Abstract:
A test chart for characterizing lenses comprises X radial lines extending from a central point and Y concentric X-sided polygons centered at the central point. Each of the X radial lines intersects one vertex of each of the polygons. Each of the X radial lines is divided into Y segments by its intersection points with the central point and with the polygons. The radial lines and the polygons define boundaries of X*Y non-overlapping patches. At least two of the patches share a boundary and are distinctly colored from each other.

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
       [0001]    This application claims the benefit under 35 U.S.C. §119 of U.S. Patent Application No. 61/893,142 filed 18 Oct. 2013 and entitled APPARATUS AND METHODS FOR CHARACTERIZING LENSES which is hereby incorporated herein by reference for all purposes. 
     
    
     TECHNICAL FIELD 
       [0002]    This application relates to apparatus and methods for characterizing lenses. 
       BACKGROUND 
       [0003]    It is often desirable to determine the properties of a lens. An important property of a lens is its “sharpness”. Sharpness can be defined in a variety of ways, but in general, it refers to the ability of a lens to form an image without excessive blurring. 
         [0004]    A “test chart” is an object (typically a flat surface with a pattern printed thereon) which may be viewed through a lens. Features of images of test charts obtained using the lens may be measured in order to determine various properties of the lens. For example, an image can be analyzed mathematically to determine a modulation transfer function (“MTF”) of the lens. The MTF of a lens refers to the ability of the lens to resolve various spatial frequencies. 
         [0005]    An example of a MTF is shown in  FIG. 1 . For low spatial frequencies, the modulation is 100%. As the spatial frequency increases, the modulation decreases to 0%. 
         [0006]    An example setup of a lens measuring system  20  is shown, schematically, in  FIG. 2 . A lens  21  is positioned between a test chart  22  and a focal plane array (“FPA”)  23 . 
         [0007]    FPA  23  may comprise an image sensor, such as a charge-coupled device or a complementary metal-oxide-semiconductor (CMOS) imaging array. FPA  23  comprises a plurality of discrete regions, typically referred to as “pixels”. Each pixel contains a light sensor. The pixels may be arranged in a regular grid. The regular grid of pixels may comprise rows of pixels in a horizontal direction and columns of pixels in a vertical direction. The horizontal and vertical directions may be orthogonal. 
         [0008]    An aperture stop  24  is positioned between lens  21  and FPA  23 . In some embodiments, aperture  24  is positioned between lens  21  and test chart  22 . A ray of light  25  originates from a point  26  on test chart  22 . Ray  25  passes through region  27  of lens  21 , through the aperture of aperture stop  24 , and strikes pixel  28  of FPA  23 . In a similar way, each pixel of FPA  23  is associated with a unique region of lens  21  and a unique region of test chart  22 . 
         [0009]    Lens measuring systems of the general arrangement shown in  FIG. 2  are typically used for measuring properties of small-aperture object-space telecentric lenses. 
         [0010]    Another example setup of a lens measuring system  30  is shown, schematically, in  FIG. 3 . A lens  31  is positioned between a test chart  32  and a FPA  33 . Lens  31  is supported by a mount  34 . Mount  34  also acts as an aperture stop (with lens  31  completely filling the aperture). 
         [0011]    A chief ray of light  35  originates from a point  36  on test chart  32 . Chief ray  35  passes through the centre  37  of lens  31  and strikes pixel  38  of FPA  33 . Other rays of light, for example rays  39 A and  39 B, also originate from point  36  and pass through lens  31  and strike pixel  38 . The relative distances between test chart  32 , lens  31 , and FPA  33  are selected so that a focused image of test chart  32  is projected onto FPA  33 . 
         [0012]    In lens measuring system  30 , test chart  32  is typically larger than lens  31 . Also, the distance between test chart  32  and lens  31  is typically greater than the distance between FPA  33  and lens  31 . 
         [0013]    Many other types of lens measuring systems are also known in the art. 
         [0014]    The entire image formed by a lens at its focal plane is called the “image circle”. When testing a lens, it is desirable to test the entire image circle because it is not known in advance which portions (if any) of the lens will be unused in the lens&#39; final application. For this reason, it is generally advantageous for the relative dimensions of the image circle and the FPA to be selected so that the image circle fits entirely within the FPA, thereby allowing all portions of the lens to be measured. Alternatively, the FPA may be moved relative to the lens during testing so that the entire image circle can measured piecewise. It is also generally advantageous for the image of the test chart to fill the entire image circle of the lens. 
         [0015]      FIG. 4  shows an example portion of a test chart  40 . Test chart  40  has a black region  41  and a grey region  42 . The boundary between black region  41  and grey region  42  defines an edge  43 . Edge  43  is as close to “perfect” as possible (in this context “perfect” means that edge  43  consists of a straight line of zero width with one side being uniformly black and the other side being uniformly grey). 
         [0016]    The image of test chart  40  through a lens can be obtained by an FPA in a lens measuring system (e.g. lens measuring system  20  or lens measuring system  30 ). The lens introduces aberrations that cause blurring of the image of edge  43 . The extent of this blurring can be calculated based on measurements obtained by the FPA. 
         [0017]    Assume that the image of a point of black region  41  far from edge  43  has a pixel value of 1 (where pixel value corresponds to the amount of light incident on the pixel), and that the image of a point of grey region  42  far from edge  43  has a pixel value of 100. If we consider a line of consecutively numbered pixels spaced apart along a line orthogonal to edge  43 , the pixel values might be as follows: 
         [0000]    
       
