related. Representative rays  202 , which pass through the calibrated microlens  201  centers and the centers of sensor pixels  203 , are depicted with solid lines. In this example, the actual position of microlens  201 B differs from the calibrated (expected) position. Actual light rays  501  that pass through the center of microlens  201 B, and also pass through the centers of certain sensor pixels  203 , are depicted with dashed lines. Thus the actual position of microlens  201 B is centered at the point where the dashed lines representing actual light rays  501  intersect microlens  201 B, rather than (as depicted) the point where the solid (representative) rays  202  intersect it. In this example, these positions differ by a distance equal to one-eighth of the microlens pitch. 
     One image artifact, herein known as geometric distortion, may result from the difference between 1) the representative ray  202  assigned to a sensor pixel  203  and 2) the actual light ray  501  that passes through the center of that sensor pixel  203  and the true center of the microlens  201  associated with that sensor pixel  203 . This situation is illustrated by sensor pixel  203 A in  FIG. 5 . The solid line passing through sensor pixel  203 A is the representative ray  202 A assigned to sensor pixel  203 A. It passes through the calibrated center of microlens  201 B, and through the center of sensor pixel  203 A. The dashed line passing through the center of sensor pixel  203 A is an actual light ray  501 A, which passes through the true center of microlens  201 B. During projection, the color of sensor pixel  203 A will be projected along the path specified by the (solid) representative ray  202 A passing through it. In actuality, however, light arrived at sensor pixel  203 A from light rays surrounding dashed ray  501 A, and should be projected along this path. This discrepancy between the projection and the actual light path causes artifacts. 
     The farther the rays are projected (that is, the greater the distance between the surface of sensor  803  and virtual projection surface  401 ) the greater the error due to divergence of each representative ray  202  from the corresponding actual ray. In the depicted example, although the distance between microlens array  802  and virtual projection surface  401  is not large (relative to the distance between microlens array  802  and the surface of sensor  803 ), representative ray  202 A that passes through sensor pixel  203 A intersects image pixel  402 A, while the actual ray that passes through sensor pixel  203 A intersects image pixel  402 B. The farther virtual projection surface  401  is from microlens array  802 , the greater the distance between the two intersections. This distance will manifest as geometric distortion in the projected image, the magnitude of the distortion being proportional to the distance between virtual projection surface  401  and microlens array  802 . If projection to a range of surfaces  401  is animated (for example, as a focus sweep), regions of the resulting images in which calibration errors exist may sweep or twist across the field of view. 
     A second form of distortion, herein referred to as ghosting, may also result from incorrect microlens-position calibration. Ghosting is illustrated by sensor pixel  203 B in  FIG. 5 . As in the case of sensor pixel  203 A, representative ray  202 B and actual light ray  501 B passing through sensor pixel  203 B follow different paths. Accordingly, geometric distortion, as described in the case of sensor pixel  203 A, will result. But the difference between ray paths is much greater than for the rays that pass through sensor pixel  203 A, because the two rays pass through different microlens centers—the representative ray passing through the pre-calibration center of microlens  201 A, and the actual light ray passing through the true center of microlens  201 B. This difference causes light passing through microlens  201 B to be aggregated with light passing through sensor pixel  203 B. The effect in projected images is adjacent duplicates of image features; hence the term “ghosting”. 
     Light field camera  800  may be designed so that small calibration errors result in geometric distortion, but do not cause ghosting. This may be accomplished, in at least one embodiment, by arranging the imaging geometry, including the geometry of sensor  803  and of the microlens array, so that disks  102  not only do not overlap, but are separated by a gap. Sensor pixels  203  are “assigned” to the nearest microlens  201  center, in calibrated coordinates, so gaps allow calibration errors up to half the gap size before a pixel&#39;s  203  assignment snaps to the incorrect microlens  201 . Such a technique limits or eliminates ghosting, since, until such snapping occurs, calibration errors may result in only geometric distortion, rather than ghosting. 
     Disk-Center Calibration 
     Microlens  201  positions can be difficult to measure directly. However, they may be inferred from pixel values in the light field image, which is readily available. Thus, in at least one embodiment, the key calibration problem is to identify the center of each disk  102  in the light field image. 
     The center of a disk  102  is formally the point where a ray from the center of the exit pupil of the light field camera&#39;s  800  main lens  813 , which passes through the center of the corresponding microlens  201 , intersects sensor  803 . Assuming that the exit pupil is round, or nearly round, and that the light field image is a modulation image, the center of a disk  102  may be approximated as the weighted-center of pixels values illuminated by rays passing through the corresponding microlens  201 . The weighted-center of pixel values in the x dimension is the solution to
 
0=Σ i   p   i ( x   i   −x   center )  (Eq. 1)
 
     for pixels  203  in the region i (those pixels  203  illuminated by rays passing through the corresponding microlens  201 ). In this equation, x i  is the x coordinate of the pixel&#39;s  203  center, p i  is the pixel&#39;s value, and x center  is the x coordinate of the disk  102  center (that is being computed). The y coordinate of the weighted-center may be computed equivalently:
 
0=Σ i   p   i ( y   i   −y   center )  (Eq. 2)
 
     With such a definition, however, it is necessary to know the disk  102  center, at least approximately, to determine which set of pixel values to consider when computing the weighted-center. (If pixels  203  corresponding to a different disk  102  are included in the weighted sum, the result will be incorrect.) In various embodiments, either of two general approaches can be used to estimate the center of a disk  102  prior to computing it more exactly using these equations. In a first embodiment, either of the following methods are performed:
         1. Grayscale image erosion is performed. Referring now to  FIG. 13 , there is shown an example of a method of grayscale image erosion, according to one embodiment. Grayscale image erosion is a morphological image processing technique that is well known in the art. A single morphological step  1301  revalues each pixel  203  in the image as a function of its pre-step value and the pre-step values of neighboring pixels  203 . This serves to reduce (erode) the values of pixels  203  that are on the edge of groups of high-value pixels  203 . A determination is made  1303  as to whether sufficient erosion has taken place; if not, step  1301  is repeated. Repeated erosion steps  1301  reliably reduce the light field image to a pattern of 2×2-pixel illuminated regions (disks), with interstitial pixel values reduced (nearly) to zero. After erosion is complete 1399, these 2×2-pixel blocks can be identified, and their centers can be evaluated. (The equations given above reduce to linear interpolation for a 2×2-pixel block.)   2. An alternative method is referred to as “stepping”. According to this method, a disk&#39;s  102  center is estimated by taking a grid step from the center of an accurately calibrated disk center. In a square tiling of microlenses  201 , a grid step changes either x or y by the known microlens pitch. In a hexagonal tiling of microlenses  201 , a grid step changes x and y in one of six directions, such that the distance moved is equal to the microlens pitch.       

