Abstract:
An X-ray inspection system incorporates an improved technique for determining, in an X-ray image of a multilayered assembly, the gray level component of a first material in the presence of a second material. The total gray level of the image is dependent upon the physical characteristics of each material comprising the assembly. The present invention accurately determines the component of the total image gray level due to the first material. In the case of circuit board inspections using X-ray images of solder connections, a calibration procedure facilitates the direct conversion of the gray level component due to the solder connection to the thickness of the solder connection.

Description:
FIELD OF THE INVENTION 
     The invention relates generally to the automated X-ray inspection of printed circuit assemblies, and in particular, to systems which use X-ray images of solder joints to provide a measured thickness of the solder joint. 
     BACKGROUND OF THE INVENTION 
     Rapid and precise quality control inspections of the soldering and assembly of electronic devices have become priority items in the electronics manufacturing industry. Many existing inspection systems for electronic devices and connections make use of penetrating radiation to form images which exhibit features representative of the internal structure of the devices and connections. These systems often utilize conventional radiographic techniques wherein the penetrating radiation comprises X-rays. Medical X-ray pictures of various parts of the human body, e.g., the chest, arms, legs, spine, etc., are perhaps the most familiar examples of conventional radiographic images. The images or pictures formed represent the X-ray shadow cast by an object being inspected when it is illuminated by a beam of X-rays. The X-ray shadow is detected and recorded by an X-ray sensitive material such as film or electronic means. Alternatively, tomographic techniques such as laminography and computed tomography (CT) may be used to produce cross-sectional images of the object being inspected. Laminography systems which are capable of achieving the speed and accuracy requirements necessary for electronics inspection are described in the following patents: 1) U.S. Pat. No. 4,926,452 entitled “A UTOMATED  L AMINOGRAPHY  S YSTEM FOR  I NSPECTION OF  E LECTRONICS ”, issued to Baker et al.; 2) U.S. Pat. No. 5,097,492 entitled “A UTOMATED  L AMINOGRAPHY  S YSTEM FOR  I NSPECTION OF  E LECTRONICS ”, issued to Baker et al.; 3) U.S. Pat. No. 5,081,656 entitled “A UTOMATED  L AMINOGRAPHY  S YSTEM FOR  I NSPECTION OF  E LECTRONICS ”, issued to Baker et al.; 4) U.S. Pat. No. 5,291,535 entitled “M ETHOD AND  A PPARATUS FOR  D ETECTING  E XCESS /I NSUFFICIENT  S OLDER  D EFECTS ”, issued to Baker et al.; 5) U.S. Pat. No. 5,621,811 entitled “L EARNING  M ETHOD AND  A PPARATUS FOR  D ETECTING AND  C ONTROLLING  S OLDER  D EFECTS ”, issued to Roder et al.; 6) U.S. Pat. No. 5,561,696 “M ETHOD  &amp; A PPARATUS FOR  I NSPECTING  E LECTRICAL  C ONNECTIONS ”, issued to Adams et al.; 7) U.S. Pat. No. 5,199,054 entitled “M ETHOD AND  A PPARATUS FOR  H IGH  R ESOLUTION  I NSPECTION OF  E LECTRONIC  I TEMS ”, issued to Adams et al.; 8) U.S. Pat. No. 5,259,012 entitled “L AMINOGRAPHY  S YSTEM AND  M ETHOD WITH  E LECTROMAGNETICALLY  D IRECTED  M ULTIPATH  R ADIATION  S OURCE ”, issued to Baker et al.; 9) U.S. Pat. No. 5,583,904 entitled “C ONTINUOUS  L INEAR  S CAN  L AMINOGRAPHY  S YSTEM AND  M ETHOD ”, issued to Adams; and 10) U.S. Pat. No. 5,687,209 entitled “A UTOMATIC  W ARP  C OMPENSATION FOR  L AMINOGRAPHIC  C IRCUIT  B OARD  I NSPECTION ”, issued to Adams. The entirety of each of the above referenced patents is hereby incorporated herein by reference. 
     In automated X-ray inspection (AXI) of printed circuit assemblies, gray-scale images of interconnects or slices thereof are examined to detect and classify improper joints and/or to provide statistical process control data relating to the manufacturing process. For reasons including but not limited to portability, reproducibility and clarity, it is desirable that measurements taken relate directly to physical characteristics of the joint under inspection. In characterizing solder joints, for example, it is preferable to deal with measured joint thickness rather than gray scale pixel values. However, extracting solder thickness from the measured gray scale pixel values is complicated by several factors. First, X-ray sources used in AXI typically emit X-rays at many wavelengths with varying intensities as a function of wavelength. Additionally, in passing through a printed circuit assembly, X-rays will typically encounter other absorbers in addition to the solder, e.g., copper power and ground planes, tantalum capacitors, etc. Each material has its own characteristic absorption spectrum as a function of wavelength. The resulting interaction is highly non-linear, and complete characterization of the thickness of solder and other shading materials in the path is generally not possible from a limited number of gray scale calibration measurements. 
     Nonetheless, useful approximations can be made when prior knowledge of the assembly under inspection is available. For example, in many cases, solder thickness may be desired and it may be known that the background shading is due almost entirely to a particular material, e.g., copper. In such cases, by measuring background (due to the copper alone) and foreground (due to both copper and solder) gray values, one may attempt to estimate solder thickness if a suitable correction for background “shading” by copper can be constructed. 
     Previous calibration procedures have encountered a number of difficulties in practice. For example, previous attempts which use polynomial regression techniques to fit a set of calibration points to a surface which approximates solder thickness have been deficient. Such fitted surfaces frequently have unwanted maxima, minima, saddle points and inflection points, and often do not accurately reflect the underlying physical process. Better fits may be obtained by using a more constrained surface (e.g. one which is linear along one or more axis) to a portion of the calibration surface. This helps avoid the problems that often plague higher order regression surfaces, but leads to its own difficulties. In particular, multiple “patches” are often required to approximate the entire calibration surface. In the presence of measurement noise, this can lead to inconsistent behavior for points lying near the borders of adjacent patches. 
     OBJECTS AND ADVANTAGES OF THE PRESENT INVENTION 
     It is the object of the present invention to circumvent the above described difficulties. In particular, the present invention: 
     a) provides a single, globally consistent calibration for any chosen material in the presence of varying amounts of shading by a second material; 
     b) is fast in terms of its computational requirements; 
     c) is compact in terms of its storage requirements; 
     d) is more accurate than previous methods; 
     e) is numerically invertible; 
     f) may be made traceable to known standards criteria, for example, the National Institute of Standards &amp; Technology (NIST) or similar standards agencies. This feature permits process engineers to relate thicknesses measured by the X-ray system to physical joint dimensions. Traceability can be achieved by constructing the calibration standard out of materials of known purity, and by measuring thicknesses of the calibration standard using instruments which themselves have a traceable calibration; 
     g) is portable, in the sense that measurement of the same joint on multiple systems will return similar or identical thicknesses. Portability requires that the calibration compensates for the physically significant sources of variation between systems; and 
     h) supports multiple calibrations. With the advent of lead-free solders, the joint and background compositions can vary from board to board, or even within a board. As a result, it is desirable to be able to store multiple calibrations simultaneously, and to permit the user to select the appropriate calibration on a pin, component, or board level. 
     SUMMARY OF THE INVENTION 
     The present invention comprises an improved system which provides more accurate determination of solder joint thicknesses derived from X-ray images of the solder joints. More generally, the present invention may be used to determine the quantities of two materials comprising a two component assembly. The configuration of the two materials in the assembly may be in any form, e.g., the two materials may be in two separate layers, multiple mixed layers, an homogenous mixture, etc. The two materials may themselves consist of complex chemical mixtures rather than pure elements or compounds. 
     Consider the special case of lead or solder shaded by copper for the purpose of simplifying the following illustration. The present invention measures the gray levels of X-ray images of a number of test coupons which contain known thicknesses of the lead or solder shaded by varying amounts of copper. By a combination of theoretical and empirical arguments, it has been found that the effect of the copper shading may be described by a particular nonlinear equation with three free parameters. Moreover, two of the three parameters are found to be characteristics of the AXI system and not functions of the amount of copper or lead/solder in the X-ray beam path. One aspect of the system calibration involves estimation and storage of these two parameters. Foreground and background gray level values from an unknown sample are adequate to fix the third parameter, completely characterizing the shading effect for that sample. As a result, it is possible to use the two stored system parameters and the known functional form of the shading equation to extrapolate to values that would have been measured under a “standard” or predetermined reference shading level. For example, “no shading”, i.e., zero background, may be used as the standard condition. However, other non-zero background shading levels may also be selected as the standard condition. Since any measured sample can be readily converted to standard conditions using this approach, there is no need for a two dimensional thickness calibration. Instead, a simple one dimensional curve suffices, since measurements can always be corrected to standard conditions. 
     In a first aspect, the present invention includes a method for calibrating an X-ray imaging system for quantitatively determining the thickness of a first absorbing material in the presence of a second absorbing material where an incident X-ray beam having an incident X-ray beam intensity is transmitted through the first and second absorbing materials, the method comprising the steps of: providing a calibration standard having: a) multiple combinations of a first known thickness of the first absorbing material (denoted by t M1,1 ) in combination with three thicknesses of the second absorbing material (denoted by t M2,1 , t M2,2  and t M2,3 ); and b) multiple combinations of a second known thickness of the first absorbing material (denoted by t M1,2 ) in combination with three thicknesses of the second absorbing material (denoted by t M2,4 , t M2,5  and t M2,6 ); determining the values of first, second and third foreground parameters (denoted by F 1 , F 2  and F 3 ) wherein: a) the first foreground parameter F 1  is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the first absorbing material having the thickness t M1,1  in combination with the second absorbing material having the thickness t M2,1 ; b) the second foreground parameter F 2  is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the first absorbing material having the thickness t M1,1  in combination with the second absorbing material having the thickness t M2,2 ; and c) the third foreground parameter F 3  is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the first absorbing material having the thickness t M1,1  in combination with the second absorbing material having the thickness t M2,3 ; determining the values of first, second and third background parameters (denoted by B 1 , B 2  and B 3 ) wherein: a) the first background parameter B 1  is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through only the second absorbing material having the thickness t M2,1 ; b) the second background parameter B 2  is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through only the second absorbing material having the thickness t M2,2 ; and c) the third background parameter B 3  is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through only the second absorbing material having the thickness t M2,3 ; determining the values of fourth, fifth and sixth foreground parameters (denoted by F 4 , F 5  and F 6 ) wherein: a) the fourth foreground parameter F 4  is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the first absorbing material having the thickness t M1,2  in combination with the second absorbing material having the thickness t M2,4 ; b) the fifth foreground parameter F 5  is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the first absorbing material having the thickness t M1,2  in combination with the second absorbing material having the thickness t M2,5 ; and c) the sixth foreground parameter F 6  is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the first absorbing material having the thickness t M1,2  in combination with the second absorbing material having the thickness t M2,6 ; determining the values of fourth, fifth and sixth background parameters (denoted by B 4 , B 5  and B 6 ) wherein: a) the fourth background parameter B 4  is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through only the second absorbing material having the thickness t M2,4 ; b) the fifth background parameter B 5  is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through only the second absorbing material having the thickness t M   2,5 ; and c) the sixth background parameter B 6  is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through only the second absorbing material having the thickness t M2,6 ; and determining a first functional form of a non-linear function, y 1 (x), which describes the value of the foreground minus the background (y 1 =F−B) as a function of background (x=B) such that the non-linear functional form: a) approximates the following values of foreground minus background: (F 1 −B 1 ), (F 2 −B 2 ), (F 3 −B 3 ), (F 4 −B 4 ), (F 5 −B 5 ) and (F 6 −B 6 ); b) supports extrapolation beyond the range of the values of foreground minus background {(F 1 −B 1 ), (F 2 −B 2 ), (F 3 −B 3 ), (F 4 −B 4 ), (F 5 −B 5 ), (F 6 −B 6 )} and/or foreground {F 1 , F 2 , F 3 , F 4 , F 5 , F 6 } and/or background {B 1 , B 2 , B 3 , B 4 , B 5 , B 6 }; and c) incorporates one or more additional constraints determined by or approximating the physical behavior of the X-ray imaging system. The steps of determining the values of the foreground and background parameters may further comprise the steps of: illuminating the calibration standard with a beam of X-rays having the incident X-ray beam intensity, wherein the beam of X-rays is produced by an X-ray source; and measuring the values of the foreground and background parameters with an X-ray detector. The steps of determining the values of the foreground and background parameters may further comprise the step of simulating the values of the foreground and background parameters using one or more of the following simulation factors: a) spectral characteristics of the X-ray source; and/or b) angular distribution of X-rays produced by the X-ray source; and/or c) stopping power and spectral sensitivity of the X-ray detector; and/or d) X-ray attenuation properties of the first and second absorbing materials as functions of X-ray energy/wavelength. The foreground parameters F i  may be described by a functional form, y F : 
      y F =y 0 −∫α(E) e   −β(E)t     1     e   −γ(E)t     2   dE 
     or its discrete approximation: 
     