         
               
               
             
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
             
           
               
                   
                   
               
               
                   
                 Pixel Number 
               
             
          
           
               
                   
                 1 
                 2 
                 3 
                 4 
                 5 
                 6 
                 7 
                 8 
                 9 
               
               
                   
                   
               
             
          
           
               
                 Pixel Value 
                 1 
                 1 
                 10 
                 20 
                 50 
                 80 
                 90 
                 100 
                 100 
               
               
                   
               
             
          
         
       
     
         [0018]    If the lens being tested was perfectly sharp, all the pixel values would be either 1 or 100, with the possible exception of a single pixel that receives light rays from both sides of edge  43  (this is possible because each pixel has a finite size). However, since no lens is perfect, there will always be some degree of blurring. 
         [0019]    The relationship between Pixel Number and Pixel Value describes the “edge step response” of a lens. The MTF of a lens can be estimated by conducting mathematical operations on the edge step response (including Fourier transformations). 
         [0020]    The number of data points in the edge step response depends on the pixel density of the FPA.  FIG. 5  shows an edge step response with a very high number of data points. 
       SUMMARY 
       [0021]    One aspect of the invention provides a test chart that may be used for lens testing. The test chart may be used with any suitable lens testing arrangement including arrangements of a type illustrated in  FIGS. 2 and 3  The test chart comprises X radial lines extending from a central point and Y concentric X-sided polygons centered at the central point. Each of the X-sided polygons comprises X tangential lines. Each of the X radial lines intersects one vertex of each of the polygons. Each of the X radial lines is divided into Y segments by its intersection points with the central point and with the polygons. The radial lines and the polygons define boundaries of X*Y non-overlapping patches. At least two of the patches share a boundary and are distinctly colored from each other. 
         [0022]    In some embodiments the X radial lines extend from the central point at equally spaced apart angular intervals and the Y concentric polygons are regular polygons. 
         [0023]    In some embodiments, each of the patches is distinctly colored from every patch with which it shares a boundary. 
         [0024]    In some embodiments X is even and the patches are colored to form a checkerboard pattern. 
         [0025]    In some embodiments X is in the range of four to twenty. In some embodiments Y is in the range of two to ten. In some embodiments X is eight and Y is three. 
         [0026]    In some embodiments every patch is either a first color or a second color, distinct from the first color. 
         [0027]    In some embodiments the test chart is printed on a surface comprising a reference line. 
         [0028]    In some embodiments the reference line comprises a line printed on the surface. 
         [0029]    In some embodiments the reference line comprises an edge of the surface. 
         [0030]    In some embodiments at least one of the radial or tangential lines has a slope relative to the reference line that has a threshold angular separation from slopes of 1/1, −1/1, 1/0, and 0/1 (where x/y represents a slope with a rise of x and a run of y). 
         [0031]    In some embodiments at least one of the radial or tangential lines has a slope relative to the reference line that has the threshold angular separation away from slopes of 2/1, −2/1, 1/2, −1/2, 3/1, −3/1, 1/3, −1/3, 3/2, −3/2, 2/3, and −2/3. 
         [0032]    In some embodiments at least one the radial or tangential lines has a slope relative to the reference line that has the threshold angular separation away from slopes of M/N where M/N represents any slope that satisfies the following properties: both M and N are integers; the fraction M/N is irreducible; and both M and N are less than K and greater than −K; for some integer value of K between two and ten, inclusive. 
         [0033]    In some embodiments the threshold angular separation is at least 1, 0.5, or 0.1 degrees. 
         [0034]    Another aspect of the invention provides a test chart comprising a plurality of isosceles trapezoids. Each of the plurality of trapezoids comprises a pair of parallel edges and a pair of non-parallel edges. The non-parallel edges of each of the plurality of trapezoids define lines which aim at a common point. At least two of the trapezoids share an edge and are distinctly colored from each other. 
         [0035]    In some embodiments, each of the plurality of trapezoids is distinctly colored from every trapezoid with which it shares an edge. 
         [0036]    In some embodiments the plurality of trapezoids includes a subset of at least three congruent trapezoids and each trapezoid belonging to the subset shares two non-parallel edges with two other trapezoids belonging to the subset. 
         [0037]    In some embodiments the test chart is printed on a surface comprising a reference line. 
         [0038]    In some embodiments the reference line comprises a line printed on the surface. 
         [0039]    In some embodiments the reference line comprises an edge of the surface. 
         [0040]    In some embodiments at least one of the parallel or non-parallel edges has a slope relative to the reference line that has a threshold angular separation away from slopes of 1/1, −1/1, 1/0, and 0/1. 
         [0041]    In some embodiments at least one of the parallel or non-parallel edges has a slope relative to the reference line that has the threshold angular separation away from slopes of 2/1, −2/1, 1/2, −1/2, 3/1, −3/1, 1/3, −1/3, 3/2, −3/2, 2/3, and −2/3. 
         [0042]    In some embodiments at least one of the parallel or non-parallel edges has a slope relative to the reference line that has the threshold angular separation away from slopes of M/N where M/N represents any slope that satisfies the following properties: both M and N are integers; the fraction M/N is irreducible; and both M and N are less than K and greater than −K; for some integer value of K between two and ten, inclusive. 
         [0043]    Another aspect of the invention provides a method for measuring a property of a lens. The method comprises providing a test chart. The test chart comprises X radial lines extending from a central point and Y concentric X-sided polygons centered at the central point. Each of the X-sided polygons comprises X tangential lines. Each of the X radial lines intersects one vertex of each of the polygons. Each of the X radial lines is divided into Y segments by its intersection points with the central point and with the polygons. The radial lines and the polygons define boundaries of X*Y non-overlapping patches. At least two of the patches share a boundary and are distinctly colored from each other. The method further comprises providing a lens and a focal plane array comprising a regular grid of pixels. The method further comprises obtaining an image of the test chart through the lens using the focal plane array. The slope of at least one of the radial and tangential lines, relative to either a horizontal or vertical direction of the regular grid of pixels, has a threshold angular separation away from slopes of 1/1, −1/1, 1/0, and 0/1. 
         [0044]    In some embodiments the X radial lines extend from the central point at equally spaced apart angular intervals and the Y concentric polygons are regular polygons. 
