Abstract:
Circuit flaws in microelectronic circuitry present regions of high resistance in which a current distribution deviates from that of a defect-free circuit. The altered current distribution emits a correspondingly altered magnetic field in accordance with Ampere&#39;s Law. When compared with the magnetic field of a defect-free circuit, the anomaly in the magnetic field of the defective device is detected and the location of the circuit flaw may be determined therefrom. As the anomaly in the magnetic field is very small in magnitude, a sensitive magnetic microscope is utilized to obtain images of the magnetic fields of a defect-free reference device and a device-under-test. The distance between the magnetic sensor and the devices being scanned is precisely controlled to minimize influences of scanning distance on the difference in measured magnetic field strength. Comparative image analysis reveals the location of the circuit flaw. Maximal image registration through image interpolation, displacement and resampling optimizes the comparative image analysis.

Description:
[0001]     This Utility patent application is based on the Provisional Patent Application No. 60/502,628, filed 15 Sep. 2005. 
     
    
     BACKGROUND OF THE INVENTION  
       [0002]     1. Field of the Invention  
         [0003]     The invention described herein is related to fault isolation in microelectronic circuitry. More specifically, the present invention is directed to locating circuit defects in integrated circuits by comparing images of the magnetic field thereof with those of standard reference integrated circuit known to be free from the defect.  
         [0004]     2. Description of the Prior Art  
         [0005]     As integrated circuitry becomes increasingly compact and the packaging associated therewith becomes more complex, the localization of circuit defects becomes correspondingly difficult. The localization of flaws is particularly difficult for those classified to the high resistance (HR) defect class of packaging and interconnect anomalies causing an increase in the impedance of a signal line to beyond the designed specification thereof. These HR defects include cracked signal traces, cracked plated through-holes, delaminated vias and improperly wetted controlled collapse chip connection (C4) bumps. Such HR defects can result in devices that do not operate at optimum speed, have an increased probability of failing in the field, or simply do not work at all.  
         [0006]     The physical size of microelectronic circuitry presents unique failure analysis challenges. In some cases, microscopic dissection of a failed circuit may yield the location of the undesired anomaly. However, the failed circuit is destroyed by the process, thus precluding any subsequent functional testing should the inspiration to do so suddenly befall the tester.  
         [0007]     The prevailing non-destructive method for localizing HR defects in integrated circuits is time domain reflectometry (TDR). The TDR technique involves applying a current pulse of short duration to an input port of a signal line of an integrated circuit and monitoring the input port for one or more reflections of the applied pulse. The time measured between the application of the pulse and the detection of the reflected signal is an indication of the distance from the input port to the location of the impedance change. However, such reflected signals may be due to normal transitions within the device and not necessarily due to an undesired anomaly. Thus, tedious analysis by comparison to reflection characteristics of a standard device must be conducted once the data are collected. Additionally, even when the analysis is performed flawlessly, the spatial resolution of TDR in microelectronic circuitry is typically 200 microns, which is greater than the size of most components of modern integrated circuits.  
         [0008]     It is thus apparent from the shortcomings of the prior art that the need exists for a non-destructive fault isolation technique for use in microelectronic circuitry implementing less tedious analysis while simultaneously providing higher spatial resolution.  
       SUMMARY OF THE INVENTION  
       [0009]     The present invention provides a method for locating defects in an electronic circuit by means of a scanning magnetic microscope for obtaining images of magnet fields in a predetermined area thereof. A reference circuit and a circuit-under-test are provided to the scanning magnetic microscope and a corresponding fiducial is respectively located on each circuit. A test current is induced in the reference circuit and a micrographic image of the magnetic field emitted therefrom is obtained. An equivalent test current is induced in the circuit-under-test and a second micrographic image is obtained from the magnetic field emitted therefrom. A spatial transform is applied to the second micrographic image for providing magnetic field pixel values for the circuit-under-test at locations other than the original image pixel locations, i.e., the second micrographic image is spatially displaced and resampled from a continuous-space representation of the magnetic field to produce the transformed micrographic image. A difference image between the transformed micrographic image and the reference micrographic image is formed and a registration error is computed from the difference image. The second micrographic image is iteratively transformed using an updated spatial displacement until the registration error of the difference image resulting therefrom is minimized.  
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0010]      FIGS. 1A-1B  illustrate the effect on current distribution from a high resistance delamination defect;  
         [0011]      FIG. 2  is a graph demonstrating the required sensitivity of a magnetic probe to detect high resistance defects;  
         [0012]      FIG. 3  is a block diagram of an exemplary system for executing the method of the present invention;  
         [0013]      FIG. 4  is a flow chart illustrating an exemplary scan procedure of the present invention;  
         [0014]      FIG. 5  is an illustration of image registration shortcomings of the prior art;  
         [0015]      FIG. 6  is an illustration of an exemplary coordinate system as implemented by the present invention;  
         [0016]      FIG. 7  is a flow chart illustrating steps of an exemplary comparative image analysis procedure of the present invention; and  
         [0017]      FIG. 8  is an illustration of a magnetic anomaly corresponding to a high resistance defect as located by the present invention.  
     