       
         y F =y 0 −Σ i α i   e   −β     i     t     1     e   −γ     i     t     2     
       
     
     where t 1  and t 2  are the thicknesses of the first absorbing material and the second absorbing material, respectively; y 0  is a fitting constant; and, in the general functional form: a) the X-ray source energy spectrum is distributed as a function of energy with weightings determined by the parameter α(E); and b) β(E) and γ(E) are the X-ray attenuation coefficients for the first and second absorbing materials, respectively, and in the discrete approximation: c) the total X-ray source energy spectrum is split up into some number of bands i, where the total source intensity is distributed among the bands as a functions of X-ray source energy and detector sensitivity with weightings for each band i determined by the parameter α i ; and d) β i  and γ i  are the effective linear attenuation coefficients for X-rays in band i for the first and second absorbing materials, respectively. The step of determining a first functional form of a smoothly varying non-linear function which expresses the value of the foreground minus the background (y 1 =F−B) as a function of background (x=B) may also comprise the step of selecting a function of the form: 
     
       
         y 1 ={square root over ((x−a) 2 +L +b 2 +L )}+c 
       
     
     where x corresponds to the background B i , y 1  corresponds to the difference between the foreground and background (F i −B i ), and a, b and c are fitting constants. The method may further comprise the steps of: selecting a reference background level (x=B R ); determining the values of foreground minus background (F Ri −B Ri ) at the reference background level (B R ) for multiple known thicknesses of the calibration standard using the smoothly varying non-linear function y 1  which expresses the value of the foreground minus the background (y 1 =F−B) as a function of background (x=B); and determining a second functional form y 2  which expresses the values of foreground minus background (F Ri −B Ri ) at the reference background level (B R ) for the multiple known thicknesses of the first absorbing material as a function of the thickness of the first absorbing material. The step of determining a second functional form y 2  may further comprise the step of selecting a function which is a sum of exponentials of the form: 
     
       
         y 2 (t)=p−Σ i q i   e   −r     i     t   
       
     
     where p, q i  and r i  are fitting constants. The method may further include the step of producing a lookup table for values of (background) vs. (foreground minus background) vs. (thickness) for one or both of the first and/or second absorbing materials. The method may also further comprise the steps of: determining the value of a seventh foreground parameter (denoted by F 7 ) which is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the first absorbing material having an unknown thickness t M1,7  in combination with the second absorbing material having an unknown thickness t M2,7 ; determining the value of a seventh background parameter (denoted by B 7 ) which is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the second absorbing material having an unknown thickness t M2,7 ; and using the lookup table and the values of F 7  and B 7  to determine one or both of the unknown thickness(es) of the first absorbing material (t M1,7 ) and/or the second absorbing material (t M2,7 ). This method may also include the step of interpolating between values in the lookup table. The step of interpolating may further comprise the step of bilinear interpolation. The method may further include the step of selecting the thicknesses of the second absorbing material (t M2,i ) such that at least one of the values of the first, second and third background parameters (denoted by B 1 , B 2  and B 3 ) is equal to at least one of the values of the fourth, fifth and sixth background parameters (denoted by B 4 , B 5  and B 6 ). Similarly, the method may further comprise the step of selecting the thicknesses of the second absorbing material (t M2,i ) such that at least two of the values of the first, second and third background parameters (denoted by B 1 , B 2  and B 3 ) are equal and/or at least two of the values of the fourth, fifth and sixth background parameters (denoted by B 4 , B 5  and B 6 ) are equal. 
     In a second aspect, the present invention includes a method for measuring the thickness of a first material in the presence of a second material comprising the steps of: providing a calibration standard having: a) multiple combinations of a first known thickness of the first material in combination with a range of thicknesses of the second material; and b) multiple combinations of a second known thickness of the first material in combination with a range of thicknesses of the second material; exposing the calibration standard to a source of transmissive energy having an incident intensity; detecting the intensity of the transmissive energy which passes through the calibration standard, the detecting step further comprising the step of: acquiring multiple pairs of image data which are representative of a portion of the transmissive energy which is measured after transmission through the first and second materials, where a foreground value (F) in each pair of image data corresponds to a portion of the incident intensity which is transmitted through the known thickness of the first material in combination with one of the multiple thicknesses of the second material, and a background value (B) in each pair of transmitted intensities corresponds to a portion of the incident intensity which is transmitted through only the corresponding thickness of the second material which was in combination with the first material when the foreground value (F) was acquired; determining fitting constants a,b and c for each member of a family of hyperbolic curves which describe delta gray values (y 1 =ΔG=F−B) as a function of background values (B), where each curve in the family represents delta gray values for a fixed known thickness of the first material in combination with a range of thicknesses of the second material, each of the hyperbolic curves having the general form of: 
      y 1 ={square root over ((x−a) 2 +L +b 2 +L )}+c 
     where x corresponds to the background values (x=B); y 1  corresponds to the delta gray values (y 1 =ΔG=F−B) for a fixed known thickness of the first material in combination with the range of thicknesses of the second material; and a, b and c are the fitting constants, wherein the fitting constants are determined such that each hyperbolic curve in the family has the same x-axis intercept (BG MAX ,O) and each hyperbolic curve in the family has a minimum value at the same value of x (x=a); determining for each known thickness of the first material, a delta gray level at a reference background level, i.e., y 1 (x=B R ), from the hyperbolic curve defined by the multiple pairs of image data for the respective known thickness of the first material; and determining fitting constants for a second functional form (y 2 ) which describes the delta gray level values at the reference background level, as a function of the known thicknesses (t) of the first material, where the functional form is: 
     
       
         y 2 (t)=BG MAX   −βe   −k     1     t −(BG MAX −β) e   −k     2     t   
       
     
     where fitting constants β, k 1  and k 2  are determined by fits to the known thicknesses of the first material and corresponding delta gray levels at the reference background level derived from the hyperbolic curves which describe the delta gray values (y 1 ) as a function of the background values (B). 
     A third aspect of the present invention is method for measuring the thickness of a first material in the presence of a second material comprising the steps of: providing a calibration standard having: a) multiple combinations of a first known thickness (t M1,1 ) of the first material in combination with a range of thicknesses (t M2,a , t M2,b , . . . , t M2,n1 ) of the second material; and b) multiple combinations of a second known thickness (t M1,2 ) of the first material in combination with a range of thicknesses (t M2,n1+1 , t M2,n1+2 , . . . , t M2,n1+n2 ) of the second material; exposing the calibration standard to a source of transmissive energy having an incident intensity; detecting the intensity of the transmissive energy which passes through the calibration standard and determining therefrom image data which are representative of a portion of the transmissive energy which is measured after transmission through the first and second materials, the detecting step further comprising the step of: acquiring multiple pairs of image data, where each pair includes a foreground value and a background value, for each known thickness of the first material (t M1,1 , t M1,2 ) in combination with multiple thicknesses (t M2,a , t M2,b , etc.) of the second material; where the foreground value (y f ) in each pair of image data corresponds to a portion of the incident intensity which is measured after transmission through the known thickness of the first material in combination with one of the multiple thicknesses of the second material, and the background value (y b ) in each pair of image data corresponds to a portion of the incident intensity which is measured after transmission through the corresponding thickness of the second material which was in combination with the first material when the foreground value (y F ) was acquired; determining fitting constants y 0 , α i  and β i  from the calibration standard background values for a functional form which approximates the measured background values (y b ) as a function of the thickness, wherein the functional form is: 
     
       
         y b =y 0 −Σ i α i   e   −β     i     t     M2     
       
     
     determining fitting constants y i , using the previously determined fitting constants y 0 , α i  and β i  from the calibration standard background values, for a functional form which approximates the measured foreground values (y f ) as a function of the thickness, wherein the functional form is: 
     