         [0045]    In some embodiments, each of the patches is distinctly colored from every patch with which it shares a boundary. 
         [0046]    In some embodiments the slope of at least one of the radial and tangential lines, relative to either the horizontal or vertical direction of the regular grid of pixels, has the threshold angular separation away from slopes of 2/1, −2/1, 1/2, −1/2, 3/1, −3/1, 1/3, −1/3, 3/2, −3/2, 2/3, and −2/3. 
         [0047]    In some embodiments the slope of at least one of the radial and tangential lines, relative to either the horizontal or vertical direction of the regular grid of pixels, has the threshold angular separation away from slopes of M/N where M/N represents any slope that satisfies the following properties: both M and N are integers; the fraction M/N is irreducible; and both M and N are less than K and greater than −K; for some integer value of K between two and ten, inclusive. 
         [0048]    In some embodiments the method comprises identifying a region of interest on the test chart, the region of interest containing at least a portion of a radial or tangential line; determining the pixel values of the pixels which correspond to the region of interest; and calculating a K-times oversampled edge function of the portion of the line using the pixel values. 
         [0049]    In some embodiments the method comprises calculating a modulation transfer function of a portion of a field of view of a lens corresponding to the region of interest using the oversampled edge function. 
         [0050]    Another aspect of the invention provides a lens measuring system comprising a test chart. The test chart comprises X radial lines extending from a central point and Y concentric X-sided polygons centered at the central point. Each of the X-sided polygons comprises X tangential lines. Each of the X radial lines intersects one vertex of each of the polygons. Each of the X radial lines is divided into Y segments by its intersection points with the central point and with the polygons. The radial lines and the polygons define boundaries of X*Y non-overlapping patches. At least two of the patches share a common boundary and are distinctly colored from each other. The system further comprises a focal plane array comprising a regular grid of pixels; a data store containing data identifying a subset of the pixels; and a processor. The subset of pixels comprises pixels that are located to image a region of interest of the test chart. The region of interest of the test chart contains a portion of a radial or tangential line. The processor is configured to receive the pixel values of the subset of pixels and to calculate an oversampled edge function of the portion of the line based on the pixel values. The slope of the line relative to either a horizontal or vertical direction of the regular grid of pixels is S, and S has a threshold angular separation away from slopes of 1/1, −1/1, 1/0, and 0/1. 
         [0051]    In some embodiments the X radial lines extend from the central point at equally spaced apart angular intervals and the Y concentric polygons are regular polygons. 
         [0052]    In some embodiments, each of the patches is distinctly colored from every patch with which it shares a boundary. 
         [0053]    In some embodiments S has the threshold angular separation away from slopes of 2/1, −2/1, 1/2, −1/2, 3/1, −3/1, 1/3, −1/3, 3/2, −3/2, 2/3, and −2/3. 
         [0054]    In some embodiments S has the threshold angular separation away from slopes of M/N, where M/N represents any slope that satisfies the following properties: both M and N are integers; the fraction M/N is irreducible; and both M and N are less than K and greater than −K; for some integer value of K between two and ten, inclusive. 
         [0055]    Another aspect of this invention provides a test chart. The test chart comprises a plurality of first areas of a first color and a plurality of second areas of a second color, distinct from the first color. The first and second areas are bounded by a plurality of radial first lines extending radially relative to a point and a plurality of polygonal second lines. Each of the polygonal second lines intersects each of the radial first lines. The test chart is printed on a surface comprising a reference line. The first and second areas are arranged such that at each location where a first area is bounded by one of the first lines and a second area lies on the other side of the first line, the first line has a slope relative to the reference line that has a threshold angular separation away from slopes of 1/1, −1/1, 1/0, and 0/1. 
         [0056]    In some embodiments the first line has a slope relative to the reference line that has the threshold angular separation away from slopes of 2/1, −2/1, 1/2, −1/2, 3/1, −3/1, 1/3, −1/3, 3/2, −3/2, 2/3, and −2/3. 
         [0057]    In some embodiments the first line has the threshold angular separation away from slopes of M/N, where M/N represents any slope that satisfies the following properties: both M and N are integers; the fraction M/N is irreducible; and both M and N are less than K and greater than −K; for some integer value of K between two and ten, inclusive. 
         [0058]    In some embodiments the first and second areas are arranged such that at each location where a third area is bounded by one of the second lines and a fourth area lies on the other side of the second line, the second line has a slope relative to the reference line that has the threshold angular separation away from slopes of 1/1, −1/1, 1/0, and 0/1. 
         [0059]    In some embodiments the first line has a slope relative to the reference line that has the threshold angular separation away from slopes of 2/1, −2/1, 1/2, −1/2, 3/1, −3/1, 1/3, −1/3, 3/2, −3/2, 2/3, and −2/3. 
         [0060]    In some embodiments the second line has the threshold angular separation away from slopes of M/N, where M/N represents any slope that satisfies the following properties: both M and N are integers; the fraction M/N is irreducible; and both M and N are less than K and greater than −K; for some integer value of K between two and ten, inclusive. 
         [0061]    Further aspects of the invention and features of example embodiments are illustrated in the accompanying drawings and/or described in the following description. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0062]    The accompanying drawings illustrate non-limiting example embodiments of the invention. 
           [0063]      FIG. 1  is graph of an example Modulation Transfer Function. 
           [0064]      FIG. 2  is a schematic diagram of an example setup of a lens measuring system. 
           [0065]      FIG. 3  is schematic diagram of another example setup of a lens measuring system. 
           [0066]      FIG. 4  shows an example portion of a test chart. 
           [0067]      FIG. 5  shows an edge step response with a very high number of data points. 
           [0068]      FIG. 6  shows a test chart according to an example embodiment of the invention. 
           [0069]      FIG. 7A  shows a portion of a regular grid of pixels of an FPA. 
           [0070]      FIG. 7B  shows example pixel values for pixels of  FIG. 7A . 
           [0071]      FIG. 7C  shows a portion of a regular grid of pixels which has been horizontally subdivided into notional subpixels. 
           [0072]      FIGS. 7D and 7E  show schematically the process of offsetting rows of subpixels and summing columns of subpixels. 
           [0073]      FIG. 8  is a diagram of the edges of the innermost zone of an example test chart. 
           [0074]      FIG. 9  shows an example region of interest containing an edge. 
           [0075]      FIGS. 10A-17  show various test charts according to alternative example embodiments of the invention. 
           [0076]      FIG. 18  is a flowchart illustrating a method for using a test chart according to an example embodiment of the invention. 
       