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS  
       [0018]     The method of the present invention constitutes a novel approach to isolating circuit defects, such as high resistance (HR) defects, using current imaging and comparative image analysis. Current imaging is based on the measurement and spatial mapping of the magnetic field generated by current carried in the structures of a device-under-test (DUT). Comparative image analysis identifies local differences between images. By way of the precise current imaging and subsequent comparative image analysis, the present invention attains localization of circuit defects to within tens of microns.  
         [0019]     As is well known, the characteristic impedance of a signal line changes at geometric transitions in the line. An HR defect may manifest itself as such a geometrical change such as through regions of delamination, cracks, voids, etc. HR defects may also result from improper processing of materials during integrated circuit construction. Such processing failures may affect the conductivity in the region of the affected material, thus leading to a corresponding change in resistance in the region.  
         [0020]     The current distribution in a given region of a microelectronic circuit is a function of the geometry of, and material formed in, the region. Clearly, the current distribution is affected by geometric anomalies associated with an HR defect. An example of such an affected current distribution is illustrated in FIGS.  1 A and  1 B. The Figures depict a typical C4 bump interface of an integrated circuit package commonly referred to in the industry as a “flip-chip” package.  FIG. 1A  illustrates the normal configuration in which a circuit die  110  is electrically coupled to the package substrate  130  through the C4 bump  120 . As is shown in  FIG. 1A , the lines of current, shown as the directed arrows, are evenly distributed through the interface at C4 bump  120 . In  FIG. 1B , however, the delamination of the metallic layer on the die  110  minimizes the available area for electric contact thereon. As such, the current distribution is skewed, as is shown in the diagram.  
         [0021]     As the current distribution is affected by the geometric alteration, a corresponding change in the magnetic field distribution can be expected. That is to say, a small change in the magnetic field distribution in the area around the defect would be observed when compared to the magnetic field distribution of a normal, defect-free device. A detailed image comparison between the good and failing parts would be capable of detecting this difference and subsequently locating the defective region.  
         [0022]     It should be apparent that other circuit defects may affect the current distribution while not presenting an increase in the resistance of a circuit element. For example, plane-to-plane shorting of the power distribution network of an integrated circuit does not exhibit characteristics of a high resistance defect, but may instead distribute undesirable current flows over relatively large areas. These undesirable flows are detectable by the method of the present invention, as will be made clear in the paragraphs that follow.  
         [0023]     As previously stated, the spatial magnetic field distribution {right arrow over (B)}({right arrow over (r)}) is related to the spatial current density distribution {right arrow over (J)}({right arrow over (r)}) by Ampere&#39;s law:  
                 B   ⇀     ⁡     (     r   ⇀     )       =         μ   0       4   ⁢   π       ⁢     ∫     ∫     ∫             J   ⇀     ⁡     (       r   ⇀     -       r   ⇀     &#39;       )       ×     (       r   ⇀     -       r   ⇀     &#39;       )                  r   ⇀     -       r   ⇀     &#39;            3       ⁢         ⅆ   3     ⁢     r   &#39;       .                       (   1   )             
 