       
         y f =y 0 −Σ i α i   e   −β     i     t     M2     e   −γ     i     t     M1     
       
     
     where t M1  and t M2  are the thicknesses of the first material and the second material, respectively; and generating a Background (y b ) vs. Delta Gray (ΔG=y f −y b ) vs. First Material Thickness (t M1 ) surface using the fitted values for y 0 , α i γ i  and β i . The step of acquiring multiple pairs of image data may include the step of simulating the intensities of the transmissive energy which passes through the calibration standard using one or more of the following simulation factors: a) spectral characteristics of the source of transmissive energy; and/or b) angular distribution of the source of transmissive energy; and/or c) stopping power and spectral sensitivity of a transmissive energy detector; and/or d) transmissive energy attenuation properties of the absorbing material as a function of energy/wavelength of the source of transmissive energy. This method may further comprise the steps of: measuring foreground and background values for a combination of the first and second materials having unknown thicknesses; and locating on the Background (y b ) vs. Delta Gray (ΔG=y f −y b ) vs. First Material Thickness (t M1 ) surface, background and Delta Gray image data values corresponding to the measured background and foreground values to determine at least one of the corresponding first and/or second material thicknesses. This method may further comprise the step of generating a Background (y b ) vs. Delta Gray (ΔG=y f −y b ) vs. First Material Thickness (t M1 ) and/or Second Material Thickness (t M2 ) look up table using the fitted values for y 0 , α i γ i  and β i . Additionally, the method may also comprise the steps of: measuring foreground and background values for a combination of the first and second materials having unknown thicknesses; and locating on the Background (y b ) vs. Delta Gray (ΔG=y f −y b ) vs. First Material Thickness (t M1 ) look up table, Background and Delta Gray intensity values corresponding to the measured background and foreground values to determine at least one of the corresponding first and/or second material thicknesses. The method may also include the step of interpolating between values in the lookup table. 
     In a fourth aspect, the invention is a method for calibrating an X-ray imaging system for quantitatively determining the thickness of a first absorbing material in the presence of a second absorbing material where an incident X-ray beam having an incident X-ray beam intensity is transmitted through the first and second absorbing materials, the method comprising the steps of: providing a calibration standard for characterizing the imaging system wherein the calibration standard includes a first known thickness of the first absorbing material (denoted by t M1,1 ) in combination with two different thicknesses of the second absorbing material (denoted by t M2,1  and t M2,2 ); determining values of first and second foreground parameters (denoted by F 1  and F 2 ) wherein: a) the first foreground parameter F 1  is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the first absorbing material having the thickness t M1,1  in combination with the second absorbing material having the thickness t M2,1 ; and b) the second foreground parameter F 2  is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the first absorbing material having the thickness t M1,1  in combination with the second absorbing material having the thickness t M2,2 ; determining values of first and second background parameters (denoted by B 1  and B 2 ) wherein: a) the first background parameter B 1  is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through only the second absorbing material having the thickness t M2,1 ; and b) the second background parameter B 2  is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through only the second absorbing material having the thickness t M2,2 ; determining a first non-linear functional form, y 1 (x), which describes values of foreground (y 1 =F) as functions of the background (x=B) such that the first non-linear functional form: a) approximates the previously determined values of the first and second foreground parameters (F 1  and F 2 ) in terms of the previously determined values of the first and second background parameters (B 1  and B 2 ); b) incorporates one or more additional constraints determined by or approximating the physical behavior of the X-ray imaging system; and c) provides means to extrapolate a third foreground parameter (F 3 ) at a corresponding third background parameter (B 3 ) to a reference background value (x=B R ), thereby determining a reference foreground value (y 1 =F R ) at the reference background value (x=B R ); and determining a second non-linear functional form, y 2 (x), which describes reference foreground values (y 2 =F Ri ) as a function of corresponding first absorbing material thicknesses (x=t M1,i ) such that the second non-linear functional form: a) approximates a reference foreground value (y 2 =F R1 ) of the calibration standard first known thickness of the first absorbing material (t M1,1 ) at the reference background value (x=B R ); and b) incorporates one or more additional constraints determined by or approximating the physical behavior of the X-ray imaging system. The step of determining a first non-linear functional form, y 1 (x), may further comprise the step of selecting hyperbolic functions as one of the additional constraints having characteristics determined by or approximating the physical behavior of the X-ray imaging system. The step of determining a second non-linear functional form, y 2 (x), may further comprise the step of inverting, either numerically or analytically, the second non-linear functional form to obtain a first material thickness (t M1,K ) corresponding to a given reference foreground value (y 2 =F RK ). The step of determining a second non-linear functional form, y 2 (x), may further comprise the step of selecting a sum of exponential functions as one of the additional constraints having characteristics determined by or approximating the physical behavior of the X-ray imaging system. In this method, the steps of determining the values of the foreground and background parameters may further comprise the step of simulating the values of the foreground and background parameters using one or more of the following simulation factors: a) spectral characteristics of an X-ray source; and/or b) angular distribution of X-rays produced by the X-ray source; and/or c) stopping power and spectral sensitivity of an X-ray detector; and/or d) X-ray attenuation properties of the first and second absorbing materials as functions of X-ray energy/wavelength. 
     A fifth aspect of the present invention is a method for calibrating an X-ray imaging system for quantitatively determining the thickness of a first absorbing material in the presence of a second absorbing material where an incident X-ray beam having an incident X-ray beam intensity is transmitted through the first and second absorbing materials, the method comprising the steps of: providing a calibration standard for characterizing the imaging system wherein the calibration standard includes a first known thickness of the first absorbing material (denoted by t M1,1 ) in combination with two different thicknesses of the second absorbing material (denoted by t M2,1  and t M2,2 ); determining values of first and second foreground parameters (denoted by F 1  and F 2 ) wherein: a) the first foreground parameter F 1  is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the first absorbing material having the thickness t M1,1  in combination with the second absorbing material having the thickness t M2,1 ; and b) the second foreground parameter F 2  is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the first absorbing material having the thickness t M1,1  in combination with the second absorbing material having the thickness t M2,2 ; determining values of first and second background parameters (denoted by B 1  and B 2 ) wherein: a) the first background parameter B 1  is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through only the second absorbing material having the thickness t M2,1 ; and b) the second background parameter B 2  is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through only the second absorbing material having the thickness t M2,2 ; and determining a functional form of a non-linear function, y(x 1 ,x 2 ), which describes the value of the thickness of the first material (y=t M1 ) as a function of the foreground and background (e.g., x 1 =F, x 2 =B) such that the non-linear functional form: a) approximates a set of calibration data points {(t M1,i ,F i ,B i )} containing the previously determined first material thicknesses (t M1,i ), foreground parameters (F i ) and background parameters (B i ); b) incorporates one or more additional constraints determined by or approximating the physical behavior of the X-ray imaging system; and c) provides means to extrapolate beyond the range of the calibration standard foreground and background parameters. The step of determining a functional form of the non-linear function, y(x 1 ,x 2 ), may further comprise the step of selecting a sum of the product of two exponentials to represent the foreground parameters and a sum of single exponentials to represent the background parameters as the additional constraints having characteristics determined by or approximating the physical behavior of the X-ray imaging system. The step of determining a functional form of the non-linear function, y(x 1 ,x 2 ), may further comprise the step of inverting, either numerically or analytically, the non-linear functional form such that any one of y, x 1  or x 2  may be expressed as a function of the remaining two variables. In this method, the steps of determining the values of the foreground and background parameters may further comprise the step of simulating the values of the foreground and background parameters using one or more of the following simulation factors: a) spectral characteristics of an X-ray source; and/or b) angular distribution of X-rays produced by the X-ray source; and/or c) stopping power and spectral sensitivity of an X-ray detector; and/or d) X-ray attenuation properties of the first and second absorbing materials as functions of X-ray energy/wavelength. 
     In a sixth aspect, the present invention is a method for calibrating an X-ray imaging system for quantitatively determining the thickness of a first absorbing material in the presence of a second absorbing material where an incident X-ray beam having an incident X-ray beam intensity is transmitted through the first and second absorbing materials, the method comprising the steps of: providing a calibration standard for characterizing the imaging system wherein the calibration standard includes first and second known thicknesses of the first absorbing material (denoted by t M1,1  and t M1,2 ) in combination with a thickness of the second absorbing material (denoted by t M2,1  and t M2,2 ); determining values of first and second foreground parameters (denoted by F 1  and F 2 ) wherein: a) the first foreground parameter F 1  is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the first absorbing material having the thickness t M1,1  in combination with the second absorbing material having the thickness t M2,1 ; and b) the second foreground parameter F 2  is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the first absorbing material having the thickness t M1,2  in combination with the second absorbing material having the thickness t M2,2 ; determining values of first and second background parameters (denoted by B 1  and B 2 ) wherein: a) the first background parameter B 1  is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through only the second absorbing material having the thickness t M2,1 ; and b) the second background parameter B 2  is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through only the second absorbing material having the thickness t M2,2 ; determining a first non-linear functional form, y 1 (x), which describes values of foreground (y 1 =F) as functions of the background (x=B) such that the first non-linear functional form: a) approximates the previously determined values of the first and second foreground parameters (F 1  and F 2 ) in terms of the previously determined values of the first and second background parameters (B 1  and B 2 ); b) incorporates one or more additional constraints determined by or approximating the physical behavior of the X-ray imaging system; and c) provides means to extrapolate a third foreground parameter (F 3 ) at a corresponding third background parameter (B 3 ) to a reference background value (x=B R ), thereby determining a reference foreground value (y 1 =F R ) at the reference background value (x=B R ); and determining a second non-linear functional form, y 2 (x), which describes reference foreground values (y 2 =F Ri ) as a function of corresponding first absorbing material thicknesses (x=t M1,i ) such that the second non-linear functional form: a) approximates a first reference foreground value (y 2 =F R1 ) of the calibration standard first known thickness of the first absorbing material (t M1,1 ) at the reference background value (x=B R ) and a second reference foreground value (y 2 =F R2 ) of the calibration standard second known thickness of the first absorbing material (t M1,2 ) at the reference background value (x=B R ); and b) incorporates one or more additional constraints determined by or approximating the physical behavior of the X-ray imaging system. The step of providing a calibration standard may further include the step of selecting the second absorbing material such that the thickness t M2,1  equals the thickness t M2,2 . The step of determining a first non-linear functional form, y 1 (x), may further comprise the step of selecting hyperbolic functions as one of the additional constraints having characteristics determined by or approximating the physical behavior of the X-ray imaging system. In this method, the step of determining a second non-linear functional form, y 2 (x), may further comprise the step of inverting, either numerically or analytically, the second non-linear functional form to obtain a first material thickness (t M1,K ) corresponding to a given reference foreground value (y 2 =F RK ). The step of determining a second non-linear functional form, y 2 (x), may further comprise the step of selecting a sum of exponential functions as one of the additional constraints having characteristics determined by or approximating the physical behavior of the X-ray imaging system. Additionally, the steps of determining the values of the foreground and background parameters may further comprise the step of simulating the values of the foreground and background parameters using one or more of the following simulation factors: a) spectral characteristics of an X-ray source; and/or b) angular distribution of X-rays produced by the X-ray source; and/or c) stopping power and spectral sensitivity of an X-ray detector; and/or d) X-ray attenuation properties of the first and second absorbing materials as functions of X-ray energy/wavelength. 
     In a seventh aspect, the present invention is a method for calibrating an X-ray imaging system for quantitatively determining the thickness of a first absorbing material in the presence of a second absorbing material where an incident X-ray beam having an incident X-ray beam intensity is transmitted through the first and second absorbing materials, the method comprising the steps of: providing a calibration standard for characterizing the imaging system wherein the calibration standard includes first and second known thicknesses of the first absorbing material (denoted by t M1,1  and t M1,2 ) in combination with a thickness of the second absorbing material (denoted by t M2,1  and t M2,2 ); determining values of first and second foreground parameters (denoted by F 1  and F 2 ) wherein: a) the first foreground parameter F 1  is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the first absorbing material having the thickness t M1,1  in combination with the second absorbing material having the thickness t M2,1 ; and b) the second foreground parameter F 2  is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the first absorbing material having the thickness t M1,2  in combination with the second absorbing material having the thickness t M2,2 ; determining values of first and second background parameters (denoted by B 1  and B 2 ) wherein: a) the first background parameter B 1  is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through only the second absorbing material having the thickness t M   2,1 ; and b) the second background parameter B 2  is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through only the second absorbing material having the thickness t M2,2 ; and determining a functional form of a non-linear function, y(x 1 ,x 2 ), which describes the values of the thickness of the first material (y=t M1 ) as a function of the foreground and background (e.g., x 1 =F, x 2 =B) such that the non-linear functional form: a) approximates a set of calibration data points {(t M1,i ,F i ,B i )} containing the previously determined first material thicknesses (t M1,i ), foreground parameters (F i ) and background parameters (B i ); b) incorporates one or more additional constraints determined by or approximating the physical behavior of the X-ray imaging system; and c) provides means to extrapolate beyond the range of the calibration standard foreground and background parameters. The step of providing a calibration standard may further comprise the step of selecting the second absorbing material such that the thickness t M2,1  equals the thickness t M2,2 . In this method, the step of determining a functional form of the non-linear function, y(x 1 ,x 2 ), may further comprise the step of selecting a sum of the product of two exponentials to represent the foreground parameters and a sum of single exponentials to represent the background parameters as the additional constraints having characteristics determined by or approximating the physical behavior of the X-ray imaging system. The step of determining a functional form of the non-linear function, y(x 1 ,x 2 ), may further comprise the step of inverting, either numerically or analytically, the non-linear functional form such that any one of y, x 1  or x 2  may be expressed as a function of the remaining two variables. Additionally, the steps of determining the values of the foreground and background parameters may further comprise the step of simulating the values of the foreground and background parameters using one or more of the following simulation factors: a) spectral characteristics of an X-ray source; and/or b) angular distribution of X-rays produced by the X-ray source; and/or c) stopping power and spectral sensitivity of an X-ray detector; and/or d) X-ray attenuation properties of the first and second absorbing materials as functions of X-ray energy/wavelength. 
     An eighth aspect of the present invention is a method for calibrating an X-ray imaging system for quantitatively determining a first thickness, T x , of an absorbing material in the presence of an additional, second thickness, T y , of the absorbing material, where an incident X-ray beam having an incident X-ray beam intensity is transmitted through the absorbing material, the method comprising the steps of: providing a calibration standard for characterizing the imaging system wherein the calibration standard provides two known thicknesses T 1  and T 2  of the absorbing material; determining values F 1  and F 2  reflective of transmitted X-ray beam intensities corresponding to transmission through thicknesses T 1  and T 2  of the absorbing material, respectively; determining a functional form of an invertible, non-linear function y(x) which describes the variation of transmitted X-ray beam intensity as a function of thickness of the absorbing material; determining values B and F reflective of transmitted X-ray beam intensities corresponding to transmission through the second thickness, T y , of the absorbing material and through the combined thickness, T x +T y , of the absorbing material, respectively; applying the previously determined functional form to determine T y  and T x +T y  from the measured values of F and B; and determining the unknown first thickness, T x , as the difference (T x +T y )−T y . The step of determining a functional form which describes transmitted beam intensity as a function of thickness may further comprise selecting a general functional form described by: 
     
       
         y=y 0 −∫α(E) e   −β(E)T dE 
       
     
     or its discrete approximation: 
     