    
    
     DESCRIPTION 
       [0077]    Throughout the following description specific details are set forth in order to provide a more thorough understanding to persons skilled in the art. However, well known elements may not have been shown or described in detail to avoid unnecessarily obscuring the disclosure. The following description of examples of the technology is not intended to be exhaustive or to limit the system to the precise forms of any example embodiment. Accordingly, the description and drawings are to be regarded in an illustrative, rather than a restrictive, sense. 
         [0078]    One embodiment of the invention comprises a test chart with X (where X&gt;3) radial lines originating from a common central point. In some embodiments, X is even to allow for the test chart to be filled to make a checkerboard pattern. The X radial lines delineate the test chart into X sectors. The test chart is also delineated into one or more zones delineated by one or more concentric polygons.  FIG. 6  shows an example of such a test chart with X=8 and with three zones delineated by three concentric regular octagons. 
         [0079]    Test chart  60  comprises eight radial lines  61  originating from a common central point  62 . Lines  61  radiate at equally spaced angular intervals relative to central point  62 . Radial lines  61  delineate test chart  60  into eight distinct sectors  63 . 
         [0080]    Test chart  60  also comprises a plurality of tangential lines  64 . Tangential lines  64  form three concentric regular octagons  65  that delineate test chart  60  into three distinct zones  66 . Each radial line  61  intersects one vertex of each of regular octagons  65 . 
         [0081]    Radial lines  61  and tangential lines  64  define twenty-four distinct trapezoidal patches  67 . The trapezoidal patches  67  that are adjacent to central point  62  are actually triangles, but they may be thought of as trapezoids with one edge of zero length. Each patch  67  is an isosceles trapezoid. 
         [0082]    Patches  67  are filled with alternating dark and light gray tones so that no adjacent patches  67  have the same tone. The boundaries between adjacent patches  67  define edges  68 . 
         [0083]    In some embodiments other colors are used instead of dark and light gray. (“Colors” for the purposes of this document, includes all colors, hues, tints, shades, tones, greys, etc.) In some embodiments more than two distinct colors are used. In some embodiments not all alternating patches have distinct colors. As long as at least two adjacent patches have distinct colors, there will be at least one edge which can be used to determine some properties of a lens, as described below. 
         [0084]    The contrast of test chart  60  is preferably sufficiently low so as to avoid clipping the image at the black or white level during image capture. For example, if test chart  60  is to be used with an FPA with sensors with a range of a to b, then the relation: a&lt;(normal exposure of dark grey patches)&lt;(normal exposure of light grey patches)&lt;b is beneficial. 
         [0085]    Test chart  60  may be created by any suitable process, for example, a printing process (e.g. inkjet printing, laser printing, lithography, etc.). The dot pitch of the printing process should be sufficiently fine so that the sharpness of the image of a lens is determined by the lens, not the printing process. Test chart  60  may be reflective or transmissive. 
         [0086]    A lens measuring system (such as lens measuring system  20  or lens measuring system  30 ), may be used with test chart  60  to determine the properties of a lens. In some embodiments, the optical axis of the lens is perpendicular to test chart  60 , and aligned with the central point  62  of test chart  60 . In some embodiments, the FPA of the lens measuring system has a desired angular alignment with test chart  60  as discussed below. 
         [0087]    Test chart  60  has a large number of edges  68  spaced throughout the chart. Test chart  60  may be sufficiently large so that when it is used in an appropriate lens measuring system, test chart  60  fills the entire image circle of the lens. An image obtained by an FPA of the lens measuring system may be used to provide information about a large and well distributed number of portions of the lens. 