         [0024]     This is a fundamental relationship between the magnetic field created at a point {right arrow over (r)} by a given current density {right arrow over (J)}({right arrow over (r)}·{right arrow over (r)}′) located thereat. Any change in the current distribution should result in a change in the magnetic field distribution of the failing device, {right arrow over (B)} f (r).  
         [0025]     In view of the superposition principle, at least locally in the neighborhood of the defect, N d , we can consider the current distribution in a failing part, {right arrow over (J)} f ({right arrow over (r)}), to differ from that of a defect-free device, {right arrow over (J)}({right arrow over (r)}), by some amount Δ{right arrow over (J)}({right arrow over (r)}), or:  
                   J   ⇀     f     ⁡     (     r   ⇀     )       =     {                 J   ⇀     ⁡     (     r   ⇀     )       -     Δ   ⁢           ⁢       J   ⇀     ⁡     (     r   ⇀     )                 r   ∈     N   d                   J   ⇀     ⁡     (     r   ⇀     )           otherwise         .               (   2   )             
 
 The expression for the magnetic field of the failing device is  
                   B   ⇀     f     ⁡     (     r   ⇀     )       =         μ   o       4   ⁢   π       ⁢     ∫     ∫     ∫               J   ⇀     f     ⁡     (       r   ⇀     -       r   ⇀     &#39;       )       ×     (       r   ⇀     -       r   ⇀     &#39;       )                  r   ⇀     -       r   ⇀     &#39;            3       ⁢         ⅆ   3     ⁢     r   &#39;       .                       (   3   )             
 
 From eq. (2) the magnetic field can be expressed s:  
                 B   ⇀     ⁡     (     r   ⇀     )       =           μ   o       4   ⁢   π       ⁢     ∫     ∫     ∫             (     ⁢     J   ⇀     ⁢     (     ⁢     r   ⇀       -         r   ⇀     &#39;     ⁢     )       -     Δ   ⁢               ⁢   J     ⇀     ⁢     (     ⁢     r   ⇀       -       r   ⇀     ⁢     )     ⁢     )     ×     (     ⁢     r   ⇀       -         r   ⇀     &#39;     ⁢     )                    r   ⇀     -       r   ⇀     &#39;            3       ⁢     ⅆ     3   ⁢     r   &#39;                   ⁢     
     =         μ   o       4   ⁢   π       ⁢     {             ∫     ∫     ∫             J   ⇀     ⁢     (     ⁢     r   ⇀       -         r   ⇀     &#39;     ⁢     )     ×     (     ⁢     r   ⇀       -         r   ⇀     &#39;     ⁢     )                    r   ⇀     -       r   ⇀     &#39;            3       ⁢       ⅆ   3     ⁢     r   &#39;               -     ∫     ∫     ∫           Δ   ⁢           ⁢   J   ⁢     (     ⁢   r     -       r   &#39;     ⁢     )     ×     (     ⁢   r     -       r   &#39;     ⁢     )                  r   -     r   &#39;            3       ⁢       ⅆ   3     ⁢     r   &#39;                       r   ∈     N   d                 ∫     ∫     ∫             J   ⇀     ⁢     (     ⁢     r   ⇀       -         r   ⇀     &#39;     ⁢     )     ×     (     ⁢     r   ⇀       -         r   ⇀     &#39;     ⁢     )                    r   ⇀     -       r   ⇀     &#39;            3       ⁢       ⅆ   3     ⁢     r   &#39;                   otherwise                       (   4   )             
 
         [0028]     It should be noted that the first term of the right hand side of the above equation is simply the magnetic field of the defect free device {right arrow over (B)}({right arrow over (r)}). Thus,  
                   B   ⇀     f     ⁢     (     ⁢     r   ⇀     ⁢     )       =     {                   B   ⇀     ⁢     (     ⁢     r   ⇀     ⁢     )       -     Δ   ⁢           ⁢       B   ⇀     d     ⁢     (     ⁢     r   ⇀     ⁢     )               r   ∈     N   d               0       otherwise         ⁢     
     ⁢   where     ,               (   5   )                 Δ   ⁢           ⁢       B   ⇀     d     ⁢     (     ⁢     r   ⇀     ⁢     )       =         μ   0       4   ⁢   π       ⁢     ∫     ∫     ∫           Δ   ⁢           ⁢     J   ⇀     ⁢     (     ⁢     r   ⇀       -         r   ⇀     &#39;     ⁢     )     ×     (     ⁢     r   ⇀       -         r   ⇀     &#39;     ⁢     )                    r   ⇀     -       r   ⇀     &#39;            3       ⁢       ⅆ   3     ⁢   r                       (   6   )             
 
 is the magnetic signature of the defect. The current density change, Δ{right arrow over (J)}({right arrow over (r)}), must vanish abruptly outside the defect region as it is constrained to the physical boundaries of the defect itself. The associated magnetic field thereof, however, may extend well beyond the geometric boundaries of the defect. 
 