       
         y=y 0 −Σ i α i   e   −β     i     T   
       
     
     where T is the thickness of the absorbing material, y 0  is a fitting constant; and, in the general functional form: a) the X-ray source energy spectrum is distributed as a function of energy with weightings determined by the parameter α(E); and b) β(E) is the X-ray attenuation coefficient for the absorbing material, and in the discrete approximation: c) the total X-ray source energy spectrum is split up into some number of bands i, where the total source intensity is distributed among the bands as a functions of X-ray source energy and detector sensitivity with weightings for each band i determined by the parameter α i ; and d) β i  is the effective linear attenuation coefficient for X-rays in band i for the absorbing material. The step of determining the values F 1  and F 2  may comprise the step of simulating the transmitted intensities using one or more of the following simulation factors: a) spectral characteristics of the incident X-ray beam; and/or b) angular distribution of X-rays comprising the incident X-ray beam; and/or c) stopping power and spectral sensitivity of an X-ray detector; and/or d) X-ray attenuation properties of the absorbing material as a function of X-ray energy/wavelength. 
     A ninth aspect of the present invention is an apparatus for calibrating an X-ray imaging system for quantitatively determining the thickness of a first absorbing material in the presence of a second absorbing material where an incident X-ray beam having an incident X-ray beam intensity is transmitted through the first and second absorbing materials, the apparatus comprising: a calibration standard for characterizing the imaging system wherein the calibration standard includes at least one known thickness t M1,i  of the first absorbing material in combination with at least one thickness t M2,i  of the second absorbing material; means for determining a value of foreground and background parameters (denoted by F and B) wherein: a) the foreground parameter F is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the first absorbing material having thickness t M1,i  in combination with the second absorbing material having a thickness t M2,i ; and b) the background parameter B is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through only the second absorbing material having the thickness t M2,i ; and means for determining a non-linear functional form which describes values of the foreground and/or the background and/or the material thicknesses such that the non-linear functional form: a) is consistent with the previously determined foreground parameter (F), background parameter (B), and thickness values; b) incorporates one or more additional constraints determined by or approximating the physical behavior of the X-ray imaging system; and c) provides means to extrapolate the foreground and/or the background and/or the material thicknesses beyond the range of the calibration standard. The means for determining a non-linear functional form may further include: means for determining a first non-linear functional form, y 1 (x), which describes values of foreground (y 1 =F) as functions of the background (x=B) such that the first non-linear functional form: a) approximates the previously determined value of the foreground parameter (F) in terms of the previously determined value of the background parameter (B); b) incorporates one or more additional constraints determined by or approximating the physical behavior of the X-ray imaging system; and c) provides means to extrapolate a measured foreground parameter (F M ) corresponding to a first absorbing material having an unknown thickness t M1,U  in combination with a second absorbing material having a thickness t M2,U  to a reference background value (x=B R ), thereby determining a reference foreground value (y 1 =F R,U ) at the reference background value (x=B R ); and means for determining a second non-linear functional form, y 2 (x), which describes reference foreground values (y 2 =F Ri ) as a function of corresponding first absorbing material thicknesses (x=t M1,i ) such that the second non-linear functional form: a) approximates a reference foreground value (y 2 =F R1 ) of the calibration standard for the known thickness of the first absorbing material (t M1,1 ) at the reference background value (x=B R ); and b) incorporates one or more additional constraints determined by or approximating the physical behavior of the X-ray imaging system. The means for determining a non-linear functional form may further comprise: means for determining a functional form of a non-linear function, y(x 1 ,x 2 ), which describes the values of the thickness of the first material (y=t M1 ) as a function of the foreground and background (e.g., x 1 =F, x 2 =B) such that the non-linear functional form: a) approximates a set of calibration data points {(t M1,i ,F i ,B i )} containing the previously determined first material thicknesses (t M1,i ), foreground parameters (F i ) and background parameters (B i ); b) incorporates one or more additional constraints determined by or approximating the physical behavior of the X-ray imaging system; and c) provides means to extrapolate beyond the range of the calibration standard foreground and background parameters. 
     A tenth aspect of the present invention is a method for calibrating an X-ray imaging system for quantitatively determining the thickness of a first absorbing material in the presence of a second absorbing material where an incident X-ray beam having an incident X-ray beam intensity is transmitted through the first and second absorbing materials, the method comprising the steps of: providing a calibration standard for characterizing the imaging system wherein the calibration standard includes at least one known thickness t M1,i  of the first absorbing material in combination with at least one thickness t M2,i  of the second absorbing material; determining a value of foreground and background parameters (denoted by F and B) wherein: a) the foreground parameter F is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through the first absorbing material having thickness t M1,i  in combination with the second absorbing material having a thickness t M2,i ; and b) the background parameter B is representative of a transmitted X-ray beam intensity corresponding to a portion of the incident X-ray beam intensity which is transmitted through only the second absorbing material having the thickness t M2,i ; and determining a non-linear functional form which describes values of the foreground and/or the background and/or the material thicknesses such that the non-linear functional form: a) is consistent with the previously determined foreground parameter (F), background parameter (B), and thickness values; b) incorporates one or more additional constraints determined by or approximating the physical behavior of the X-ray imaging system; and c) provides means to extrapolate the foreground and/or the background and/or the material thicknesses beyond the range of the calibration standard. The step of determining a non-linear functional form may further include the steps of: determining a first non-linear functional form, y 1 (x), which describes values of foreground (y 1 =F) as functions of the background (x=B) such that the first non-linear functional form: a) approximates the previously determined value of the foreground parameter (F) in terms of the previously determined value of the background parameter (B); b) incorporates one or more additional constraints determined by or approximating the physical behavior of the X-ray imaging system; and c) provides means to extrapolate a measured foreground parameter (F M ) corresponding to a first absorbing material having an unknown thickness t M1,U  in combination with a second absorbing material having a thickness t M2,U  to a reference background value (x=B R ), thereby determining a reference foreground value (y 1 =F R,U ) at the reference background value (x=B R ); and determining a second non-linear functional form, y 2 (x), which describes reference foreground values (y 2 =F Ri ) as a function of corresponding first absorbing material thicknesses (x=t M1,i ) such that the second non-linear functional form: a) approximates a reference foreground value (y 2 =F R1 ) of the calibration standard for the known thickness of the first absorbing material (t M1,1 ) at the reference background value (x=B R ); and b) incorporates one or more additional constraints determined by or approximating the physical behavior of the X-ray imaging system. The steps of determining the values of the foreground and background parameters may further comprise the step of simulating the values of the foreground and background parameters using one or more of the following simulation factors: a) spectral characteristics of the X-ray beam; and/or b) angular distribution of X-rays comprising the X-ray beam; and/or c) stopping power and spectral sensitivity of an X-ray detector; and/or d) X-ray attenuation properties of the first and second absorbing materials as functions of X-ray energy/wavelength. In this method, the foreground parameters F i  may be described by a general functional form, y F:   
     
       
         y F =y 0 −∫α(E) e   −β(E)t     1     e   −γ(E)t     2   dE 
       
     
     or its discrete approximation: 
     
       
         y F =y 0 −Σ i α i   e   −β     i     t     1     e   −γ     i     t     2     
       
     
     where t 1  and t 2  are the thicknesses of the first absorbing material and the second absorbing material, respectively; y 0  is a fitting constant; and, in the general functional form: a) the X-ray beam energy spectrum is distributed as a function of energy with weightings determined by the parameter α(E); and b) β(E) and γ(E) are the X-ray attenuation coefficients for the first and second absorbing materials, respectively, and in the discrete approximation: c) the total X-ray beam energy spectrum is split up into some number of bands i, where the total source intensity is distributed among the bands as a functions of X-ray beam energy and detector sensitivity with weightings for each band i determined by the parameter α i ; and d) β i  and γ i1  are the effective linear attenuation coefficients for X-rays in band i for the first and second absorbing materials, respectively. The step of determining a non-linear functional form may further comprise the step of selecting a function of the form: 
     
       
         y 1 ={square root over ((x−a) 2 +L +b 2 +L )}+c 
       
     
     where x corresponds to the background B, y 1  corresponds to the difference between the foreground and background (F−B), and a, b and c are fitting constants. The method may further comprise the steps of: selecting a reference background level (x=B R ); determining the values of foreground minus background (F Ri −B Ri ) at the reference background level (B R ) for multiple known thicknesses of the calibration standard using the smoothly varying non-linear function y 1  which expresses the value of the foreground minus the background (y 1 =F−B) as a function of background (x=B); and determining a second functional form y 2  which expresses the values of foreground minus background (F Ri −B Ri ) at the reference background level (B R ) for the multiple known thicknesses of the first absorbing material as a function of the thickness of the first absorbing material. The step of determining a second functional form y 2  may further comprise the step of selecting a function which is a sum of exponentials of the form: 
     
       
         y 2 (t)=p−Σ i q i   e   −r     i     t   
       
     
     where p, q i  and r i  are fitting constants. This method may further comprise the step of producing a lookup table for values of (background) vs. (foreground minus background) vs. (thickness) for one or both of the first and/or second absorbing materials. The step of determining a non-linear functional form may further comprise the step of: determining a functional form of a non-linear function, y(x 1 ,x 2 ), which describes the values of the thickness of the first material (y=t M1 ) as a function of the foreground and background (e.g., x 1 =F, x 2 =B) such that the non-linear functional form: a) approximates a set of calibration data points {(t M1,i ,F i ,B i )} containing the previously determined first material thicknesses (t M1,i ), foreground parameters (F i ) and background parameters (B i ); b) incorporates one or more additional constraints determined by or approximating the physical behavior of the X-ray imaging system; and c) provides means to extrapolate beyond the range of the calibration standard foreground and background parameters. 
     These and other characteristics of the present invention will become apparent through reference to the following detailed description of the preferred embodiments and accompanying drawings. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1A is a graphical representation of the gray scale image intensity versus solder thickness for an X-ray image of solder material. 
     FIG. 1B shows a calibration step wedge of solder material used for calibrating the gray scale image intensity versus thickness relationship for X-ray images of the solder material. 
     FIG. 1C is a graphical representation of the gray scale image intensity versus thickness relationship for the solder material calibration step wedge shown in FIG.  1 B. 
     FIG. 2 is a schematic representation of a laminography system illustrating the principles of the technique. 
     FIG. 3A shows an object having an arrow, a circle and a cross embedded in the object at three different planar locations. 
     FIG. 3B shows a laminograph of the object in FIG. 3A focused on the plane containing the arrow. 
     FIG. 3C shows a laminograph of the object in FIG. 3A focused on the plane containing the circle. 
     FIG. 3D shows a laminograph of the object in FIG. 3A focused on the plane containing the cross. 
     FIG. 3E shows a conventional, two-dimensional X-ray projection image of the object in FIG.  3 A. 
     FIG. 4A is a diagrammatic cross-sectional view of a circuit board inspection laminography system showing how the laminographic image is formed and viewed by a camera. 
     FIG. 4B shows a top view enlargement of an inspection region shown in FIG.  4 A. 
     FIG. 4C is a perspective view of the circuit board inspection laminography system shown in FIG.  4 A. 
     FIG. 5 shows a schematic cross sectional representation of a portion of a two component assembly  300  comprising a first material  310  (e.g., solder) in combination with a second material  320  (e.g., copper, plastic, etc). 
     FIG. 6 shows a plan view representation of an X-ray image of the two component assembly  300  shown in FIG.  5 . 
     FIG. 7 illustrates linear plots of Delta Gray Level due to solder having a constant solder thickness (ΔG) vs. Background Gray Level (BG). 
     FIG. 8 shows two sets of calibration data and hyperbolic fits to the data illustrating conditions of the non-linear shading correction technique of the present invention. 
     FIG. 9 shows a plot of measured delta gray vs. background levels for 9 sets of calibration data for 9 known solder thicknesses in combination with 15 different known background levels. 
     FIG. 10A shows the results of solder thickness vs. background determined by applying a linear shading correction to the data illustrated in FIG.  9 . 
     FIG. 10B shows the results of solder thickness vs. background determined by applying a non-linear shading correction to the data illustrated in FIG.  9 . 
     FIG. 11A shows an example of a Background (BG) vs. Delta Gray (ΔG) vs. Solder Thickness (t) surface (generated from the calibration data illustrated in FIG. 9) in accordance with the present invention. 
     FIG. 11B shows a graphical representation of a Look Up Table (LUT) for Background (BG) vs. Delta Gray (ΔG) vs. Solder Thickness (t) (generated from the calibration data illustrated in FIG. 9) in accordance with the present invention. 
     FIG. 12 shows the results of solder thickness vs. background determined from a lookup table (generated from the calibration data illustrated in FIG. 9) in accordance with the present invention. 
     
       
         
               
             
               
               
             
           
               
                   
               
               
                 Reference Numerals in Drawings 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 4 
                 step wedge 
               
               
                 8 
                 step wedge steps 
               
               
                 8′ 
                 step wedge image intensities 
               
               
                 10 
                 object under inspection 
               
               
                 20 
                 source of X-rays 
               
               
                 30 
                 X-ray detector 
               
               
                 40 
                 common axis of rotation 
               
               
                 50 
                 central ray 
               
               
                 60 
                 image plane in object 10 
               
               
                 60a 
                 arrow image plane 
               
               
                 60b 
                 circle image plane 
               
               
                 60c 
                 cross image plane 
               
               
                 62 
                 plane of source of X-rays 
               
               
                 64 
                 plane of X-ray detector 
               
               
                 70 
                 point of intersection 
               
               
                 81 
                 arrow test pattern 
               
               
                 82 
                 circle test pattern 
               
               
                 83 
                 cross test pattern 
               
               
                 100 
                 image of arrow 81 
               
               
                 102 
                 blurred region 
               
               
                 110 
                 image of circle 82 
               
               
                 112 
                 blurred region 
               
               
                 120 
                 image of cross 83 
               
               
                 122 
                 blurred region 
               
               
                 130 
                 image of arrow 81 
               
               
                 132 
                 image of circle 82 
               
               
                 134 
                 image of cross 83 
               
               
                 200 
                 X-ray tube 
               
               
                 210 
                 printed circuit board 
               
               
                 212 
                 electronic components 
               
               
                 214 
                 electrical connections 
               
               
                 220 
                 support fixture 
               
               
                 230 
                 positioning table 
               
               
                 240 
                 rotating X-ray detector 
               
               
                 250 
                 fluorescent screen 
               
               
                 252 
                 first mirror 
               
               
                 254 
                 second mirror 
               
               
                 256 
                 turntable 
               
               
                 258 
                 camera 
               
               
                 260 
                 feedback system 
               
               
                 262 
                 input connection 
               
               
                 263 
                 sensor 
               
               
                 264 
                 output connection 
               
               
                 265 
                 position encoder 
               
               
                 270 
                 computer 
               
               
                 276 
                 input line 
               
               
                 278 
                 output line 
               
               
                 280 
                 rotating source spot 
               
               
                 281 
                 deflection coils 
               
               
                 282 
                 X-rays 
               
               
                 283 
                 region of circuit board 
               
               
                 284 
                 X-rays 
               
               
                 285 
                 rotating electron beam 
               
               
                 286 
                 light 
               
               
                 287 
                 target anode 
               
               
                 290 
                 granite support table 
               
               
                 292 
                 load/unload port 
               
               
                 294 
                 operator station 
               
               
                 300 
                 two component assembly 
               
               
                 310 
                 first material 
               
               
                 320 
                 second material 
               
               
                 330 
                 incident X-rays 
               
               
                 350 
                 X-ray image 
               
               
                 360 
                 foreground image region 
               
               
                 370 
                 background image regions 
               
               
                 410 
                 (BG,ΔG) calibration points 
               
               
                 420 
                 linear fit to t 1  calibration data 
               
               
                 430 
                 linear fit to t 2  calibration data 
               
               
                 440 
                 linear fit to t 3  calibration data 
               
               
                 450 
                 unknown thickness line 
               
               
                 510 
                 hyperbolic calibration curve 
               
               
                 512 
                 calibration data points 
               
               
                 520 
                 hyperbolic calibration curve 
               
               
                 514 
                 calibration data points 
               
               
                 530 
                 hyperbolic data curve 
               
               
                   