         [0088]    A plurality of Regions of Interest (“ROI&#39;s”)  69  are identified relative to test chart  60 . ROI&#39;s  69  are not actually printed on test chart  60  and are shown for illustrative purposes only. Each ROI  69  corresponds to a region of pixels on an FPA imaging test chart  60 . When the FPA captures an image of test chart  60  through a lens, each region of pixels corresponding to an ROI contains an image of a portion of the test chart corresponding to the ROI  69 . ROI&#39;s are selected to include edges. 
         [0089]    The image of an edge  68  within an ROI  69  may be analyzed to determine the MTF of a region of a lens corresponding to that ROI  69 . A plurality of ROI&#39;s  69  may be used to calculate the MTF of many different parts of a lens. An MTF may be converted into other metrics of sharpness, such as MTF50. 
         [0090]    The sharpness of an edge imaged by a lens will typically differ depending on whether the edge is radial or tangential relative to the lens. This difference may be caused by various types of lens aberrations such as astigmatism and lateral chromatic aberration. Test chart  60  provides both radial and tangential edges so that the MTF can be measured in both of these directions. 
         [0091]      FIG. 7A  shows a portion of a regular grid of pixels  71  of a FPA  70 . For convenience, the pixels of regular grid of pixels  71  are identified by horizontal numerical coordinates and vertical alphabetical coordinates. 
         [0092]    An image of a test chart (not shown) produced by a lens (not shown) is incident on FPA  70 . Line  72  represents the image of an edge on the test chart separating a region of a first color with a region of a second color, distinct from the first color. In this example line  72  has a slope of 3/1. Pixels far to the left of line  72  have relatively high pixel values (i.e. they receive a relatively high number of photons per exposure) corresponding to the relatively lighter region of the test chart. Pixels far to the right of line  72  have relatively low pixel values, corresponding to the relatively darker region of the test chart. 
         [0093]    In  FIG. 7B  the pixel value of a particular pixel is represented by P XY  where X and Y are the horizontal and vertical coordinates, respectively, of the pixel (e.g. the pixel value of pixel  1 A is represented by P 1A ).  FIG. 7B  shows each of the pixel values of regular grid of pixels  71  (line  72  is not shown in  FIG. 7B ). 
         [0094]    An MTF can be calculated based on the pixel values from a single row (or column) of pixels which crosses line  72 . For example, an MTF can be calculated based on the pixel values in row A of  FIG. 7B . 
         [0095]    A technique called “oversampling” can be used to generate an “oversampled edge function” that can be used to obtain a more accurate measurement of an MTF. An example oversampling calculation is described below. In this example, an oversampling factor of four will be applied in the horizontal direction. Other oversampling factors may be used in other embodiments. 
         [0096]    The oversampled edge function is obtained by combining the pixel values from a plurality of rows (or the pixel values from a plurality of columns), in this example all of rows A-F will be combined. An oversampled edge function has a higher resolution, than any individual row of pixels, and thus it can be used to calculate a relatively more accurate MTF. 
         [0097]    Each pixel is horizontally subdivided into four notional subpixels to create a subpixel grid  75 , as illustrated in  FIG. 7C . The pixel value of each pixel is assigned to one of its subpixels. In this example, the third subpixel is used. The choice of subpixel is arbitrary. 
         [0098]    To generate the oversampled edge function, each of rows B-F of subpixel grid  75  is horizontally offset from row A. The purpose of these offsets is to vertically align the portion of each row which is intersected by line  72 . The appropriate offset for a particular row Θ may be calculated using the following formula: 
         [0000]    
       
      
       H={V*K/S} 
      
     
         [0000]    where:
       H is the horizontal offset between row A and row Θ (measured in units of horizontal subpixels);   V is the vertical distance between row A and row Θ (measured in units of vertical pixels);   K is the oversample factor (measured in units of horizontal subpixels per horizontal pixel);   S is the slope of line  72  (measured in units of vertical pixels per horizontal pixel); and   { } means “rounded to the nearest subpixel”.       
 