     In at least one embodiment, a technique referred to as gridded calibration is performed. Referring now to  FIG. 14 , there is shown an example of a method of gridded calibration, according to one embodiment. Each disk  102  in the light field image is reduced  1401  to a roughly 2-pixel by 2-pixel point on a black background, using grayscale image erosion. A least-squares algorithm is applied  1402  to fit a small number of lines to the disk centers along orthogonal directions; these lines may be horizontal and vertical for a square grid, or at 60-degree angles for a hexagonal grid. Step  1402  may be implemented by fitting each line incrementally, first to a single disk center, then to additional disk centers at substantial incremental distances (for example, at 10-disk separations). In at least one embodiment, disks from defective microlenses are also detected and ignored during the mapping in order to prevent inaccurate mapping. Defective disks can include, but are not limited to, those that are decentered by greater than half of the disk pitch or have low transmission. 
     From the fitted lines generated in step  1402 , a rigid square or hexagonal grid (as appropriate) is fitted  1403  to the entire light field image. 
     For each grid region (such as a square or hexagonal region), the corresponding disk center is computed  1405  by passing all values for pixels  203  within that grid region into the weighted-center equations. 
     For each grid region, a vector distance is computed  1406  from the geometric center of the region to the computed disk center. This vector is assigned as the displacement associated with the corresponding disk center. A spatial filter is then applied  1407  to the spatial array of displacement vectors. Any of a number of known and suitable filters, such as Gaussian and Box, can be used. Optimizations such as bucketing may also be employed. In at least one embodiment, a square filter is employed. 
     For each grid region, the final disk center is computed  1408  as the vector sum of the grid&#39;s geometric center and the filtered displacement vector. The method ends  1499 . 
     Referring now to  FIG. 16 , there is shown an example of gridded calibration in a hexagonal grid, according to one embodiment. A weighted center  1601  has been computed for each disk  102 . Application of grid  1602  (using, for example, a least-squares algorithm) yields ideal position  1603  for each disk  102 . A similar technique can be used for square grids. 
     In at least one embodiment, several steps of gridded calibration can be performed in parallel, allowing for efficient and high-performance implementation. In at least one embodiment, the primary limitation of the above-described gridded calibration method is that only small errors (displacement-vector magnitude less than half the pitch of microlens array  802 ) may be computed accurately. If manufacturing tolerances cannot be held to this tight standard, gridded calibration may fail to compute a correct calibration. 
     A second form of calibration, herein referred to as incremental calibration, may also operate on a modulation image, as is described in the above-cited related U.S. Provisional Application. Incremental calibration overcomes the primary shortcoming of gridded calibration, which is its inability to handle error vectors with magnitudes greater than half the microlens pitch. It does so by first calibrating a single disk, and then growing a region of calibrated disks around this initial disk, taking incremental (e.g., one-microlens) steps. As long as the relative error of adjacent disks remains below a (fairly large) threshold, the correct 1-to-1 mapping of pre-calibration and post-calibration disks is ensured. After all disk centers have been calibrated, the calibrated centers are filtered with steps equivalent to steps  1406  to  1408  described above, wherein the displacement vectors may have magnitudes greater than half the microlens pitch. 
     Modulation images may vary as a function of camera parameters such as zoom, focus, and f-stop. Thus, in at least one embodiment, calibration is based on an appropriate modulation image. 
     In at least one embodiment, both gridded calibration and incremental calibration assume that the true center of a disk  102  corresponds to its pixel-weighted center. This may not be true in the case of vignetting, especially vignetting that is due to occlusion of the main-lens exit pupil. Such occlusion rarely occurs for disks  102  near the center of the light field image, but may be common for disks  102  near the edge (or, especially, the corner) of the light field image. Because such occlusion may vary slowly across the light field image, and may vary little from one camera of the same design to another, it may be modeled as an invariant field for cameras of the same design. Both algorithms (gridded calibration and incremental calibration) may be modified to account for such occlusion vignetting by adding a step prior to step ( 5 ) in which each calibrated disk center is perturbed to account for vignetting, by 1) resampling the vignetting field (which may itself be defined as a regular pattern of displacement vectors), and adding the sampled displacement vector to the disk center. 
     Application of Calibration Data 
     After calibrated disk centers are computed, they may be employed in at least two ways:
         1. Projection. Once calibrated representative rays  202  have been computed, they may be projected as described above. Representative rays  202  may be computed from calibrated disk centers in at least two different ways:
           a. Perspective. For each disk  102 , the corresponding microlens  201  center may be computed by casting a ray  202  from the disk center to the center of the main-lens exit pupil, and then finding the intersection of this ray  202  with the surface of microlens array  802 . True representative rays  202  for each pixel  402  in the light field image may then be determined as the ray  202  that extends from the corresponding microlens  201  center through the center of the sensor pixel  203 .   b. Orthographic. Each disk  102  may be treated as though it is at the center of microlens array  802 , or equivalently, that its center is coincident with the optical axis of main lens  813 . In this approximation, disk centers and microlens centers are the same in two coordinates; they differ only in the dimension that is parallel to the main lens optical axis. Equivalently, for each disk  102 , the corresponding microlens center may be computed by casting a ray  202  from the disk center directly up toward microlens array  802 , and then finding the intersection of this ray  202  with the surface of microlens array  802 . Orthographic representative rays  202  for each image pixel  402  in the light field image may then be found as the ray that extends from the corresponding microlens  201  center through the center of the sensor pixel  203 .   c. Warping. Projection may also be computed using representative rays  202  that pass through pre-calibration microlens centers. In this case, regions in the image will be distorted as a predictable function of their depth. A depth map may be computed using techniques that are known in the art. Using this depth map, and the known divergence between corrected and uncorrected representative rays  202 , a distortion vector may be estimated for each pixel  402  in the projected image. The projected image may than be warped to correct this distortion at each pixel  402 . This algorithm may be used to correct geometric distortion.
 
Influence
   
               

     In at least one embodiment, representative rays  202  that pass through the centers of pixels  203  which are themselves near the centers of disks  102  may be given more influence in the reconstructed 2-D image than representative rays that pass through pixels  203  that lie near the edge of disks  102 . An influence value may be assigned to each representative ray  202 . This influence value may be computed as a function of sensor-pixel location and of other parameters. In such an embodiment, each pixel  402  in the 2-D image may include an influence value, in addition to the values of its color components. During reconstruction, color components are multiplied by the filter coefficient (as described above) and also by the ray&#39;s influence value, before they are summed into the 2-D image pixel  402 . The product of the filter coefficient and the ray&#39;s  202  influence value is then summed to the 2-D pixel&#39;s influence value. When all representative rays have been processed, the color components in each 2-D image pixel are normalized, meaning that they are divided by the 2-D pixel&#39;s influence value. After normalization of a pixel is complete, that pixel&#39;s influence value may be discarded. 
     Any of a number of different functions may be employed to compute influence values. In at least one embodiment, for example, each representative ray&#39;s  202  influence value is set to the value of a corresponding pixel in the modulation image. This corresponding pixel is the pixel  203  through which the representative ray  202  passes. 
     Influence Based on Noise Function 
     In at least one embodiment, the influence value is determined based on a function that takes noise into account. In the projection process, if all sensor pixels  203  {L i |i=1, . . . , N} reaching the same image pixel  402  are assumed to come from the same physical point in the scene, and if they are equally affected by the optical and electrical system and thus have identical signal strength and noise level, an estimate of the pixel value of that image pixel  402 , say p, is 
     
       
         
           
             
               
                 