               
             
          
         
       
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     R ELATIONSHIP  B ETWEEN  S OLDER  T HICKNESS AND  X-R 
     AY I MAGE  G RAY  L EVEL    
     While the following description is presented in terms of a two component assembly comprising a layer of solder and a layer of copper, it is to be understood that the present invention also applies to any two component assembly. It is to be further understood that the present invention applies equally to a three component assembly where one of the three components is unchanging (e.g., the G10 substrate of a printed circuit assembly). Since the effect of the unchanging third component is simply to alter the source intensity spectrum, it is not explicitly treated in the following description. Furthermore, the two components need not be in distinct layers but may be intermixed. One skilled in the art will recognize that the terms “gray level” and “intensity”, as used throughout, are closely related and are often interchangeable. In general, “gray level” refers to an X-ray detector measured X-ray intensity which is converted to an arbitrary scale of gray levels. Thus, a specific gray level is functionally related to a corresponding X-ray intensity. Similarly, one skilled in the art will recognize that the terms “attenuation” and “absorption” with reference to X-rays, are closely related and are often used interchangeably in the literature. Generally, “attenuation” usually includes both “absorption” and “scattering” of X-rays, and is the parameter of interest herein, without regard to whether it is caused by absorption or scattering. However, since the terms are frequently interchanged in the art, “absorption” may sometimes also be used to include both “absorption” and “scattering” of X-rays. If a distinction is significant, one skilled in the art will generally be able to determine the correct intention by reference to the context in which the terms are used. 
     In an X-ray image of solder material, typically a combination of lead and tin, there is a relationship between the intensities comprising the X-ray image and the thicknesses of the solder material forming the X-ray image. FIG. 1A illustrates an example of this general relationship. In this example, it is seen that the image intensity increases from values corresponding to lighter shades of gray (white) to values corresponding to darker shades of gray (black) as the thickness of the solder material increases. That is, the image of a thin section of solder will have a gray level that is less than the gray level of the image of a thicker section of solder. The image of the thin section will appear to be a lighter shade of gray than the image of the thicker section. (This convention is typically used in electronic image representation of X-ray images, however, the opposite convention may also be used, i.e., where the image of a thin section of solder has a gray level that is greater than the gray level of the image of a thicker section of solder. The latter convention has traditionally been followed in film radiography where the X-ray images are recorded on X-ray film. The present invention may be implemented using either convention.) Additionally, in the following description, the gray scale ranges from zero to a maximum value where the lower values correspond to the lighter shades of gray (white) and the values near the maximum value correspond to darker shades of gray (black). It is to be understood that other conventions for representing the gray scale may also be used. For example, lower values may be selected to correspond to the darker shades of gray (black) and the values near the maximum value may be selected to correspond to lighter shades of gray (white). 
     S OLDER  T HICKNESS  D ETERMINATION  U SING  C ALIBRATION  S TEP  W EDGE    
     The relationship between solder thickness and image gray level may be calibrated using a calibration step wedge comprising multiple steps of differing thickness. An example of such a step wedge  400  is shown in FIG.  1 B. Step wedge  400  is constructed of solder material and comprises ten steps  8  having thicknesses ranging from 0.001 inch to 0.010 inch in increments of 0.001 inch. It is possible to construct the step wedge  4  with other dimensions (e.g., in 2 mil increments from 2 mils to 20 mils, etc.), depending upon the thicknesses of the solder joints and the type of circuit board that is to be inspected. An X-ray image of the step wedge  4  exhibits an image intensity  8 ′ versus solder thickness relationship as shown in FIG.  1 C. Since the thicknesses of the steps  8  are known, the corresponding intensities  8 ′ may be compared to intensities of other X-ray images of solder material where the thicknesses are not known to determine the unknown thicknesses. Alternative methods of calibrating the solder thickness of a step wedge to correspond to various image intensities may yield more accurate results than this technique. 
     In the case of circuit board assemblies, the solder is attached to a circuit board. Thus, gray scales displayed in the X-ray images include contributions from the solder as well as the material comprising the circuit board. Typically the circuit board substrate is a plastic or resin type material and may further include ground planes and circuit traces made of a conducting material, e.g., copper. In these cases, determination of the solder thickness is complicated by the presence of the circuit board and associated materials which contribute to a background in the X-ray images. Background shading correction techniques for removing the contribution due to a background are described below. 
     An alternative calibration standard for solder thickness calibration measurements comprises multiple isolated dots or circular regions of solder of differing known thicknesses attached to an epoxy/plastic substrate typical of a circuit board, e.g., a G-10 material. Typically, the gray level of the portion of the X-ray image of the calibration standard corresponding to the central region of each dot/circular region of solder is selected as being representative of the gray level of the entire dot/circular region to eliminate possible errors due to edge effects, etc. 
     X-R AY  I MAGE  F ORMATION    
     FIG. 2 shows a schematic representation of a typical laminographic geometry which may be used with the present invention. An object  10  under examination, for example, a circuit board, is held in a stationary position with respect to a source of X-rays  20  and an X-ray detector  30 . Synchronous rotation of the X-ray source  20  and detector  30  about a common axis  40  causes an X-ray image of the plane  60  within the object  10  to be formed on the detector  30 . The image plane  60  is substantially parallel to the planes  62  and  64  defined by the rotation of the source  20  and detector  30 , respectively. The image plane  60  is located at the intersection  70  of a central ray  50  from the X-ray source  20  and the common axis of rotation  40 . This point of intersection  70  acts as a fulcrum for the central ray  50 , thus causing an in-focus cross-sectional X-ray image of the object  10  at the plane  60  to be formed on detector  30  as the source and detector synchronously rotate about the intersection point  70 . Structure within the object  10  which lies outside of plane  60  forms a blurred X-ray image on detector  30 . 
     In the laminographic geometry shown in FIG. 2, the axis of rotation of the radiation source  20  and the axis of rotation of the detector  30  are coaxial. However, it is not necessary that these axes of rotation of the radiation source  20  and the detector  30  be coaxial. The conditions of laminography are satisfied and a cross-sectional image of the layer  60  will be produced as long as the planes of rotation  62  and  64  are mutually parallel, and the axes of rotation of the source and the detector are mutually parallel and fixed in relationship to each other. Coaxial alignment reduces the number of constraints upon the mechanical alignment of the apparatus. It is to be understood that the present invention is not limited to any specific laminographic configuration. One skilled in the art will recognize that there are numerous alternative configurations for generating laminographic images which may also be used. Furthermore, the present invention is not limited to cross-sectional images of a two component assembly, but may be practiced with any type of X-ray image of the assembly, including but not limited to laminographic images, CT images, shadow graph images, etc. 
     FIGS. 3A-3E show laminographs produced by the above described laminographic technique. The object  10  shown in FIG. 3A has test patterns in the shape of an arrow  81 , a circle  82  and cross  83  embedded within the object  10  in three different planes  60   a ,  60   b  and  60   c , respectively. 
     FIG. 3B shows a typical laminograph of object  10  formed on detector  30  when the point of intersection  70  lies in plane  60   a  of FIG.  3 A. The image  100  of arrow  81  is in sharp focus, while the images of other features within the object  10 , such as the circle  82  and cross  83  form a blurred region  102  which does not greatly obscure the arrow image  100 . 
     Similarly, when the point of intersection  70  lies in plane  60   b , the image  110  of the circle  82  is in sharp focus as seen in FIG.  3 C. The arrow  81  and cross  83  form a blurred region  112 . 
     FIG. 3D shows a sharp image  120  formed of the cross  83  when the point of intersection  70  lies in plane  60   c . The arrow  81  and circle  82  form blurred region  122 . 
     For comparison, FIG. 3E shows an X-ray shadow image of object  10  formed by conventional projection radiography techniques. This technique produces sharp images  130 ,  132  and  134  of the arrow  81 , circle  82  and cross  83 , respectively, which overlap one another. FIG. 3E vividly illustrates how multiple characteristics contained within the object  10  may create multiple overshadowing features in the X-ray image which obscure individual features of the image. 
     FIG. 4A illustrates a schematic diagram of a typical laminographic apparatus usable with the present invention. In this configuration, an object under inspection is a printed circuit board  210  having multiple electronic components  212  mounted on the board  210  and electrically interconnected via electrical connections  214  (See FIG.  4 B). Typically, the electrical connections  214  are formed of solder. However, various other techniques for making the electrical connections  214  are well known in the art and even though the invention will be described in terms of solder joints, it will be understood that other types of electrical connections  214  including, but not limited to, conductive epoxy, mechanical, tungsten and eutectic bonds may be inspected utilizing the invention. FIG. 4B, which is a top view enlargement of a region  283  of the circuit board  210 , more clearly shows the components  212  and solder joints  214 . 
     The laminographic apparatus acquires cross-sectional images of the solder joints  214  using the previously described laminographic method or other methods capable of producing equivalent cross-sectional images. The cross-sectional images of the solder joints  214  are automatically evaluated to determine their quality and physical characteristics, including, e.g., solder thickness. Based on the evaluation, a report of the solder joint quality and physical characteristics is presented to the user. 
     The laminographic apparatus, as shown in FIG. 4A, comprises an X-ray tube  200  which is positioned adjacent printed circuit board  210 . The circuit board  210  is supported by a fixture  220 . The fixture  220  is attached to a positioning table  230  which is capable of moving the fixture  220  and board  210  along three mutually perpendicular axes, X, Y and Z. A rotating X-ray detector  240  comprising a fluorescent screen  250 , a first mirror  252 , a second mirror  254  and a turntable  256  is positioned adjacent the circuit board  210  on the side opposite the X-ray tube  200 . A camera  258  is positioned opposite mirror  252  for viewing images reflected into the mirrors  252 ,  254  from fluorescent screen  250 . A feedback system  260  has an input connection  262  from a sensor  263  which detects the angular position of the turntable  256  and an output connection  264  to X and Y deflection coils  281  on X-ray tube  200 . A position encoder  265  is attached to turntable  256 . The position sensor  263  is mounted adjacent encoder  265  in a fixed position relative to the axis of rotation  40 . The camera  258  is connected to a computer  270  via an input line  276 . The computer  270  includes the capability to perform high speed image analysis. An output line  278  from the computer  270  connects the computer to positioning table  230 . 
     A perspective view of the laminographic apparatus is shown in FIG.  4 C. In addition to the X-ray tube  200 , circuit board  210 , fluorescent screen  250 , turntable  256 , camera  258 , positioning table  230  and computer  270  shown in FIG. 4A, a granite support table  290 , a load/unload port  292  and an operator station  294  are shown. The granite table  290  provides a rigid, vibration free platform for structurally integrating the major functional elements of the laminographic apparatus, including but not limited to the X-ray tube  200 , positioning table  230  and turntable  256 . The load/unload port  292  provides a means for inserting and removing circuit boards  210  from the machine. The operator station  294  provides an input/output capability for controlling the functions of the laminographic apparatus as well as for communication of inspection data to an operator. 
     In operation of the laminographic apparatus as shown in FIGS. 4A and 4C, high resolution, cross-sectional X-ray images of the solder joints  214  connecting components  212  on circuit board  210  are acquired using the X-ray laminographic method previously described in reference to FIGS. 2 and 3. Specifically, X-ray tube  200 , as shown in FIG. 4A, comprises a rotating electron beam spot  285  which produces a rotating source  280  of X-rays  282 . The X-ray beam  282  illuminates a region  283  of circuit board  210  including the solder joints  214  located within region  283 . X-rays  284  which penetrate the solder joints  214 , components  212  and board  210  are intercepted by the rotating fluorescent screen  250 . 
     Dynamic alignment of the position of the X-ray source  280  with the position of rotating X-ray detector  240  is precisely controlled by feedback system  260 . The feedback system correlates the position of the rotating turntable  256  with calibrated X and Y deflection values stored in a look-up table (LUT). Drive signals proportional to the calibrated X and Y deflection values are transmitted to the steering coils  281  on the X-ray tube  200 . In response to these drive signals, steering coils  281  deflect electron beam  285  to locations on a target anode  287  such that the position of the X-ray source spot  280  rotates in synchronization with the rotation of detector  240  in the manner previously discussed in connection with FIG.  2 . 
     X-rays  284  which penetrate the board  210  and strike fluorescent screen  250  are converted to visible light  286 , thus creating a visible image of a single plane within the region  283  of the circuit board  210 . The visible light  286  is reflected by mirrors  252  and  254  into camera  258 . Camera  258  typically comprises a low light level closed circuit TV (CCTV) camera which transmits electronic video signals corresponding to the X-ray and visible images to the computer  270  via line  276 . The image analysis feature of computer  270  analyzes and interprets the image to determine the quality of the solder joints  214 . 
     Computer  270  includes one or more processors, one or more memories and various input and output devices including but not limited to monitors, disk drives, printers and keyboards. It is to be understood that the image analysis methods of the present invention may be implemented in a variety of ways by one skilled the art, however, implementation with the computer or specially dedicated image processor is preferred. Additionally, it is to be understood that the term “image” is not limited to formats which may be viewed visually, but may also include digital or analog representations which may be acquired, stored and analyzed by the computer. 
     Computer  270  also controls the movement of positioning table  230  and thus circuit board  210  so that different regions of circuit board  210  may be automatically positioned within inspection region  283 . 
     The laminographic geometry and apparatus shown and described with reference to FIGS. 2-4 are typical of that which may be used in conjunction with the present invention. However, specific details of these systems are not critical to the practice of the present invention, which addresses the accurate measurement of the thickness of a solder joint positioned on a circuit board  210 . For example, the number of computers and delegation of tasks to specific computers may vary considerably from system to system as may the specific details of the X-ray source, detector, circuit board positioning mechanism, etc. More detailed descriptions of laminography systems may be found in the following U.S. Pat. Nos.: 4,926,452; 5,097,492; 5,081,656; 5,291,535; 5,621,811; 5,561,696; 5,199,054; 5,259,012; 5,583,904; and 5,687,209, previously incorporated herein by reference. 
     One skilled in the art will also recognize that other techniques, for example computed tomography, may be used to produce cross sectional images of specific planes within a solder joint. It is also to be understood that the present invention may be practiced using conventional X-ray shadowgraph images (See FIG. 3E) of solder joints on circuit boards or other multiple component assemblies. Furthermore, specific details of various techniques and equipment for creating the cross-sectional or shadowgraph X-ray images of the multiple component assemblies being inspected may be utilized. The present invention is applicable to any type of system which derives the thickness or relative quantity of a first material in the presence of a second material from an analysis of the gray levels comprising an X-ray image of the assembly. 
     P HYSICS OF  X-R AY  A TTENUATION    
     FIG. 5 shows a schematic cross sectional representation of a portion of a two component assembly  300  comprising a first material  310  (e.g., solder) in combination with a second material  320  (e.g., copper, plastic, etc). U.S. Pat. No. 5,291,535 discusses various techniques for calibrating such configurations including 1) a background subtraction (additive component); and 2) a combination background subtraction followed by a multiplicative component. While these approaches may be adequate for certain applications, other applications require more accurate techniques for deriving a solder thickness measurement from an X-ray image in the presence of a background material. The linear shading correction method described below has been found to provide significant improvement over the additive and additive/multiplicative corrections described in U.S. Pat. No. 5,291,535. 
     As shown in FIG. 5, X-rays  330  having a incident intensity I 0 , are directed upon the assembly  300  from a first side and encounter regions of the assembly  300  which include the first material  310  having a thickness t 1  in combination with the second material  320  having a thickness t 2 , and other regions of the assembly  300  which include only the second material  320 . In regions where the X-rays have passed through only the second material  320 , the incident intensity I 0  is attenuated to an intensity I 1 . Similarly, in regions where the X-rays have passed through both the first material  310  and the second material  320 , the incident intensity I 0  is attenuated to an intensity I 2 . The absorption of monochromatic X-rays in the region including only the second material  320  is governed by the following relation: 
     