         [0030]     In the presence of noise, the expression of (5) should be modified as follows:  
                   B   ⇀     f     ⁢     (     ⁢     r   ⇀     ⁢     )       =     {               B   ⇀     ⁢     (     ⁢     r   ⇀     ⁢     )       -     Δ   ⁢           ⁢       B   ⇀     d     ⁢     (     ⁢     r   ⇀     ⁢     )       +         σ   ⇀     B     ⁡     (     r   ⇀     )               r   ∈     N   d                     σ   ⇀     B   &#39;     ⁡     (     r   _     )           otherwise                   (   7   )             
 
 where σ B  is the magnetic noise, which is assumed to have a white noise distribution. 
 
         [0032]     It is thus evident that careful measurement of the magnetic field distribution of both defect-free and defective devices would allow the extraction of the residual magnetic field due to the defect. This may be achieved by subtracting the measured magnetic field distributions i.e.,  
                   B   ⇀     ⁡     (     r   ⇀     )       -         B   ⇀     f     ⁡     (     r   ⇀     )         =     {                   B   ⇀     d     ⁡     (     r   ⇀     )       +         σ   ⇀     B     ⁡     (     r   ⇀     )               r   ∈     N   d                     σ   ⇀     B   &#39;     ⁡     (     r   _     )           otherwise         .               (   8   )             
 