                   
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     This assumes that sensor pixels  203  are demodulated to begin with. Let m i  denote the modulation factor for i-th sample. m i  can be obtained from the modulation image. The imaging system can apply an analog or digital gain factor g to the sensed signal, before storing pixel values in digital form. Each sensor pixel  203  may also be affected by noise N i . Combining all these factors together, the sensor pixel value E i  is related to the ideal sensor pixel value L i  by the equation:
 
 E   i   =gm   i   L   i   +N   i   (Eq. 4)
 
     Given the noisy and scaled signal, the task is to find the optimal estimate of p. The “optimality” of the estimate can be defined as the expected difference between the estimate and the true value. To compute the estimate or measure its optimality, the noise characteristics of the system can be modeled. In the imaging system, the noise N i  usually has zero-mean, and its variance can be decoupled into two main components, including one that depends on the ideal sensor pixel value L i , and another that is signal-independent, as follows:
 
 v   E     i     2   =g   2 ( m   i   L   i )+ v   C   2   (Eq. 5)
 
     Given this model, the estimate of L i  and its variance can be calculated: 
     
       
         
           
             
               
                 
                   
                     
                       
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     Note that this calculation is the so-called demodulation process. Then, using the statistical estimation technique, the optimal estimate of p can be calculated from an estimated {L i } as 
     
       
         
           
             
               
                 
                   
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     Thus, samples with lower variance tend to have higher influence in the estimation process. If the influence of each sensor pixel  203  is defined as w i , the optimal influence value can be expressed as: 
     
       
         
           
             
               
                 
                   
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     This particular formulation is merely exemplary. In various other embodiments, the system of the present invention can use other techniques for taking into account noise in determining influence for pixels  203  at different positions within disks  102 . Different techniques may be applicable for different imaging systems. For example, if a sensor pixel  203  is defective or is clamped due to saturation, there may be no way to infer the original pixel L i  value from the corrupted data E i . In this case, the variance of this sensor pixel  203  can be modeled as infinite, and thus the influence would be zero. Alternatively, if there is no signal-dependent component in the noise, the optimal influence would be: 
                     w   i     =         g   2     ⁢     m   i   2         v   C   2               (     Eq   .           ⁢   9     )               
Infilling
 
     After projection of rays to the 2-D image is complete, but prior to normalization of influence values, it may be determined that some pixels  402  in this 2-D image have reconstructed influence values that are either zero (because no rays  202  contributed to this pixel  402 ) or are substantially lower than the influence values of other pixels  402 . Such a finding indicates that there are gaps, or “holes”, in the reconstructed 2-D image. These holes (which may be due to insufficient sampling, perhaps as a result of irregularities in the sampling pattern due to variations in the positions of microlenses) may be eliminated by a process of infilling: using the values of nearby pixels  402  to estimate the value of pixels  402  in the “hole”. 
     Pull-Push Infilling 
     Infilling may be implemented by first pulling 2-D image values up into an array of images of gradually reduced dimensions, then pushing values from these reduced-dimension images back down into the original 2-D image. Identifying the original 2-D image as image  0 , each image n may have pixel dimensions that are half those of image n−1. For example, if image  0  has dimensions w×h, then image  1  would have dimensions w/2×h/2, and image  2  would have dimensions w/4×h/4. (We visualize the reduced-dimension images as being stacked above image  0  in their indexed order, hence values are pulled up and pushed down.) The total number of images may be k+1 for any integer k such that both w/2 k  and h/2 k  are integer values. For example, w and h may be 3280, and k may be 4, because 3280/2 4 =205, an integer value. In other embodiments, different ratios may exist between the pixel dimensions of adjacent image levels; thus, any n×n square of pixels in a given level can be combined into a single pixel in the next level up. For example, each 3×3 square of a level might be combined into an individual pixel in the next level up. In yet other embodiments, the groups of pixels to be combined need not be squares; thus any group of pixels of any shape can be combined into a single pixel in the next level up. 
     Referring now to  FIG. 10 , there is shown a flow diagram depicting an example of a method for pulling 2-D image values up into array of images of gradually reduced dimensions, according to one embodiment. Pulling may proceed one image level at a time, such that all pixels in image n−1 are pulled up to image n, before any pixels in image n are pulled up to image n+1. During the pull of pixels from image n−1 to image n, four pixels in level n−1 are pulled to each single pixel in level n. In at least one embodiment, each 2×2 block of pixels in image n−1 corresponds to a single pixel in image n. Mathematically, a pixel in image n with coordinates [i,j] corresponds to pixels in image n−1 with coordinates [2i,2j], [2i,2j+1], [2i+1,2j], and [2i+1,2j+1]. 
     Initial image (designated as image 1), is received  1001 . Steps  1002  to  1006  are performed for successively higher levels on images having successively reduced resolution. At each level, each pixel in image n is set to the sum of its corresponding pixels in image n−1, so as to combine  1002  multiple values of pixels in the current-level image n−1 into a single pixel value in the next-higher-level image n. For a color image, each value within the pixel at a given level, typically red, green, blue, and influence, takes the sum of the corresponding values in the corresponding pixels at a lower level. The sums in each pixel in image n are then normalized  1003  such that they represent average values of the corresponding pixels in image n−1. This normalization may be accomplished by dividing each value by four, or may alternatively be implemented by dividing each value by the number of corresponding pixels in image n−1 with non-zero influence values. (Division by zero may be avoided by not normalizing in this case.) Other normalization algorithms are also possible. 
     After all pixels in image n are normalized, a clamping algorithm may be applied  1004  to image n, as follows. A maximum pull influence I pull  may be specified. For each pixel in image n, if I n  (the influence value of that pixel) is greater than I pull , then each value in the pixel (including I n ) is multiplied by I pull /I n . Thus, the pixel&#39;s influence value is effectively set equal to I pull . Other clamping algorithms are possible. 
     Finally, a low-pass filter may be applied  1005  to image n. Pulling proceeds one image level at a time, and is complete when all the pixels in image k have their final values. Thus, a determination is made  1006  as to whether the pull operation is complete, by determining whether all the pixels in image k have their final values. If so, image k is output  1007  and the method ends  1099 . Otherwise, the method returns to step  1002 . 
     Referring now to  FIG. 11 , there is shown a flow diagram depicting an example of a method for pushing reduced-dimension 2-D images down into higher-resolution images, according to one embodiment. In at least one embodiment, pushing down begins after pulling up is complete, working one image at a time, starting with image k and completing with image  1 . 
     Image k is received  1101 . Steps  1102  through  1105  are performed for successively lower levels on images having successively higher resolution. In step  1102 , for each pixel at the current level, target pixels in the next lower-level image are identified. The image-to-image pixel correspondence is the same as for pulling. For example, each pixel in image n is pushed down to four pixels in image n−1. In at least one embodiment, each pixel-to-pixel push may be conditioned on a test. For example, the push may happen only if I n-1 , the influence value of the pixel in image n−1, is less than a threshold influence value I push . Thus, a determination is made  1103  as to whether this condition is satisfied. If so, then each value v n-1  in the pixel in image n−1 is replaced  1104  with a weighted sum of its value and the value of the pushed pixel in image n. For example, pixel values v n-1  may be replaced as follows: 
     
       
         
           
             
               
                 