       
         I 1 =I 0   e   −α     2     t     2     (1) 
       
     
     where α 2  is is the X-ray attenuation coefficient for the second material  320 . The absorption of monochromatic X-rays in the region including both the first material  310  and the second material  320  is governed by the following relation: 
     
       
         I 2 =I 0   e   −α     1     t     1     e   −α     2     t     2     (2) 
       
     
     where α 1  is the X-ray attenuation coefficient for the first material  310 . 
     FIG. 5 illustrate s the X-rays  330  passing through the assembly  300  in a direction which is perpendicular to the first and second layers  310  and  320 , thus, t 1  and t 2  represent the thicknesses of the first and second layers  310  and  320 , respectively. In the event the X-rays pass through the assembly at some other angle, t 1  and t 2  represent the distances the X-rays have travelled through the first and second layers  310  and  320 , respectively. 
     S OLDER  T HICKNESS  D ETERMINATION  U SING  L INEAR  S HADING  C ORRECTIONS    
     As previously discussed, X-ray inspection of printed circuit assemblies typically produces gray-scale images of interconnects or slices thereof which are analyzed and examined to detect and classify improper joints and/or to provide statistical process control data relating to the manufacturing process. It is desirable that measurements taken relate directly to physical characteristics of the joint under inspection. For example, in characterizing solder joints, it is preferable to deal with measured joint thickness, i.e., solder thickness, rather than gray scale pixel values. The following described linear shading correction technique has previously been used for converting the solder image gray scale pixel values to solder thicknesses. Since the present invention is an enhancement and extension of this linear shading correction method, a summary description is presented to facilitate the understanding of the present invention. 
     Shown in FIG. 6 is a plan view representation of an X-ray image  350  of the two component assembly  300  shown in FIG. 5 where the first material  310  is solder and the second material  320  is copper or a combination of copper and circuit board materials. A foreground image region  360  is representative of a portion of a typical X-ray image of a solder pad, i.e., the first material  310  (e.g., solder) in combination with the second material  320  (e.g., copper). Similarly, background regions  370  are representative of portions of a typical X-ray image of a circuit board substrate, i.e., the second material  320  (e.g., copper, plastic, etc). A gray scale level which is representative of the gray level due to the solder, ΔG i , is obtained by subtracting a background (copper) gray level B i , i.e., the gray level of the X-ray image in regions  370 , from a foreground (solder+copper) gray level F i , i.e., the gray level of the X-ray image in region  360 , as follows: 
     
       
         ΔG i =F i −B i   (3) 
       
     
     The linear shading correction technique is based on the following two assumptions: 
     1) Plots of Delta Gray level due to solder at constant solder thickness (ΔG) vs. Background Gray Level (BG) may be approximated by a series of straight lines intersecting the Background axis at a single point; and 
     2) At a “nominal” or reference background gray level (e.g., zero) the Delta Gray level due to solder (ΔG) vs. Solder Thickness (t), function may be approximated by a fitted curve of known form, e.g., a sum of exponentials. In the following examples, a reference background gray level of zero was selected for convenience. However, it is to understood that other non-zero reference background gray levels may be selected. 
     In accordance with assumption 1) of the linear shading correction technique, three values of ΔG i  for a constant value of solder thickness in the presence of three different thicknesses of copper are obtained. Shown in FIG. 7 is a plot of an example where a solder thickness of 4 mils is shielded by copper having thicknesses of 5, 10 and 15 mils. Calibration data points  410   a ,  410   b  and  410   c  correspond to (BG,ΔG) coordinates (5,ΔG 1 ), (10,ΔG 2 ) and (15,ΔG 3 ), respectively. These values of solder and copper thicknesses are selected for purposes of illustration only. Different thickness values and materials may be used for particular applications. Fitting a straight line  420  to the points  410   a ,  410   b  and  410   c  determines a BG-axis intercept of BG MAX  and a ΔG-axis intercept of ΔG 0 (Cal). The linear function describing straight line  420  is:                Δ                 G     =         -       Δ                     G   0          (   CAL   )           BG   MAX            BG     +     Δ                     G   0          (   CAL   )                   (   4   )                                
     where the constants BG MAX  and ΔG 0 (CAL) are determined from the linear fit to calibration points  410   a ,  410   b  and  410   c.    
     As stated previously, the linear shading correction technique assumes that plots of Delta Gray level due to solder at different constant solder thicknesses will form a series of straight lines intersecting the Background-axis at the same point, BG MAX . In accordance with this assumption, straight lines  430  and  440  represent plots of ΔG vs. BG for solder thicknesses of 2 mils and 10 mils, respectively (individual data points not shown for lines  430  and  440 ). 
     Thus, for any set of measured coordinates, (BG,ΔG), corresponding to an unknown solder thickness, t U , the solder delta gray level at a “nominal” or reference background gray level (e.g., zero) ΔG 0 (UNKNOWN), is the ΔG-axis intercept of a straight line  450  determined by the measured coordinates (BG,ΔG) and the BG-axis intercept, (BG MAX ,0). The linear function describing straight line  450  is:                Δ                 G     =         -       Δ                     G   0          (   UNKNOWN   )           BG   MAX            BG     +     Δ                     G   0          (   UNKNOWN   )                   (   5   )                                
     Using the measured data (BG,ΔG), corresponding to the unknown solder thickness t U , the unknown ΔG-axis intercept, ΔG 0 (UNKNOWN), may be determined by rearrangement of equation (5) as follows:                Δ                     G   0          (   UNKNOWN   )         =       Δ                 G       1   -     BG     BG   MAX                   (   6   )                                
     Applying assumption 2) of the linear shading correction technique, the unknown solder thickness t U  may then be determined by using the solder delta gray level at a “nominal” or reference background gray level (e.g., zero) for the unknown solder thickness, ΔG 0 (UNKNOWN), in the following functional relationship: 
     
       
         ΔG 0 (UNKNOWN)=A(1− e   −k     1     t     U   )+B(1 −e   −k     2     t     U   )  (7) 
       
     
     where fitting constants A, B, k 1  and k 2  have been previously determined using calibration data. 
     In summary, the measured data point (BG,ΔG) corresponding to the unknown solder thickness t U , is used in equation (6) to calculate the unknown ΔG-axis intercept, ΔG 0 (UNKNOWN), for the unknown thickness t U . Equation (7) is then used to calculate the value of the unknown thickness t U . Alternatively, equation (7) may be used to generate a look up table (LUT) of solder delta gray levels (at a “nominal” or reference background gray level of zero) vs. thickness, to speed up the computation. Since the LUT created from equation (7) comprises multiple pairings of gray levels for solder of various thicknesses corrected to zero background, ΔG 0  and corresponding thicknesses, t, it is a simple matter to find the thickness corresponding to the ΔG 0 (UNKNOWN) for any measured data point. That is, once the solder delta gray level at the “nominal” or reference background gray level (zero in this example) ΔG 0 (UNKNOWN) for a measured unknown point (BG,ΔG) is determined using equation (6), the thickness of solder represented by the value of ΔG 0 (UNKNOWN) is found in the LUT, where it is paired with the corresponding solder thickness, t. Interpolation between values in the LUT may be used if the value of ΔG 0 (UNKNOWN) does not exactly match an entry in the LUT. 
     It has been found that the linear shading correction method described above may only be accurate over a limited range of thicknesses. The limited accuracy is due to the fact that the actual plots of Delta Gray level due to solder at constant solder thickness (ΔG) vs. Background Gray Level (BG) curves are only approximately linear. Additionally, the BG-axis intercept, BG MAX , may change when the X-ray camera settings are changed, (e.g., camera gain, field of view, etc.) thereby requiring a new calibration. 
     S OLDER  T HICKNESS  D ETERMINATION  U SING A  N ON -L INEAR  S HADING  C ORRECTION    
     The present invention uses a non-linear shading correction procedure to improve the accuracy and repeatability of the linear shading correction method described previously. In order to simplify the following discussion of the non-linear shading correction procedure, the special case of solder shaded by copper will be considered. However, it is to be understood that the invention is not limited to this combination of materials and also applies to assemblies having more than two components. 
     In the present invention, the gray levels of X-ray images of a number of test coupons which contain known thicknesses of solder shaded by varying amounts of copper are measured. By a combination of theoretical and empirical arguments, it has been found that the effect of the shading may be described by a particular nonlinear equation with three free parameters. Moreover, two of the three parameters are found to be characteristics of the AXI system and not functions of the amount of copper or solder in the X-ray beam path. One aspect of the system calibration involves estimation and storage of these two parameters. Foreground and background gray level values from an unknown sample are adequate to fix the third parameter, completely characterizing the shading effect for that sample. As a result, it is possible to use the two stored system parameters and the known functional form of the shading equation to extrapolate to values that would have been measured under “standard” shading conditions. (Typically, “no shading”, i.e., zero background, is used as the standard condition). Since any measured sample can be readily converted to standard conditions using this approach, there is no need for a two dimensional thickness calibration. Instead, a simple one dimensional curve suffices, since measurements can always be corrected to zero background. 
     The non-linear shading correction technique of the present invention is based on the following two assumptions: 
     1) Plots of Delta Gray Level (y=ΔG=F−B) due to solder at constant solder thickness (y-axis) vs. Background Gray Level (x=B, x-axis) may be approximated by points located on a left branch of a series of hyperbolic curves having two common parameters as follows: 
     A) a common x-axis (i.e., BG-axis) value at which each hyperbolic curve assumes its minimum value of y (i.e., ΔG); and 
     B) a common x-axis intercept at a maximum background gray level x=BG MAX ; and 
     2) At a “nominal” or reference background gray level (e.g., zero), the Delta Gray level due to solder (y 0 ) vs. Solder Thickness (t), function may be approximated by a fitted curve of known form, e.g., a sum of exponentials. In the following examples, a reference background gray level of zero was selected for convenience. However, it is to understood that other non-zero reference background gray levels may be selected. 
     In accordance with assumption 1) of the non-linear shading correction technique, each of multiple sets of calibration data are jointly fit to hyperbolic functions of the following form: 
     