         [0104]    Applying this formula for row B we obtain: 
         [0000]      horizontal offset between row  A  and row  B={ 1*4/3}={4/3}=1 
         [0105]    Thus, we shift row B by one subpixel to the right. Performing similar calculations for rows C-F yields  FIG. 7D . Rows C-F are shifted 3, 4, 5, and 7 subpixels to the right, respectively. Note that it is not necessary to physically shift each row. The rows may be shifted notionally. 
         [0106]    Next, the pixel values of each column of subpixels in  FIG. 7D  are summed. Then, each sum is divided by the number of contributing subpixels to obtain an oversampled edge function  77  as shown in  FIG. 7E . 
         [0107]    Oversampled edge function  77  contains several subpixels with no pixel values. Such subpixels are known as “empty bins”. Pixel values may be assigned to these empty bins to facilitate performing an MTF calculation. It is possible to use interpolated values for the empty bins, but this causes a loss of accuracy of the MTF. It is desirable to use an oversampled edge function which has a relatively low number of empty bins (or even no empty bins). 
         [0108]    A few subpixels (including empty bins) on either end of the oversampled edge function may be ignored in the MTF calculation. In some embodiments, the number of pixels ignored on each end is equal to half of the maximum horizontal offset between adjacent rows. Since the subpixels on either end of the oversampled edge function are relatively far from edge  72 , ignoring them does not significantly affect the accuracy of the MTF. 
         [0109]    For any oversampling factor, an edge with a slope of 1/1, −1/1, 1/0, or 0/1 will produce a relatively large number of empty bins. 
         [0110]    For an oversampling factor of four, edges with a slope of 2/1, 1/2, 3/1, 1/3, 3/2, or 2/3, and the negatives of these slopes (approximately 63.43, 26.57, 71.57, 18.43, 56.31, and 33.69 degrees, and the negatives of these angles, respectively) will produce relatively large numbers of empty bins. 
         [0111]    For an oversampling factor of K (for values of K up to at least 10), slopes of M/N, where M and N are both integers, the fraction M/N is irreducible, and both M and N are less than K and greater than −K, will produce relatively large numbers of empty bins. 
         [0112]    By selecting an appropriate value for the slope of an edge, the incidence of empty bins in the edge&#39;s oversample edge function can reduced or even eliminated. A test chart may be rotated so that its edges have slopes that provide relatively few empty bins for the oversampling factor to be applied. 
         [0113]    Test chart  60 , as shown in  FIG. 6 , can be described as being rotated by 7.6 degrees anticlockwise away from “straight”, where “straight” means one of radial edges  61  is at zero degrees relative to a reference alignment to an FPA. A reference line may be used to achieve the desired alignment. The reference line (not shown) may be an edge of a page on which test chart  60  is printed, or it may be a line printed on the page. When test chart  60  is used with a lens measuring system, the reference line and the regular grid of pixels of the FPA should be parallel. 
         [0114]    Radial edges  61  of test chart  60  have angles of 7.6 degrees plus or minus all integer multiples of 45 degrees (e.g. 7.6, 52.6, 97.6, 142.6, 187.6, 232.6, 277.6, and 322.6 degrees). Tangential edges  64  of test chart  60  have angles of 75.1 degrees, plus or minus all integer multiples of 45 degrees (e.g. 30.1, 75.1, 120.1, 165.1, 210.1, 255.1, 300.1, and 345.1 degrees). 
         [0115]    The rotation of test chart  60  by 7.6 degrees relative to the FPA creates angular separation between radial edges  61  and the “bad angles” associated with four times oversampling identified above. Thus when four times oversampled edge functions are calculated for radial edges  61  of test chart  60 , there will be relatively few empty bins, and the MTF&#39;s calculated using test chart  60  can be relatively accurate. 
         [0116]    In general, a test chart can be created by determining several “bad angles” for the oversampling factor that will be applied, and then rotating the test chart such that the radial and tangential edges are separated from the “bad angles”. In some embodiments, an optimization calculation may be performed to determine a rotation that will maximize the angular separation of the radial and tangential edges from the identified “bad angles”. 
         [0117]    Alternatively, the angles of the radial and tangential edges of a given test chart can be readily calculated as a function of the angle of rotation of the test chart, and the number of empty bins for each edge can be readily calculated as a function of the edge angle for a given oversampling factor. Using these functions, the angle of rotation can be varied to determine an angle which results in a relatively low number of empty bins. 
         [0118]    If a test chart has a design similar to test chart  60 , but has N radial lines forming N equal sectors and is rotated by X degrees anticlockwise from “straight”, then:
       the radial lines will be at X+I(360/N) degrees, where I can be integer; and   the tangential lines will be at X+90−(180/N)+I(360/N) degrees where I can be any integer.       
 
         [0121]    The validity of the second formula can be illustrated by reference to  FIG. 8 , which is a diagram of the edges of the innermost zone of a test chart  80 . For the sake of clarity, the shading of test chart  80  is not shown. Test chart  80  has been rotated from “straight” by X degrees anticlockwise. 
         [0122]    Test chart  80  has eight radial lines forming eight equal sectors, but for the purposes of the discussion below, we will refer to test chart  80  as having N radial lines and N sectors. Test chart  80  has angles U, V, W, X, Y, and Z. We are interested in Z, the angle formed between horizontal line  81  and tangential line  82 . 
         [0123]    By algebra and geometry we obtain Z=X+90−(180/N). Thus the angle between horizontal line  81  and tangential line  82  is X+90−(180/N) degrees. 
         [0124]    Since there are a total of N tangential lines that form a closed loop, the general formula for the angles of the tangential lines is X+90−(180/N)+I(360/N) where I can be any integer. 
         [0125]    There are several different ways to compute an MTF (see, for example, “Modulation Transfer Function in Optical and Electro-optical Systems”, G. D. Boreman, 2001, SPIE Press). Depending on the nature of a test chart, different types of MTF calculations may be more or less advantageous. For example, for a slanted edge, it is advantageous to use a slanted edge MTF calculation that includes a correction for the change in effective sampling frequency due to the slanted edge. 
         [0126]    Where oversampling is used, it is advantageous to use an MTF calculation that takes into account the possibility of empty bins. 
         [0127]    It may be advantageous to use a monochrome sensor as an FPA so that the MTF is not degraded by a color filter array pattern such as a Bayer filter. 
         [0128]    It may be advantageous for the MTF calculation to include a correction to account for the fact that each pixel of an FPA has a finite area. The MTF of a square pixel is: 
         [0000]    
       
         
           