                   
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     Pushing down is complete when all pixels in image  1  have been (conditionally) pushed to image  0 . If, in step  1105 , level 0 has not yet been reached, the push operation is not yet complete, and the method returns to step  1102 . If, in step  1105 , the push operation is complete, the color components in each image- 0  pixel are normalized  1106 , meaning that they are divided by the image- 0  pixel&#39;s influence value. After normalization  1106  of a pixel is complete, that pixel&#39;s influence value may be discarded. Image  0  can then be output  1107 , and the method ends  1199 . 
     Referring now to  FIG. 6 , there is shown an example of a pull-push image stack containing three levels: image  601 A at level 0, image  601 B at level 1, and image  601 B at level 2. During the pull of pixels from each image n−1 to each higher-level image n, four pixels  402  in level n−1 are pulled to each single pixel  402  in level n. Thus, the sixteen pixels  402  of image  601 A at level 0 are reduced to four pixels  402  of image  601 B at level 1, and the four pixels  402  of image  601 B at level 1 are then reduced to one pixel  402  of image  601 C at level 2. 
     In the push operation, each pixel  402  in each image n is pushed down to four pixels  402  in image n−1. Thus, the value of single pixel  402  of image  601 C at level 2 is conditionally pushed to all four pixels  402  of image  601 E at level 1. Subsequently, the values of each of the four pixels  402  of image  601 C at level 2 is conditionally pushed to an associated group of four pixels of image  0 , thus populating all sixteen pixels in level 0. 
     Depth Map Generation 
     A depth map is a set of image-side points (points on the image side of main lens  813 ), each of which corresponds to a visible point in the scene. A point in the scene is visible if light emitted from it reaches the anterior nodal point of main lens  813 , either directly or by being reflected from a highly specular surface. The correspondence is such that light emitted from the scene point would be in best focus by main lens  813  at the corresponding image-side point. In at least one embodiment, the optical effects of microlens array  802  (primarily refraction and reflection) and of sensor  803  (primarily occlusion) are not taken into account for the calculation of best focus; it is as though these optical elements were not present. 
     The position of an image-side point in a depth map may be specified in Cartesian coordinates, with x and y indicating position as projected onto sensor  803  (x positive to the right, y positive up, when viewing toward the scene along the optical axis of main lens  813 ), and depth d indicating perpendicular distance from the surface of microlens array  802  (positive toward the scene, negative away from the scene). The units of x and y may be pixels—the pixel pitch of sensor  803 . The units of d may be lambdas, where a distance of one lambda corresponds to the distance along which a cone of light from any scene point changes its diameter by a value equal to the pitch of microlens array  802 . (The pitch of microlens array  802  is the average distance between the centers of adjacent microlenses  201 .) 
     Depth maps are known in the art. See, for example: J. Sun, H.-Y. Shum and N.-N. Zheng, “Stereo Matching using Belief Propagation,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 25, no. 7, pp. 787-800, 2003; and C.-K. Liang, T.-H. Lin, B.-Y. Wong, C. Liu, and H. Chen, “Programmable Aperture Photography: Multiplexed Light Field Acquisition,”  ACM TRANS. GRAPH.  27, 3, Article 55, 2008. 
     The following observations can be made concerning scene-side points that are directly visible to main lens  813 :
         Points at scene depths on the plane of best focus in the scene correspond to an image depth at the (microlens) surface, or plane, of sensor  803 .   Points at scene depths that are farther from camera  800  than the plane of best focus correspond to points with image depths that are closer to the main lens  802  than the plane of best focus, and therefore that have positive depth values.   Points at scene depths that are nearer to the camera  800  than the plane of best focus correspond to points with image depths that are further from the main lens  802  than the plane of best focus, and therefore that have negative depth values. A depth map may be computed using techniques that are known in the art. The depth map may include image depth values at points with a regular distribution in x and y, and thus may be treated as a height field. The sample density of this height field, in the x and y dimensions, may roughly correspond to the distribution of microlens centers, or it may be greater or smaller.
 
Assigning Depth Values to Representative Rays
       

     A depth value may be assigned to each representative ray  202  by intersecting that ray  202  with the image depth height field. If the representative ray  202  intersects the height field at multiple locations, the intersection that is farthest behind microlens  802  surface is selected. (This corresponds to the scene intersection that is nearest to camera  800 .) The depth value at this intersection is assigned to representative ray  202 . If there is no intersection, representative ray  202  may be discarded, or it may be assigned a depth value by another means. For example, it may be assigned a depth value computed from depth values assigned to nearby rays  202 . 
     Algorithms for computing the intersection of a ray  202  with a height field are known in the art. For example, a ray  202  may intersect a height-field point if it passes within a threshold distance of that point. Alternatively, the height-field points may be interpreted as control points for a surface, and ray  202  can be intersected with that surface. 
     Extended Depth-of-Field Projection 
     During projection to a refocused 2-D image, representative rays  202  are intersected with virtual projection surface  401  that is parallel to microlens array  802 , but displaced from it. If virtual projection surface  401  is ahead of microlens array  802  (closer to the scene), then the reconstructed 2-D image is focused at a virtual projection surface  401  that is ahead of the best focus scene plane of the light field camera  800 . (It is farther from the light field camera  800 .) Likewise, if virtual projection surface  401  is behind microlens array  802  (farther from the scene) then the reconstructed 2-D image is focused at a virtual projection surface  401  that is behind the best-focus scene plane of the light field camera  800 . Objects in the scene whose scene depths correspond to the image depth of virtual projection surface  401  are in exact focus; other objects in the scene are projected with blur that increases with their displacement from that scene depth. 
     In some situations, it may be desirable to construct an image with extended depth of field. In at least one embodiment, this may be accomplished by projecting representative rays  202  to a (typically non-planar) image-side surface that corresponds to the depths of objects in the scene. The depth map height field, as described in the previous section, is such a surface. Although the surface of projection may be non-planar, the resulting 2-D image is planar. 
     Let [x,y] be the (calibrated) coordinates of the point that a representative ray intersects the microlens array, and [u,v] be the slopes of that ray as a function of depth. Let d be the depth assigned to that representative ray, as described above. Let the units of positions, slopes, and depth values be consistent with each other. (That is, if the units of x and y are x unit , and the units of depth are d unit , then the units of u and v are x unit /d unit . As a specific example, the units of x and y may be the pixel pitch of sensor  803 , and the units of depth may be lambdas. In this case, the units of u and v are pixels/lambda.) Then [x′,y′], the spatial coordinates of the intersection of a representative ray  202  with the depth map, are computed as
 
 x′=x−du   (Eq. 11)
 
 y′=y−dv   (Eq. 12)
 