       
         y=ΔG={square root over ((x−a) 2 +L +b 2 +L )}+c  (8) 
       
     
     where a is the x-axis coordinate at which y has a minimum value. 
     By way of example, shown in FIG. 8 are a first calibration curve  510  and a second calibration curve  520 . The first calibration curve  510  includes multiple calibration data points  512  and the second calibration curve  520  includes multiple calibration data points  514 . On the first calibration curve  510 , each calibration data point  512  represents a delta gray level for a solder thickness of 7.7 mils in combination with an unknown background material (e.g., unknown thicknesses of circuit board materials including copper) and differing thicknesses of copper background. For example, calibration data point  512   a  is the delta gray level corresponding to a solder thickness of 7.7 mils of solder in combination with a background copper thickness of 0.0 mils, while calibration data point  512   f  is the delta gray level corresponding to the 7.7 mils of solder in combination with a copper thickness of 25 mils, etc. Similarly, on the second calibration curve  520 , each calibration data point  514  represents a delta gray level for a solder thickness of 1.2 mils in combination with differing background thicknesses of copper. For example, calibration data point  514   a  is the delta gray level corresponding to a solder thickness of 1.2 mils in combination with a background copper thickness of 0.0 mils, while calibration data point  514   f  is the delta gray level corresponding to the 1.2 mils of solder in combination with a background copper thickness of 25 mils, etc. Analysis of the first and second calibration curves  510  and  520  by any of a variety of empirical and/or analytical techniques may be used to arrive at a best fit for each hyperbolic calibration curve included in the family of calibration data, with the constraints that each hyperbolic curve has its minimum y-value at the same value of x, i.e., x=a, and each curve has the same x-axis intercept. For the data points shown in FIG. 8, it is seen that curves  510  and  520  which fit the data points  512  and  514 , respectively, are obtained by using a value of x=a=455 and an x-axis intercept at x=BG MAX =222. As shown in FIG. 8, the first calibration curve  510  is a fit to data points  512  of a hyperbolic function of the form shown in equation (8) where the hyperbolic function represented by calibration curve  510  has a minimum y-value at x=a and intercepts the x-axis at x=BG MAX . In accordance with assumption (1), the second calibration curve  520  is a fit to data points  514  of a hyperbolic function of the form shown in equation (8) where the hyperbolic function represented by calibration curve  520  also has a minimum y-value at x=a and also intercepts the x-axis at x=x 0 =BG MAX . The y-axis intercepts of the fitted calibration curves  510  and  520  are determined by extrapolation of the fitted curves. As shown in this example, the extrapolated y-axis intercept of the first calibration curve  510  is located at approximately y 0 =128 and the extrapolated y-axis intercept of the second calibration curve  520  is located at approximately y 0 =30. As previously discussed, the y-axis intercepts of the curves  510  and  520  correspond to “nominal” or reference background gray levels of zero for the respective solder thicknesses represented by the curves  510  and  520 . It is noted that the thicknesses of the background copper used to shade the known solder calibration thicknesses need not be known to determine unknown solder thicknesses. However, if it is desired to determine both unknown solder and unknown copper thicknesses, both the solder and copper thicknesses used in the calibration should be known. 
     After fitting the calibration data  512 , 514  and obtaining from these fits the values for x=a and x=x 0 =BG MAX  (a detailed description of two procedures for determining the values for x=a and x=x 0 =BG MAX  is presented below), which define curves  510  and  520 , the value of a delta gray level at a “nominal” background value of zero for an unknown data point (x,y) is obtained as follows. Recall that each calibration curve  510  and  520  is represented by an equation of the form of equation (8) where x=a is the x-axis coordinate at which y (for each calibration curve) has a minimum value. Additionally, all of the calibration curves share a common x-axis intercept at x=x 0 =BG MAX . Thus, at x=x 0 =BG MAX , equation (8) becomes 
     
       
         (BG MAX −a) 2 +b 2 =c 2   (9) 
       
     
     or 
     
       
         b 2 =c 2 −(BG MAX −a) 2 =(y−c) 2 −(x−a) 2   (10) 
       
     
     Expanding and collecting terms in equation (10) yields the following expression for “c” in terms of “x”, “y”, “a” and “BG MAX ”:              c   =         y   2     -       (     x   -     BG   MAX       )          (     x   +     BG   MAX       )       +     2        a        (     x   -     BG   MAX       )             2      y               (   11   )                                
     Thus, for any given unknown data point (x,y), values for “c” and “b” may be calculated from equations (10) and (11). Using the known values of “a”, “b” and “c” in equation (8) at x=0 yields the value of y 0 , i.e., the delta gray level at a “nominal” background value of zero for the unknown data point (x,y). For example, consider an unknown (x,y) data point measurement located at x=BG=100 and y=ΔG=35 for the system having the calibration data  512  and  514  shown in FIG.  8 . The delta gray level at a “nominal” background value of zero, y 0 , for the unknown data point (x,y) is calculated as follows. As previously described in accordance with assumption 1) of the non-linear shading correction technique, a=455 and BG MAX =222 for this system. Using these values in equations (11) and (10) yields the values of c=−1,007.3 and b=960,364. Thus, the equation of a hyperbolic curve which includes the measured data point (100,35) on its left branch and has a minimum at x=a=455 and an x-intercept at x=222 is as follows: 
     
       
         y={square root over ((x−455+L ) 2 +960,364+L )}−1,007  (12) 
       
     
     Hyperbolic curve  530  described by equation (12) is shown in FIG.  8 . Equation (12) has a y-axis intercept, i.e., delta gray level at a “nominal” background value of zero, of y 0 =73.5. 
     As previously stated, analysis of the first and second calibration curves  510  and  520  by any of a variety of empirical and/or analytical techniques may be used to arrive at a best fit for each hyperbolic calibration curve included in the family of calibration data, with the constraints that each hyperbolic curve has its minimum y-value at the same value of x, i.e., x=a, and each calibration curve has the same x-axis intercept, i.e., x=x 0 =BG MAX . A first procedure utilizes trial and error to combine least squares fits to individual curves, while a second preferred procedure fits all the data simultaneously. 
     In the trial and error procedure, hyperbolic curves of the form defined by equation (8) are fit to individual sets of calibration data. X-axis intercepts (x=x 0 =BG MAX ) for each of the individual calibration curves are compared and a common value determined by trial and error. Similarly, the x-axis value at which each hyperbolic curve has its minimum y-value, i.e., x=a, may be found by trial and error. Using this value of x=a and a dummy data point at the x-intercept (x=x 0 =BG MAX ), each set of calibration data is re-fit to a calibration curve of the form defined by equation (8). While this procedure may be effective for some applications, it is iterative and somewhat empirical and may not be adequate for production use. Alternatively, a second procedure using non-linear least squares to fit all the data at once may be employed. 
     In the second procedure, “x 0 ” and “a” are determined by a non-linear least squares fit to the calibration data sets. In this approach, equation (8) is rewritten such that the variables “b 2 ” and “c” are represented in terms of “x 0 ”, “a” and “y 0 ”, where “y 0 ” is the y-axis intercept of a particular hyperbolic curve. Thus, “y 0 ” varies with each curve in the family while the same values of “x 0 ” and “a” are shared by all of the curves in the family. Thus, at the y-axis intercept (0,y 0 ) of a particular hyperbola, equations (11), (10) and (8) become:              c   =         x   0   2     +     y   0   2     -     2        ax   0           2        y   0                 (   13   )                 b   2     =         c   2     -       (       x   0     -   a     )     2       =         [         x   0   2     +     y   0   2     -     2        ax   0           2        y   0         ]     2     -       (       x   0     -   a     )     2                 (   14   )                     y   =                    [         (     x   -   a     )     2     +       (         x   0   2     +     y   0   2     -     2        ax   0           2        y   0         )     2     -       (       x   0     -   a     )     2       ]       1   /   2       +                                  x   0   2     +     y   0   2     -     2        ax   0           2        y   0                       (   15   )                                
     Thus, the derivatives of “y” with respect to “a”, “x 0 ” and “y 0 ” are:                     y          a       =           (       x   0     -   x     )          y   0       -       x   0        y           y   0          (     y   -   c     )                 (   16   )                      y            x   0         =         (       x   0     -   a     )          (     y   -     y   0       )           y   0          (     y   -   c     )                 (   17   )                      y            y   0         =       y        (       y   0   2     -     x   0   2     +     2        ax   0         )         2          y   0   2          (     y   -   c     )                   (   18   )                                
     Note that since “c” is given by a function of “a”, “x,” and “y 0 ” in equation (13), equations (16), (17) and (18) express the derivatives of “y” as functions of “a”, “x 0 ” and “y 0 ”. Thus, equations (8), (16), (17) and (18) express “y” and its derivatives as functions of “a”, “x 0 ” and “y 0 ”. Using these expressions, fitted values for “a”, “x 0 ” and “y 0 ” can be obtained from the calibration data sets using a non-linear least squares fitting technique, for example, the Levenberg-Marquardt technique or other standard non-linear optimization technique. The optimization techniques are used to minimize the sum of square errors (or X 2  if variances are known) between the fitted function and the entire set of calibration data points for all of the curves in the family. Examples of optimization techniques may be found in a book entitled “Numerical Recipes for C” authored by Press et al., published by Cambridge University Press in 1992, the entirety of which is hereby incorporated herein by reference. 
     Applying assumption 2) of the non-linear shading correction technique, the Delta Gray level due to solder (y 0 ) vs. Solder Thickness (t), function may be approximated by a fitted curve of known form, e.g., a sum of exponentials: 
     
       
         y 0 (t)=p−Σ i q i   e   −r     i     t   (19) 
       
     
     where p, q i  and r i  are fitting constants. For example, when the sum of two exponentials is selected, the unknown solder thickness (t) may be determined by using the solder delta gray levels at a “nominal” or reference background gray level of zero (y 0 ) in the following functional form for the sum of two exponentials: 
     
       
         y 0 (t)=α−β e   −k     1     t   −γe   −k     2     t   (20) 
       