             
               MTF 
               pixel 
             
             = 
             
               
                 sin 
                  
                 
                     
                 
                  
                 π 
                  
                 
                     
                 
                  
                 fw 
               
               
                 π 
                  
                 
                     
                 
                  
                 fw 
               
             
           
         
       
     
         [0000]    where f is spatial frequency and w is the width of the pixel. The MTF of a region of a lens associated with a pixel can be approximated as: 
         [0000]    
       
         
           
             
               MTF 
               lens 
             
             = 
             
               
                 MTF 
                 measured 
               
               
                 MTF 
                 pixel 
               
             
           
         
       
     
         [0129]    Once an MTF is calculated, other metrics (e.g. MTF50) can be determined based on the MTF. 
         [0130]    An oversampled edge function may be calculated based on the pixel values of the pixels corresponding to an ROI containing the edge. The oversampled edge function can be used to calculate an MTF for a region of a lens corresponding to the ROI. Computer hardware and software may be used to perform these calculations. 
         [0131]    Various considerations dictate the optimal size and shape of the ROI&#39;s. Ideally, each ROI should be dimensioned such that the edge it contains traverses at least one whole pixel in both the horizontal and vertical directions of the alignment of the regular grid of pixels of the FPA. 
         [0132]    An appropriate ROI size may be determined empirically by repeating an MTF measurement while gradually increasing the ROI size until the MTF becomes insensitive to further increases in the ROI size. 
         [0133]    In some embodiments, the ROI&#39;s may range between 50 and 200 pixels in length. In embodiments where a relatively higher resolution FPA is used, the absolute size of the ROI&#39;s (measured in mm or inches) may be relatively lower. Smaller ROI&#39;s may accommodate greater numbers of zones and sectors, and thus permit MTF measurements of a greater number of regions of a particular lens. 
         [0134]    In some embodiments, the ROI&#39;s are rectangular in shape. In some embodiments, the rectangular ROI&#39;s are aligned so that their edges are parallel with the regular grid of pixels of an FPA. 
         [0135]    In test chart  60 , the ROI&#39;s  69  are rectangular in shape. ROI&#39;s  69  have an aspect ratio of 2:1. The long edge of each ROI  69  is oriented to maximize the distance from the corners of the ROI to the edge running through the ROI. If this distance is too small, errors may be introduced into the MTF calculation. 
         [0136]    In some embodiments the ROI&#39;s are non-rectangular in shape. In some embodiments differently shaped and/or dimensioned ROI&#39;s are used at different locations of a test chart. In some embodiments the shape and/or dimension of each ROI depends on the slope of the edge it contains. 
         [0137]    In some embodiments, every edge of a test chart contains at least one ROI. In some embodiments, every trapezoid of a test chart contains an ROI on each of its four edges. 
         [0138]    In some embodiments, information identifying the sets of pixels which correspond to each ROI for a particular test chart/lens combination are saved in a data store. In some embodiments, these sets of pixels may be modified depending on a selected angle of rotation of a test chart. 
         [0139]      FIG. 9  shows an example rectangular ROI  90  containing an edge  91  of a test chart. Opposite sides of edge  91  have distinct colors. ROI  90  has a height H and a width W. 
         [0140]    If height H is relatively large, then there will be a relatively large number of rows of pixels available to use to generate an oversampled edge function. In general, when more lines of pixels are used (e.g. rows or columns of pixels), the resulting oversampled edge function (and the corresponding MTF) will be more accurate. 
         [0141]    If width W is not large enough to ensure that the pixel values of the pixels at the ends of each row achieve the values that fully correspond, respectively, to the distinct colors on either side of line  91 , then the calculated MTF associated with ROI  90  may be inaccurate. Also, if width W is too narrow, other sources of inaccuracy may be introduced into the MTF calculation (e.g. a “window function” used in the MTF calculation may result in reduced accuracy). 
         [0142]    Some embodiments of the invention comprise test charts with greater or fewer numbers of sectors and/or zones compared to test chart  60 . Test charts with more sectors and/or zones have more edges that may be used to make MTF measurements of more regions of a lens. 
         [0143]      FIG. 10A  shows a test chart  100 A with four zones and ten sectors. The number of sectors affects the angles of both the radial and tangential edges, and therefore the number of sectors affects the rotation of test chart  90  required in order to have the radial and tangential edges separated from any “bad angles”. Test chart  100 A has been rotated so that one of its radial edges is at a 5.4 degree angle relative to the horizontal edge of the page. 
         [0144]      FIG. 10B  shows a test chart  100 B, which is an alternative embodiment of test chart  100 A. The outermost zone  101  of test chart  100 B has been extended to fill a square boundary  102 . Test chart  100 B may be dimensioned to completely fill a square sheet of paper. 
         [0145]    In some embodiments a test chart may be adapted for use with lenses that are expected to yield images distorted in known ways by changing the relative sizes of the sectors and/or zones. The sizes of the sectors and/or zones may be changed in order to achieve an even distribution of ROI&#39;s on the image projected onto the FPA. The sizes of the sectors and/or zones may also be changed in order to ensure that ROI&#39;s can fit on all of the edges. 
         [0146]    In some embodiments a test chart may be adapted for use with lenses that are expected to yield images distorted in known ways by changing the shape of the boundaries between sectors and/or zones. For example the edges of test chart  60  may be curved so that when they are imaged by a particular “distorted” lens, the boundaries appear straight in the image. 