     (This is not derivative notation; rather, du implies the product of d and u.) Each projected representative ray now terminates at its plane of best focus and is associated with a new 2D spatial coordinate [x′,y′]. These 2D coordinates make up a reconstructed extended depth of field 2-D image. 
     Referring now to  FIGS. 17 and 18 , there is shown an example of extended depth-of-field projection according to one embodiment. Different representative rays  202  terminate at different planes  1701  of best focus. In this example, some of the rays  202  terminate at plane  1701 A, which is behind MLA plane  1702 ; other rays  202  terminate at plane  1701 B or  1701 C, which are ahead of MLA plane  1702 . Rays  202  terminating at plane  1701 A come from objects that are farther away from camera  800 ; these are represented with a positive lambda value (assuming lambda is calibrated with respect to MLA plane  1702 , which is assigned a lambda value of zero). Rays  202  terminating at plane  1701 B or  1701 C come from objects that are closer to camera  800 ; these are represented with a negative lambda value. The determination of which plane  1701  is of best focus for a particular representative ray  202  is specified by the depth map. 
     In  FIG. 18 , the projections at different planes  1701  as generated in  FIG. 17  are combined with one another to generate 2D output image  1801 A. For illustrative purposes, the center of perspective is considered to be parallel to the optical axis (i.e., perpendicular to MLA plane  1702 ). 
     In this example, output image  1801 A is generated from the superposition of the projections at different planes  1701 . Projections having lower lambda values are considered to be closer to camera  800  and are thus given precedence over projections having higher lambda values. Thus, where two projections overlap, the pixel values from the lower-lambda projection are used in generating 2D output image  1801 A. 
     In  FIG. 18 , for example, the projection at plane  1701 C overlaps a portion of the projection at plane  1701 A. Since the projection at plane  1701 C has a lower lambda value than does the projection at plane  1701 A, the portion  1802 C of output image  1801 A corresponding to the overlap takes its pixel values from the projection at plane  1701 C. 
     Output image  1801 A thus has four portions:
         Portion  1802 A, taking pixel values from the projection at plane  1701 B;   Portion  1802 B, taking pixel values from the projection at plane  1701 A;   Portion  1802 C, taking pixel values from the projection at plane  1701 C;   Portion  1802 D, taking pixel values from the projection at plane  1701 A.
 
Normalization of Image Noise
       

     When reconstructing images from a light field, unstructured high frequency error (referred to herein as “noise”), may be present in varying amounts depending on many factors of the reconstructed image. The reconstructed images may be any 2D output image created from the light field, including but not limited to, image(s) having a different virtual plane of focus, image(s) having an extended depth of field image, and/or image(s) with shifted perspective. The noise in the 2D output image may vary depending on many factors related to processing, including, but not limited to, the distance between the virtual focus and the optical focus of the physical system, whether or not a subject is in-focus or out-of-focus at the virtual focus computed, and/or the type of 2D output image created. 
     In composition (where composition refers to one or more 2D image(s) intended to be viewed together), varying amounts of noise may be considered more aesthetically displeasing than if the entire composition has a higher overall amount of noise that is more evenly distributed, or “normalized”. In other words, the inconsistency in the amount of noise from one image to another is undesirable. In at least one embodiment, therefore, the noise content of the image(s) is analyzed, and some amount of noise is added to areas or images with lower noise in order to better match the portion of the image or other images with more noise. The result is that the entire composition of images may be normalized such that all images have a similar amount of noise that is evenly distributed throughout each image. 
     In at least one embodiment, the noise performance of the system is fully characterized as part of a calibration routine. Referring now to  FIG. 12A , there is shown a flow diagram depicting an example of a method for noise calibration, according to one embodiment. The calibration data may be gathered  1201  by capturing light fields of a flat field under a variety of conditions, including but not limited to, differing exposure times and/or sensor gains. For each captured light field, a variety of 2D output images are generated  1202  by varying the processing parameters; this generates, for example, images having different virtual plane(s) of focus, extended depth(s) of field, and/or perspective shift(s). The resulting images from this calibration are then analyzed  1203  for noise, and the results are recorded  1204 . The noise may be analyzed and characterized in any manner. 
     In one example, a local area average is computed for each region in the image. Deviations from the local are average are recorded and classified by size (for example, number of pixels), magnitude, and frequency of occurrence. 
     Referring now to  FIG. 12B , there is shown a flow diagram depicting an example of a method for normalizing image noise, according to one embodiment. Such a method can be used, for example, when images are computed. For each set of processing parameters to be used in creating the composition, an appropriate noise distribution is loaded  1205  based on the capture conditions. A noise ceiling distribution is generated  1206  from all the input noise and taking the highest frequency of occurrence for each magnitude and size. When creating output images based on a set of processing parameters, noise may be added  1207  to each computed 2D image, in proportion to the difference between the noise ceiling distribution and the noise distribution for the processing parameters. In one case, the proportion applied is equal to one, and the image set may be considered to have the noise fully normalized. In other cases, the proportion may be less than one, so that the noise content is more similar, but not fully normalized. 
     Varying Center-of-Perspective Projection 
     In some situations, it may be desirable to project a light field image with differing centers of perspective. According to various embodiments, any of a number of techniques can be used for varying the center of perspective for projected images. 
     Sub-Aperture Projection 
     The use of sub-aperture images is described in Ng et al. and in related U.S. Utility application Ser. No. 13/027,946 for “3D Light Field Cameras, Images and Files, and Methods of Using, Operating, Processing and Viewing Same”, filed on Feb. 15, 2011, the disclosure of which is incorporated herein by reference in its entirety. In at least one embodiment, representative rays  202  are culled, such that only rays  202  that pass through a contiguous sub-region of main lens aperture  812  are projected to the 2-D image. The contiguous sub-region of main-lens aperture  812  is referred to herein as a sub-aperture, and the resulting image is referred to as a sub-aperture image. The center of perspective of a sub-aperture image may be approximated as the center of the sub-aperture. Such a determination is approximate because the meaning of “center” is precise only if the sub-aperture is rotationally symmetric. The center of an asymmetric sub-aperture may be computed just as the center of gravity of an asymmetric object would be. Typically, the aperture of the main lens is rotationally symmetric, so that the center of perspective of a 2-D image that is projected with all representative rays  202  (i.e., the sub-aperture is equal to the aperture) is the center of main-lens aperture  812 , as would be expected. 
     Sub-aperture projection works well in principle, but has limitations in practice. Because fewer rays are projected to the 2-D image, its resolution is reduced and its noise level is increased. Also, the range of possible centers of perspective is limited by the dimensions of the sub-aperture. Large sub-apertures project more rays, but their centers cannot deviate substantially from the center of main-lens aperture  812 . Small sub-apertures increase the allowable deviation of the center of perspective, but compromise the quality of the projected 2-D image. 
     Shearing the EDOF Projection 
     The above-described algorithm for computing [x′,y′] intersections for extended depth of field may itself be extended to introduce shear, so as to approximate the effect of changing the center of perspective of the projected 2-D image. Let constants u cop  and v cop  specify the displacement of the center of perspective from the center of the main-lens aperture. Then by extensions of Equations 11 and 12, the new coordinates may be defined as:
 
 x′=x−d ( u−u   cop )
 
 y′=y−d ( v−v   cop )  (Eqs. 13, 14)
 
     Referring now to  FIG. 19 , there is shown an example of sheared EDOF projection. Here, the center of perspective is not parallel to the optical axis. Pixel values are combined as described above in connection with  FIG. 18 , although here the off-axis angle is taken into account. Collapsing the projections at different planes  1701  yields 2D output image  1801 B where each projection has been shifted horizontally depending on its lambda value (i.e., depending on its vertical distance from MLA plane  1702 ). 
     Output image  1801  thus has four portions:
         Portion  1802 E, taking pixel values from the projection at plane  1701 A;   Portion  1802 F, taking pixel values from the projection at plane  1701 B;   Portion  1802 G, taking pixel values from the projection at plane  1701 A;   Portion  1802 H, taking pixel values from the projection at plane  1701 C.       