     
     where fitting constants α, β, γ, k 1  and k 2  are determined by fitting to the calibration data. The 5 fitting parameters can be reduced to three based on the following physical characteristics of the data. At zero solder thickness, the solder delta gray level at a “nominal” background gray level of zero (y 0 ) is zero, i.e., y 0 (0)=0. The maximum value of the solder delta gray level at a “nominal” background gray level of zero (y 0—MAX ) is BG MAX , i.e., y 0—MAX (t→∞)=BG MAX  (based on theoretical and empirical observations). Using this information, the five fitting parameters in equation (20) can be reduced to three since y 0 (0)=α−β−γ=0 and y 0 (t→∞)=α=BG MAX , thus equation (20) becomes 
      y 0 (t)=BG MAX   −βe   −k     1     t −(BG MAX −β) e   −k     2     t   (21) 
     where fitting constants β, k 1  and k 2  are determined by fits to the known solder thicknesses and corresponding delta gray levels at a “nominal” background gray level of zero derived from the hyperbolic fits to the calibration data. In applications requiring greater throughput, it may be advantageous to use the above described calibration procedure to generate a look up table (LUT) or surface map of Background (x) vs. Delta Gray (y) vs. Solder Thickness (t). 
     T YPICAL  C ALIBRATION  D ATA AND  C OMPARISON OF  L INEAR VS . N ON-LINEAR  S HADING  C ORRECTION  T ECHNIQUES    
     The linear and non-linear shading correction techniques described above were applied to multiple solder thickness calibration coupons. The results of these calculations and a comparison of the two techniques is presented below. A solder thickness calibration panel having nine known solder thicknesses was measured with varying thicknesses of copper. The nine solder thicknesses, in mils, were 1.2, 1.6, 3.6, 5.7, 7.7, 9.7, 13.7, 15.6 and 20.0. Delta gray levels as a function of background levels were measured for fifteen different known thicknesses of copper in combination with each of the known solder thicknesses. The fifteen known thicknesses of copper used in these measurements, in mils, were 0, 5, 10, 15, 20, 25, 30, 41, 51, 61, 73, 83, 93, 99 and 110. 
     FIG. 9 shows a plot of the measured delta gray vs. background levels for these 9 sets of calibration data where the background level was varied by applying the 15 different thicknesses of copper to each solder thickness coupon described above. Also shown in FIG. 9 are nine hyperbolic curves fit to the data points in accordance with assumption 1) of the non-linear shading correction technique. 
     A comparison of the linear shading correction technique to the non-linear shading correction technique is illustrated in FIGS. 10A and 10B. FIG. 10A shows calculated solder thickness vs. copper background thickness for the 9 sets of calibration data, where the solder thicknesses were calculated with the linear shading correction technique. It is evident from FIG. 10A that the linear shading correction technique results in overestimating the solder thicknesses as the background copper thicknesses increase. FIG. 10B shows calculated solder thickness vs. copper background thickness for the 9 sets of calibration data, where the solder thicknesses were calculated with the non-linear shading correction technique (hyperbolic fits). The non-linear shading correction technique clearly results in more accurate determinations of solder thicknesses, especially as the background copper thicknesses increase. 
     T WO  D IMENSIONAL  (2-D) S OLDER  T HICKNESS  D ETERMINATION    
     It is often advantageous, in terms of calculation speeds, etc., to represent solder calibration information in terms of a surface or look up table (LUT) defined in terms of Background (x or BG) vs. Delta Gray (y or ΔG) vs. Solder Thickness (t). Such a surface or LUT may be generated by the following procedure: 
     1) Measure a number of calibration points, e.g., the 9 sets of data shown in FIG. 9; and 
     2) Construct a DeLaunay Triangulation of the x vs. y (i.e., BG vs. ΔG) plane and use linear or polynomial interpolation to fill in the thickness values on a regular grid of x vs. y (i.e., BG vs. ΔG) which results in a 2D lookup table (LUT) of solder thickness (t) as a function of Background (x or BG) vs. Delta Gray (y or ΔG). 
     However, this approach may result in several problems, including: 1) a coordinate singularity at x=BG=BG MAX =x 0 ; 2) artifactual “ripples”, extrema, ridges, etc. in the surface; and 3) no physically meaningful surface outside the samples region unless extrapolation is employed, in which case it may be highly inaccurate and unreliable. 
     Many of these problems may be overcome by the following procedure. Recalling the physics of X-ray attenuation in connection with a 2 component assembly as described FIG. 5, a foreground intensity (i.e., image gray level) y f  is described by the general functional form: 
     
       
         y f =y 0 −∫α(E) e   −β(E)t     1     e   −γ(E)t     2   dE  (22) 
       
     
     or its discrete approximation: 
     
       
         y f =y 0 −Σ i α i   e   −β     i     t     1     e   −γ     i     t     2     (23) 
       
     
     where t 1  and t 2  are the thicknesses of the first material and the second material, respectively. In the general functional form: 1) the X-ray source energy spectrum is distributed as a function of energy with weightings determined by the parameter α(E); and 2) β(E) and γ(E) are the X-ray attenuation coefficients for the first and second materials, respectively. In the discrete approximation: 1) the total X-ray source energy spectrum is split up into some number of bands i, where the total source intensity is distributed among the bands as a functions of X-ray source energy and detector sensitivity with weightings for each band i determined by the parameter α i ; and 2) β i  and γ i  are the effective linear attenuation coefficients for X-rays in band i for the first and second materials, respectively. The following discussion is in terms of the discrete approximation, however, one skilled in the art will understand that a similar process also applies to the general functional form. A background intensity (i.e., image gray level) y b  (t 2 =0) is described in the discrete approximation form by: 
     
       
         y b =y 0 −Σ i α i   e   −β     i     t     1     (24) 
       
     
     Delta gray, the difference between the foreground and the background, is given by: 
      ΔG=y f −y b =Σ i α i   e   −β     i     t     1   −Σ i α i   e   −β     i     t     1     e   −γ     i     t     2     (25) 
     Using measured values of foreground (y f ) and background (y b ), or equivalently, ΔG, for a series of calibration standards with known values of t 1  and/or t 2  for each calibration standard, the background measurements are used to do a least squares fit to y 0 , α i  and β i  for i=1 to n, where n, the number of bands is specified in advance, according to equation (24). Using these fitted values of y 0 , α i  and β i  for i=1 to n, the foreground measurements are used to do a least squares fit to the γ i &#39;s for i=1 to n, according to equation (23). 
     An internally consistent approximation to the actual Background (BG) vs. Delta Gray (ΔG) vs. Solder Thickness (t) surface or look up table (LUT), which is free of ripples and supports consistent extrapolation, can be generated using these fitted values of the y 0 , α i , β i  and γ i  parameters. Alternatively, it is noted that these parameters may also be obtained by simulation rather than regression, or by a combination of the two methods. For example, one could simulate the α i , β i  and γ i  parameters and fit the y 0  or fit y 0  and scale α i , β i  and γ i . One may also utilize the non-linear shading correction procedure described above to generate a surface or LUT which is consistent and free of ripples. FIG. 11A shows an example of such a Background (BG) vs. Delta Gray (ΔG) vs. Solder Thickness (t) surface (generated from the calibration data illustrated in FIG. 9) in accordance with the above discussion. 
     Regardless of how the parameters y 0 , α i , β i  and γ i  are obtained, an internally consistent look up table can be generated. For each background value desired in the look up table, equation (24) is solved for t 1 . This can be done using Newton&#39;s Method or a simple Golden Section Search. Since it is known that there is a solution and the function is convex, a binary search is better than a Golden Section Search. Throughput is not critical since this is done only to construct the look up table. For each foreground value desired in the look up table, equation (23) is solved for t 2  using the previously determined value of t 1 . Thus, in the 2D lookup table, the entry t 1  is placed in Row=y b  at Col=y f . Note that only half of a square array is needed in most cases. However, if it is desired to have the ability to read out values of t 2 , then the values of t 2  can be stored in the other half of the array. 
     In operation, the look up table is used as follows. Assume that the look up table is constructed using integer gray values from 0 to 255 for foreground and background entries. To look up the thickness t corresponding to a specific background/foreground pair, (BG,FG), let: 
     
       
         R 1 =└BG┘ 
       
     
     
       
         R 2 =┌BG┐ 
       
     
     
       
         C 1 =└FG┘ 
       
     
     
       
         C 2 =┌FF┐ 
       
     
     where └x┘ is equal to the greatest integer ≦x and ┌x┐ is equal to the smallest integer ≧x. The thickness corresponding to (BG,FG) can then be estimated by bilinear interpolation. For example, let: 
     
       
         t a =t[R 2 ,C 1 ] 
       
     
     
       
         t b =t[R 2 ,C 2 ] 
       
     
     
       
         t c =t[R 1 ,C 2 ] 
       
     
     
       
         t d =t[R 1 ,C 1 ] 
       
     
     and 
     
       
         u=(BG−R 1 )/(R 2 −R 1 ) 
       
     
     
       
         v=(FG)/(C 2 −C 1 ) 
       
     
     Then, 
     
       
         t(BG,FG)≈uvt b +(1−u)vt c +(1−u)(1−v)t d +u(1−v)t a   
       
     
     Other interpolation schemes may be used, including linear interpolation from the three nearest points, or higher order schemes. Also note that if either FG or BG is an integer, interpolation in that axis (row or column) may be skipped for greater throughput. If both FG and BG are integers, the corresponding thickness value may be looked up directly. FIG. 11B shows a graphical representation of a Look Up Table (LUT) for Background (BG) vs. Delta Gray (ΔG) vs. Solder Thickness (t) (generated from the calibration data illustrated in FIG. 9) in accordance with the present invention. Shown in FIG. 12 are the results of solder thickness vs. background determined from a lookup table (generated from the calibration data illustrated in FIG. 9) in accordance with the above discussion. 
     The above embodiments of the present invention have been described in terms of generating a procedure, look up table or surface from which an unknown thickness which corresponds to known values of the background and delta gray may be determined. However, these techniques are invertible in that: 1) an unknown background value which corresponds to known values of the thickness and delta gray may also be determined; and 2) an unknown delta gray value which corresponds to known values of the thickness and background may also be determined. 
     S INGLE  M ATERIAL  C ALIBRATION    
     The previous descriptions have been in the context of a first material, for example solder, shaded by a second material, for example, G10 circuit board material. However, the invention also applies to a single material calibration, for example, solder shaded by solder. An example where this might occur is the inspection of the solder joints on a BGA component, where a significant portion of the background surrounding the images of specific solder joints is due to the solder comprising surrounding solder joints. 
     The procedure for this situation is similar to the procedures described above. First, the gray levels of a plurality of different, known thicknesses of solder T i  mounted on an appropriate substrate are measured. A curve of the following form (or its equivalent): 
      y F=y   0 −Σ i α i   e   −β     i     T     i     (26) 
     is fit to the measured values of known thicknesses. In many cases, two energy bands are probably sufficient, however, additional energy bands can be used if required to obtain the desired accuracy. The fit may be accomplished by fitting all of y 0 , α i  and β i . However, if y 0  is known, only the remaining parameters need to be fit. Thus, given a solder background measurement B and a solder foreground measurement F, equation (26) may be inverted to find the two corresponding thicknesses T F  and T B . The thickness of interest, i.e., the thickness of the solder joint, is then given by T F−T   B . 
     As before, this procedure may also be implemented with simulated calibration data. Simulation factors may include: a) spectral characteristics of the X-ray source; and/or b) angular distribution of X-rays produced by the X-ray source; and/or c) stopping power and spectral sensitivity of the X-ray detector; and/or d) X-ray attenuation properties of the absorbing materials as functions of X-ray energy/wavelength. Additionally, the procedure may be implemented with the construction of a look up table. 
     SUMMARY, RAMIFICATIONS AND SCOPE 
     Accordingly, the reader will see that the present invention solves many of the specific problems encountered when inspecting solder connections on circuited board assemblies. Particularly important is that it improves the accuracy of solder thickness measurements derived from X-ray images of solder connections. 
     Furthermore, the present invention has the additional advantages in that 
     it provides a single, globally consistent calibration for any chosen material in the presence of varying amounts of shading by a second material; 
     it is fast in terms of its computational requirements; 
     it is compact in terms of its storage requirements; 
     it is more accurate than previous methods; 
     it is numerically invertible such that in a three parameter system, any one of the parameters may be determined from known values of the other two parameters; 
     it may be made traceable to known standards criteria, for example, the National Institute of Standards &amp; Technology (NIST) or similar standards agencies. This feature permits process engineers to relate thicknesses measured by the X-ray system to physical joint dimensions. Traceability can be achieved by constructing the calibration standard out of materials of known purity, and by measuring thicknesses of the calibration standard using instruments which themselves have a traceable calibration; 
     it is portable, in the sense that measurement of the same joint on multiple systems will return similar or identical thicknesses. Portability requires that the calibration compensates for the physically significant sources of variation between systems; and 
     it supports multiple calibrations. With the advent of lead-free solders, the joint and background compositions can vary from board to board, or even within a board. As a result, it is desirable to be able to store multiple calibrations simultaneously, and to permit the user to select the appropriate calibration on a pin, component, or board level. 
     Although the description above contains many specificities, these should not be construed as limiting the scope of the invention but as merely providing illustrations of some of the presently preferred embodiments of this invention. For example, alternative techniques for fitting the calibration data may be used; alternative techniques may be used to determine fitting parameters; alternative interpolation techniques may be used; alternative techniques may be used to acquire the cross sectional images; shadowgraph X-ray images (non-cross sectional) images may be employed; simulation may employed to determine some of the fitting parameters; the invention may be applied to assemblies having more than two layers; etc. 
     It is to be understood that the methods of the present invention may be implemented in a variety of ways by one skilled the art, however, implementation with the computer or specially dedicated image processor is preferred. A typical computer used for such analysis includes one or more processors, one or more memories and various input and output devices including but not limited to monitors, disk drives, printers and keyboards. Additionally, it is to be understood that the term “image” is not limited to formats which may be viewed visually, but may also include digital or analog representations which may be acquired, stored and analyzed by the computer. 
     Thus, the scope of the invention should be determined by the appended claims and their legal equivalents, rather than by the foregoing description and examples given. All changes which come within the meaning and range of equivalency of the claims are to be embraced within their scope.