         [0147]      FIG. 11  shows a test chart  110  which has been pre-distorted for use with a lens with barrel distortion. The zones decrease in radial thickness toward the centre of the chart. (A test chart could also be distorted for use with a lens with pincushion distortion by having the zones increase in radial thickness toward the centre of the chart.) The circular symmetry of the chart ensures that the basic structure of the chart is not changed by radial distortion and so suitable computer based target registration methods can readily function with the distortion. 
         [0148]    To measure an MTF of a distorted lens, it is not always necessary to use a distorted test chart. 
         [0149]    If test chart  110  is used with a lens with radial distortion (e.g. barrel distortion or pin-cushion distortion), the tangential edges of test chart  110  will appear curved in the resulting image. To correct for this, the tangential edges can be curved in an opposite direction on the chart so that they appear straight in the image produced by the lens. Alternatively, a slanted edge MTF calculation can incorporate a correction for curved edges. 
         [0150]    In an example “curved edge correction”, the location of the edge along each row (or column) of pixels in an ROI is estimated by finding the centroid of the derivative of the row of pixel values. A line or polynomial is then fit to the estimated edge locations. In some embodiments a least-squares method is used to determine the line or polynomial. In some embodiments the polynomial is a 2 nd  order polynomial. The calculated line or polynomial is then used to determine the horizontal offsets of each row for the purpose of calculating the oversampled edge function, as described above. 
         [0151]    In some embodiments the sizes of the zones are adjusted in order to provide the possibility of more ROI&#39;s close to the perimeter of the chart.  FIG. 12  shows a test chart  120  with a relatively larger central zone. 
         [0152]    In some embodiments, ROI&#39;s of the desired size cannot fit into the central zone. In such embodiments the central zone may be excluded from the MTF calculation, and it may be replaced by other patterns.  FIG. 13  shows a test chart  130  in which the central zone is replaced with a Siemens™ star  131  as an aid for focusing. In other embodiments, other patterns could be placed in the center of a lens test chart, such as a feature to aid with automated image processing for recognizing the center of the chart. 
         [0153]      FIG. 14  shows a test chart  140  with three zones and eighteen sectors. The sectors of the innermost two zones have been shaded such that there are only six radial edges  141  within each of these zones. Also, the tangential edges at the boundary of the two innermost zones have been modified so that there are only six such tangential edges. 
         [0154]      FIG. 15  shows a test chart  150  with four zones and twelve sectors. The sectors of the innermost zone have been shaded such that there are only four radial edges  151  within the zone. 
         [0155]      FIG. 16  shows a test chart  160  with three zones and twelve sectors. The sectors of the innermost zone have been shaded such that there are only six radial edges  161  within the zone. 
         [0156]    In some embodiments, not all sectors are equal in size.  FIG. 17  shows a test chart  170  with four zones and eight sectors. Box  171  is not part of test chart  170 , but is included to illustrate the portion of test chart  170  which is imaged by an FPA with a 4:3 aspect ratio. Four diagonal radial edges  172  run along the diagonals of box  171  (at 36.87, 143.13, 216.87, and 323.13 degrees). Diagonal radial edges  172  have an angular separation of at least 3.18 degrees from the “bad angles” for an oversampling factor of four. Thus when an oversampling factor of four is applied to ROI&#39;s on diagonal radial edges  172  there are relatively few empty bins. 
         [0157]    Four other radial edges  173  of test chart  170  are rotated 5 degrees anticlockwise from vertical and horizontal (relative to box  171 ). This rotation avoids the generation of an excessively high number of empty bins when an oversampling factor of four is applied. In an alternative embodiment of test chart  170 , radial edges  173  are rotated by 6.5 degrees anticlockwise from vertical and horizontal (relative to box  171 ). In this embodiment radial edges  173  have an angular separation of at least 3.21 degrees from the “bad angles” for an oversampling factor of four. 
         [0158]      FIG. 18  is a flowchart illustrating a method  180  for determining characteristics (e.g. measuring the sharpness) of a lens according to an embodiment of the invention. In block  181 , an image of a test chart is obtained using a FPA. In some embodiments, the lens is focused on the chart prior to obtaining the image. In some embodiments, method  180  is carried out multiple times and the focus is adjusted each time to increase the measured sharpness. 
         [0159]    The test chart is a chart as described above. The test chart may be rotated relative to a regular grid of pixels of the FPA as describe above. The image may be obtained for example by using a lens measuring system like lens measuring system  20  or lens measuring system  30  described above. The image of the test chart comprises a plurality of pixel values, each corresponding to one of a plurality of pixels of the FPA. 
         [0160]    In block  182 , one or more regions of interest are selected. Each region of interest corresponds to some subset of pixels. Each region of interest contains pixels which represent an image of an edge of the test chart. Block  182  may comprise retrieving information identifying the regions of interest associated with the test chart from a computer readable memory. 
         [0161]    In block  183 , for each region of interest, a calculation is performed, using the pixel values as inputs, to determine a characteristic (e.g. the MTF) of a portion of the lens corresponding to the region of interest. 
         [0162]    In some embodiments, some or all of the steps of method  180  are carried out by a computer. 
         [0163]    While a number of exemplary aspects and embodiments have been discussed above, those of skill in the art will recognize certain modifications, permutations, additions and sub-combinations thereof. It is therefore intended that the following appended claims and claims hereafter introduced are interpreted to include all such modifications, permutations, additions and sub-combinations as are within their true spirit and scope.