     In this example, output image  1801 B contains a gap, or hole  1901 , where there are no ray intersections and therefore no data is available. In other words, no projection maps to that area on MLA plane  1702 . Push-pull infilling can be used, as described above, to fill in pixel values in hole  1901 . Alternatively, another mechanism can be employed. For example, the height field can be used to identify the holes, and then the holes can be filled with the image from sub-aperture projection. 
     Reducing Artifacts in EDOF Projection 
     In some situations, the above-described EDOF projection methods (whether or not shearing is included) can introduce undesirable artifacts. For example, samples with different depth values d may be projected to the same [x′,y′] intersection. Since those samples come from different objects, this means their colors are mixed and the contrast of the projection image is lost. To address this problem, in at least one embodiment, a mechanism for detecting occlusions is applied, and occluded samples are rejected. The depth value of [x′,y′] in the depth map is compared to the depth value of the incoming samples; the sample is accepted only if the two depth maps are sufficiently close to one another. In the case of a shearing projection, the depth map varies with u cop  and v cop ; those depth maps of different u cop  and v cop  can be estimated jointly, or generated from analyzing the height field described above. 
     Warping the EDOF 2-D Image 
     As described above, disk-center calibration may be applied either during projection or as a warping of the projected 2-D image. Likewise, change in center of perspective may be introduced during projection (as described in “Sub-Aperture Projection”, above, and in “Shearing the EDOF Projection”, below) or as a warping of the projected 2-D image. The warp vector [Δx,Δy] at any 2-D image location [x,y] may be approximated as
 
Δ x=D ( x,y ) u   cop   (Eq. 15)
 
Δ y=D ( x,y ) v   cop   (Eq. 16)
 
     where D(x,y) returns an estimate of the depth map value at location [x,y]. Techniques for warping an image, as a function of an array of warp vectors, are well known in the art. 
     Varying Center of Perspective Interactively 
     In at least one embodiment, the system and method of the present invention provide mechanisms for interactive projection of 2-D images with varying center of perspective. For example, a person may view the 2-D image on a screen, adjust the desired center of perspective by moving a mouse or other input device, and see the effects of that motion immediately on the screen, as a sequence of 2-D images with different centers of perspective. 
     In at least one embodiment, the range of motion of the center of perspective may be limited to the entrance pupil of the main-lens  813  of light field camera  800 . In at least one embodiment, the diameter of the entrance pupil (assuming a circular aperture) cannot be larger than the physical diameter of lens  813 , and will typically be somewhat smaller. Thus, the range of motion of the center of perspective may be small in comparison to objects in the scene being photographed. 
     During such interaction, it may be desirable that the viewer have the impression that the center of perspective is moving more than it is. Referring now to  FIG. 15 , there is shown a flow diagram depicting an example of a method of interactive projection of 2-D images with varying center of perspective, according to one embodiment. The depicted method operates as follows to create the impression that the center of perspective is moving more than it is. 
     The 2-D image is scaled  1501  to be somewhat larger than the screen (or the window on the screen) that it is being displayed in, such that only a window-sized sub-region of the 2-D image may be viewed at any moment. The range of motion of the 2-D image is defined  1502 , relative to the screen, as the maximum allowed while ensuring that the image is visible at all locations of the screen. Thus, the edges of the 2-D image remain out of view. 
     The position of the 2-D image is coupled  1503 , relative to the screen, with the position of the center of perspective, such that full-range motion of the center of perspective corresponds to full-range motion of the 2-D image. The coupling may be linear or non-linear. The polarity of the coupling may ensure that, for example, when the center of perspective is at its lower-left extreme (in the coordinates of the viewer), the lower-left corner of the 2-D image is visible. 
     Then, user input is received  1504 , and the center of perspective is adjusted  1505  according to the received input. If, in step  1506 , additional user input is received, the method returns to step  1504 . Otherwise, the method ends  1599 . 
     Referring now to  FIG. 7 , there is shown an example of exaggerated center-of-perspective movement, according to one embodiment. The Figure depicts the display of an unscaled 2-D image  702  (containing object  703 ) in an on-screen window  701 . In the top part of the Figure, the size of window  701  is the same as the size of 2-D image  702 , so there is no room for movement to give the impression of a change in the center of perspective. Although object  703  changes in appearance at the center of perspective shifts from the lower left to the upper right, there is no change in the position of object  703  within window  701 . 
     In the bottom part of the Figure, 2-D image  702  has been scaled to 1.25 times the size of window  701 . Thus, at any given time, a portion of 2-D image  702  is outside window  701  and is not visible. This allows for a range of movement of 2-D image  702  with respect to window  701 , giving the impression of a change in the center of perspective. In particular, as the center of perspective shifts from the lower left to the upper right, object  703  moves within the displayed window  701  to reinforce the change in center of perspective. 
     Varying Center of Perspective by Tilting a Handheld Device 
     Hand-held devices, such as mobile phones and tablets, often include accelerometers to measure the magnitude and direction of an instantaneous acceleration vector, including the effect of gravity. If the device is held still, except for slow changes to its orientation, the accelerometers provide an accurate indication of the direction of gravity relative to the device, and thus of the tilt of the device. 
     In at least one embodiment, the measured time-varying tilt angle of a device is used to vary the center of perspective of a light field picture being projected to a screen on the device. Any of a number of algorithms can be used to convert the time-varying tilt angle to parametric specification of the 2-D image projection. For example, in at least one embodiment, the u cop  and v cop  from Equations 15 and 16 can be mapped to the tilt angle output by the accelerometer on the device. In another embodiment, the accelerometer output can be linearly or quadratically scaled for effect. 
     In at least one embodiment, tilt angles are coupled in a directly proportional manner to the position of the center of perspective. For example, tilting to the left or right controls left or right displacement of the center of perspective, respectively; forward-backward tilt controls forward or backward displacement of the center of perspective, respectively. 
     In at least one embodiment, inertia is introduced, so that changes in tilt angle are not coupled directly to the position of the center of perspective; rather a previous position is taken into account, so that changes to the position are effectively damped. The introduction of inertia adds smoothness to the response. 
     Whichever methodology is used for coupling changes in tilt angle (and other motions detected by accelerometers), such changes can be used as user input for the technique described above in connection with  FIG. 7 . 
     Varying Center of Perspective in a Slideshow 
     One common method for viewing digital images is as a slide show. The slide show may be viewed on any type of digital display, including, for example a display associated with a computer, tablet computer, laptop computer, mobile phone, television, and/or any other type of display. According to at least one embodiment, in displaying a slideshow, the perspective may be automatically changed to produce more engaging effects. 
     For example, in at least one embodiment, the display may automatically zoom in on the image, and then pan across the image. Such operations may be performed in an automated manner, either according to specifications provided by an author of a slideshow, or based on default settings. While panning, the perspective may be changed in tandem with the pan. With the pan and perspective change synchronized, an effect similar to viewing a scene while moving one&#39;s head may be created. 
     In at least one embodiment, the change in perspective is performed in a manner that is consistent with the pan. In at least one embodiment, this effect is accomplished using the method described above in connection with  FIG. 15 , although instead of responding to user input, shifts in the center of perspective are performed automatically based on the desired effect for the slide show. 
     Caching Images for Varying Center of Perspective 
     In at least one embodiment, any number of images having different centers of perspective can be pre-computed and stored in a cache. Such a technique is useful, for example, when the processing power of the playback device is insufficient for on-the-fly generation of such images. During presentation of images using varying centers of perspective, the system can retrieve images as needed from the cache. If a desired center of perspective is not available in the cache, in at least one embodiment, the system can retrieve two or more images having centers of perspective close to the desired center of perspective in the cache file, and interpolate or blend them together to generate an image that approximates the desired center of perspective. Images from the cache file can also be used in connection with techniques for sharing images, such as those described in related U.S. Utility application Ser. No. 13/155,882 for “Storage and Transmission of Pictures Including Multiple Frames,”, filed Jun. 8, 2011, the disclosure of which is incorporated herein by reference in its entirety. In particular, images in the cache file can be shared in lieu of (or in addition to) sharing the light field picture. 
     In at least one embodiment, the number of images to be stored in the cache file is determined based, at least in part, on the particular characteristics of the image content. A particular scene may dictate storage of more (or fewer) images based on, for example, the depth variation of the scene being depicted. For example, if the scene is a flat wall, it is sufficient to store only one image in the cache, because all images of different perspective are identical. By contrast, images having greater amounts of depth variation of the scene, more images can be stored so as to avoid artifacts that may result from insufficient images in the cache. Such an approach improves overall quality, while avoiding overtaxing resources such as preprocessing time and storage space by pre-computing more images than necessary. 
     In at least one embodiment, the number of images to be stored, and their centers of perspective, are determined from the depth map and/or image content. 
     Any suitable structure and range for the distribution of centers of perspective can be used. Referring now to  FIG. 20 , there is shown an example of a distribution of centers of perspective  2000  within a bounding aperture  812 , according to one embodiment, wherein an image for each center of perspective  2000  has been pre-computed and stored in a cache. In this example, a hexagonal grid is used, and the range of [u cop , v cop ] is limited to the range of aperture  812 . Central center of perspective  2000 A is an on-axis point of view, while all others are off-axis by an amount indicated by the distance from that center of perspective  2000  and central center of perspective  2000 A. The spacing L in-between neighboring centers of perspective  2000  can be chosen such that the disparity of an object with certain depth D T  is below some threshold T:
 
 L|D   T   |S&lt;T,   (Eq. 17)
 
     where S is a scaling factor that grows linearly with the image resolution. D T  can be computed in various ways, like the maximal absolute value in the depth map. Given the grid structure, range, and L, the number of the centers of perspective and their values can be calculated. In at least one embodiment, the lower bound of L can be constrained to limit the maximal number of images in the cache file. 
     Automatically Generated Animation from User Interaction 
     In at least one embodiment, images generated from a light field picture may be presented to a viewer in an interactive manner. For example, a set of images computed using a set of virtual focal planes may be animated to change the focus. In order to make an aesthetically pleasing animation, it may help to understand which objects and/or subjects are salient to viewers and which are not. 
     In at least one embodiment, an interactive image may include an automatically generated animation. A user can interactively change the image (for example, by clicking on an area of the image to select the best focus for that object). In at least one embodiment, the automatic animation may execute until the user attempts to directly interact with the image (for example, by clicking a mouse); at that point, the animation stops and the user controls the interaction. 
     In such embodiments, the automatic animation may be guided by user input. In at least one embodiment, user actions are recorded (for example, each click of the mouse and the location of the click) and stored (for example, in a database). When the automatic animation is generated, the user action data is analyzed and used to create an animation that is aesthetically pleasing to users. 
     For example, in one embodiment, a series of images, referred to as a refocus stack, each corresponding to a different virtual focal plane, is made publicly available on the Internet. This set of images also includes a mapping of location to the image containing the best focus for the location. Each time a user interacts with the refocus stack, the click and location are recorded to a database. Subsequently, when an animation is to be created, the set of clicks are grouped based on image location and/or corresponding depth. In one example, the animation may be generated by selecting the three most popular locations on the image, and cycling the focus through these three positions on some interval (for example, one second at each position, followed by a 0.5-second transition). In this manner, an animation is automatically generated based on the recorded user interactions. 
     Generalized Projection 
     The above description sets forth mechanisms for generating projections given a light field and a depth map for normal refocusing, EDOF, and shearing EDOF. In other embodiments, other types of projections can be generated. 
     The general equations for representing a projection operation can be described as:
 
 x′=f   x ( x,u,D,d )
 
 y′=f   y ( y,v,D,d ),  (Eqs. 18, 19)
 
     where d is the depth of the representative ray, and D is the depth parameter, which can be for example the target refocus depth. For regular refocusing, f x =x−Du, and for shearing EDOF projection, f x =x−d(u−u cop ). Note that u cop  is actually a variable that can be adjusted or even function of other variable. 
     The following are three examples of generalized projections. One skilled in the art will recognize that the techniques described herein can be extended to many other types of projections. 
     Dolly Zoom Effect 
     The dolly zoom effect, also referred to as the “Vertigo effect” (after its use in the film “Vertigo” by Alfred Hitchcock) is a common technique in cinematography. The field-of-view of an image is adjusted while the camera moves toward or away from the subject in such a way that the subject is kept at the same size throughout the effect. The effect causes the relative size of foreground background elements to change while foreground elements such as the subject remain the same size. 
     In at least one embodiment, generalized projection is used to simulate a similar effect using a single lightfield. Specifically, the projection equation for the dolly zoom effect is an extension of the EDOF projection:
 
 x ′=( x−du )(1+α( D−d )),
 
 y ′=( y−dv )(1+α( D−d )),  (Eq. 20, 21)
 
where α is a parameter adjusting the effect, d is the depth of the light ray, and D is the depth of the subject that is to be fixed in the image. By changing α, the size of each object in the projection image scales depending on its relative depth as compared with the depth of the target subject.
 
Varying Aperture Size
 
     If a ray of coordinate (x,y,u,v) has depth d, then a ray of coordinate (x+s, y+t, u+s/d, v+t/d) also comes from the same point in the world, until the point is being occluded when viewed from (u+s/d, v+t/d). By exploiting this fact, one can adjust the 4D coordinate before projection to redefine the shape of the aperture for projection. Specifically, (u, v) can be scaled, and (x, y) can be adjusted, as follows:
 
 û=βu  
 
 {circumflex over (v)}=βv  
 
 {circumflex over (x)}=x−d ( u−û )
 
 ŷ=y−d ( v−{circumflex over (v)}.   (Eq. 22, 23, 24, 25)
 
     As in the above-described shearing EDOF projection, rays from different sources may have identical transformed coordinates. As described, occlusion identification and sample rejection can be performed. 
     This transformation generates a new light field image that appears as if captured by a camera of aperture size β times to the original aperture size. Regular refocusing projection can be performed on this light field image. Combining the refocusing projection equations and the transformation above, the generalized projection equations for varying aperture size are as follows:
 
 x′=x−d (1−β) u−Dβu,  
 
 y′=y−d (1−β) v−Dβv,   (Eqs. 26, 27)
 
     Note that if β is one, these equations devolve into the normal refocus projection equations; if β is zero, these equations devolve into the regular EDOF projection equations. By changing β, the aperture size and the effective depth-of-field can be smoothly adjusted. 
     Correct Field Curvature 
     Field curvature is a well-known problem. In a real lens system, the focus image plane may be curved. This means a flat subject that is in focus at the image center may appear to be out of focus at the image edges. Similar problems can arise in refocusing if the light field image is captured using a lens with strong field curvature. 
     In at least one embodiment, field curvature distortion is corrected by changing the projection equations to:
 
 x′=x− ( Df ( x,y )) u,  
 
 y′=y− ( Df ( x,y )) v,  
 
 f ( x,y )= a   +a   1   x+a   2   y+a   3   xy+a   4   x   2   +a   5   y   2   (Eqs. 28, 29, 30)
 
     where f is a 2-degree polynomial of x and y. 
     The modified equations change the constant refocusing depth to a smooth spatially-variant field. This allows the image to be refocused to different depths for different regions. Distortion can thereby be corrected by matching the shape of the field to the particular field curvature indicated by the characteristics of the capture equipment. 
     One skilled in the art will recognize that these generalized projection techniques can be extended in many different ways. Various projection results can be accomplished by various types of linear and/or nonlinear processing and/or filtering of the 4D coordinates and the depth values. In at least one embodiment, the projection equations described above can be concatenated to combine different effects with one another. 
     The present invention has been described in particular detail with respect to possible embodiments. Those of skill in the art will appreciate that the invention may be practiced in other embodiments. First, the particular naming of the components, capitalization of terms, the attributes, data structures, or any other programming or structural aspect is not mandatory or significant, and the mechanisms that implement the invention or its features may have different names, formats, or protocols. Further, the system may be implemented via a combination of hardware and software, as described, or entirely in hardware elements, or entirely in software elements. Also, the particular division of functionality between the various system components described herein is merely exemplary, and not mandatory; functions performed by a single system component may instead be performed by multiple components, and functions performed by multiple components may instead be performed by a single component. 
     In various embodiments, the present invention can be implemented as a system or a method for performing the above-described techniques, either singly or in any combination. In another embodiment, the present invention can be implemented as a computer program product comprising a nontransitory computer-readable storage medium and computer program code, encoded on the medium, for causing a processor in a computing device or other electronic device to perform the above-described techniques. 
     Reference in the specification to “one embodiment” or to “an embodiment” means that a particular feature, structure, or characteristic described in connection with the embodiments is included in at least one embodiment of the invention. The appearances of the phrase “in at least one embodiment” in various places in the specification are not necessarily all referring to the same embodiment. 
     Some portions of the above are presented in terms of algorithms and symbolic representations of operations on data bits within a memory of a computing device. These algorithmic descriptions and representations are the means used by those skilled in the data processing arts to most effectively convey the substance of their work to others skilled in the art. An algorithm is here, and generally, conceived to be a self-consistent sequence of steps (instructions) leading to a desired result. The steps are those requiring physical manipulations of physical quantities. Usually, though not necessarily, these quantities take the form of electrical, magnetic or optical signals capable of being stored, transferred, combined, compared and otherwise manipulated. It is convenient at times, principally for reasons of common usage, to refer to these signals as bits, values, elements, symbols, characters, terms, numbers, or the like. Furthermore, it is also convenient at times, to refer to certain arrangements of steps requiring physical manipulations of physical quantities as modules or code devices, without loss of generality. 
     It should be borne in mind, however, that all of these and similar terms are to be associated with the appropriate physical quantities and are merely convenient labels applied to these quantities. Unless specifically stated otherwise as apparent from the following discussion, it is appreciated that throughout the description, discussions utilizing terms such as “processing” or “computing” or “calculating” or “displaying” or “determining” or the like, refer to the action and processes of a computer system, or similar electronic computing module and/or device, that manipulates and transforms data represented as physical (electronic) quantities within the computer system memories or registers or other such information storage, transmission or display devices. 
     Certain aspects of the present invention include process steps and instructions described herein in the form of an algorithm. It should be noted that the process steps and instructions of the present invention can be embodied in software, firmware and/or hardware, and when embodied in software, can be downloaded to reside on and be operated from different platforms used by a variety of operating systems. 
     The present invention also relates to an apparatus for performing the operations herein. This apparatus may be specially constructed for the required purposes, or it may comprise a general-purpose computing device selectively activated or reconfigured by a computer program stored in the computing device. Such a computer program may be stored in a computer readable storage medium, such as, but is not limited to, any type of disk including floppy disks, optical disks, CD-ROMs, magnetic-optical disks, read-only memories (ROMs), random access memories (RAMs), EPROMs, EEPROMs, flash memory, solid state drives, magnetic or optical cards, application specific integrated circuits (ASICs), or any type of media suitable for storing electronic instructions, and each coupled to a computer system bus. Further, the computing devices referred to herein may include a single processor or may be architectures employing multiple processor designs for increased computing capability. 
     The algorithms and displays presented herein are not inherently related to any particular computing device, virtualized system, or other apparatus. Various general-purpose systems may also be used with programs in accordance with the teachings herein, or it may prove convenient to construct more specialized apparatus to perform the required method steps. The required structure for a variety of these systems will be apparent from the description provided herein. In addition, the present invention is not described with reference to any particular programming language. It will be appreciated that a variety of programming languages may be used to implement the teachings of the present invention as described herein, and any references above to specific languages are provided for disclosure of enablement and best mode of the present invention. 
     Accordingly, in various embodiments, the present invention can be implemented as software, hardware, and/or other elements for controlling a computer system, computing device, or other electronic device, or any combination or plurality thereof. Such an electronic device can include, for example, a processor, an input device (such as a keyboard, mouse, touchpad, trackpad, joystick, trackball, microphone, and/or any combination thereof), an output device (such as a screen, speaker, and/or the like), memory, long-term storage (such as magnetic storage, optical storage, and/or the like), and/or network connectivity, according to techniques that are well known in the art. Such an electronic device may be portable or nonportable. Examples of electronic devices that may be used for implementing the invention include: a mobile phone, personal digital assistant, smartphone, kiosk, server computer, enterprise computing device, desktop computer, laptop computer, tablet computer, consumer electronic device, television, set-top box, or the like. An electronic device for implementing the present invention may use any operating system such as, for example: Linux; Microsoft Windows, available from Microsoft Corporation of Redmond, Wash.; Mac OS X, available from Apple Inc. of Cupertino, Calif.; iOS, available from Apple Inc. of Cupertino, Calif.; and/or any other operating system that is adapted for use on the device. 
     While the invention has been described with respect to a limited number of embodiments, those skilled in the art, having benefit of the above description, will appreciate that other embodiments may be devised which do not depart from the scope of the present invention as described herein. In addition, it should be noted that the language used in the specification has been principally selected for readability and instructional purposes, and may not have been selected to delineate or circumscribe the inventive subject matter. Accordingly, the disclosure of the present invention is intended to be illustrative, but not limiting, of the scope of the invention, which is set forth in the claims.