         [0033]     From (8) it is clear that magnetic noise limits the detectability of the residual field. If the magnetic noise σ B  is sufficiently large, it will mask the residual magnetic field coming from the defect, ΔB d . This problem is overcome through the use of highly sensitive magnetic sensors.  
         [0034]     Referring now to  FIG. 2 , there is shown a graph in which the maximum intensity of a magnetic anomaly in Tesla (T) as a function of the z-distance of a magnetic sensor from the defect for three different defect sizes. The graph was created through a simple one dimensional model of an infinitely long wire of finite width. A “mouse bite” in the wire was simulated by changing the width in a predetermined segment thereof. As is shown by the legend, several defects, each of different size, are shown in the graph (the first numeral in the legend indicates the amount of the wire remaining from the total width, which is indicated by the second numeral). The horizontal dashed line represents a noise level of about 20 nT. The graph of  FIG. 2  illustrates that the residual field vanishes gradually as the detector is moved from the defective region. As discussed hereinabove, the intensity of the residual field must be larger than the noise level for a detection technique to be capable of discriminating the defect. A highly sensitive magnetic sensor, such as a thin-film super conducting quantum interference device (SQUID) is implemented by the present invention to detect the residual field emitted in the defective region.  
         [0035]     A simplified block diagram of a magnetic microscope for performing certain aspects of the present invention is illustrated in  FIG. 3 . For purposes of the present discussion, an exemplary embodiment is assumed by which the magnetic microscope is a scanning SQUID microscope such as is well known in the magnetic microscopy field. Such a SQUID microscope may include a microscope probe head  310  selectively positioned over a translation stage  320 . Translation stage  320  may be a two dimensional translational stage in which motion is confined to the x-y plane. In such case, microscope probe head  310  must be mounted on a movable platform such that the z-distance between translation stage  320  and magnetic probe head  310  may be selectively controlled. Alternatively, translation stage  320  may be a three dimensional translation stage movable along the x, y and z axes.  
         [0036]     As is shown in  FIG. 3 , a defect-free reference device  330   R  and the device-under-test  330   T  are affixed to the translation stage  320 . In certain embodiments of the present invention, the devices are scanned sequentially so as to obtain corresponding images of the respective magnetic fields. Thus, the position of SQUID sensor  313  relative to the current carrying elements must be precisely maintained. The scan condition for each device should be as near to identical as realizable so as to improve the effectiveness of the magnetic anomaly localization method.  
         [0037]     To implement precision control of the scanning operation certain embodiments magnetic microscope probe head  310  have installed thereon a z-distance probe, such as touch probe  317 , which is electrically coupled to z-distance detector  353  in scan controller  350 . Touch probe  317  provides a reference in the z-distance of SQUID detector  313  by providing an electric signal to z-distance detector  353  upon coming in contact with a surface in the x-y plane. In essence, touch probe  317  “touches” a point on, for example, reference device  330   R  and, after subsequent translation, touches the same point on DUT  330   T . Any discrepancy in the z-distance between the two points on the respective devices is compensated for via the scan controller  350 , as will be described below. By ensuring that touch probe  317  is able to discriminate sub-micron distances, control of the z-distance from the current carrying element to the SQUID detector  313  is precisely maintained.  
         [0038]     Translation stage  320  is coupled to translation mechanisms (not shown) known in the field of scanning microscopy, (e.g., stepper motors, piezoelectric actuators, etc.), electrically coupled to translation stage controller  357 . Translation stage controller  357  provides precision control to the translation stage  320  and is coupled to z-distance detector  353 . By way of the present invention, z-distance detector  353  provides z axis positioning data to translation controller  357 , which, in turn, uses that information to maintain a consistent z-distance from the working surface throughout the scanning process.  
         [0039]     As is shown in the  FIG. 3 , SQUID sensor detector  313  is electrically coupled to image capture/image processor  340  for obtaining and processing images of the magnetic fields detected by SQUID sensor  313 . Separate images are obtained for each of reference device  330   R  and device-under-test  330   T . Image processing methods of the present invention compensate for image misregistration during comparative image analysis, as described below.  
         [0040]     Certain principal method steps for an exemplary scanning procedure are given by way of the flow chart of  FIG. 4 . The process is initiated at start block  410  and flow is transferred to block  415  whereby reference device  330   R  and DUT  330   T  are secured to the translation stage  320 . The user then selects a region of interest of the reference circuit, as shown at block  420 , through a user interface (not shown). A common fiducial is respectively selected on each device, as indicated at block  425 . The fiducial on each device  330   R ,  330   T , provides a reference point common to both devices from which subsequent measurements may be referred.  
         [0041]     The fiducials may be structures specifically designed for the purpose of providing a reference point or may be any predetermined feature of the device. Once common fiducials have been selected, flow is transferred to block  430  in which a translation vector, T=(T x , T y , T z ), is computed for each point in the region of interest of the reference circuit to a corresponding point on the device under test  330   T . It is significant to note that the translation vector T z  not only provides information as to the location in the x-y plane of the point on the device under test  330   T , but also provides the z-distance coordinate of the point. As previously stated, touch probe  317  is used to determine differences in height between points on reference device  330   R  and the corresponding point on DUT  330   T  and the z-distance is maintained via movement of translation stage  320  as controlled by controller  357 .  
         [0042]     In certain embodiments of the present invention, the fiducials are selected and the translation vectors for an entire device, or a set of ROIs for a device, are computed a priori, i.e., after securing the devices to the translation stage and prior to image acquisition. This allows for greater throughput when more than one DUT is to be tested at once. In such instances, the method steps of blocks  435  and  430  may be performed prior to the iteration loop.  
         [0043]     Once the movement between devices has been coordinated, flow is transferred to block  435 , whereby a region of interest of the device-under-test is located that corresponds to the region of interest of the reference circuit. The process continues at block  445  in which the devices are scanned sequentially to obtain the magnetic field images. This is achieved by taking a magnetic field measurement with SQUID sensor  313  at a point on reference device  330   R  and moving the translation stage  320  so as to place the same point on device under test  330   T  under SQUID detector  313 . This process is then repeated until measurements for all points in the region of interest have been collected from both devices  330   R ,  330   T .  
         [0044]     During the process of sequential scanning, level compensation/distance fine adjustment is performed to ensure that SQUID detector  313  is maintained at a consistent z-distance from the working surface of each part as indicated at block  440 . In certain embodiments of the present invention, the level of the surface of both devices is sampled via touch probe  317 . The level data are maintained in scan controller  350  and are used in positioning translation stage  320  during scanning operations so as to maintain a consistent relative distance from the current carrying element to SQUID sensor  313 .  
         [0045]     Once the images from both devices have been obtained, the comparative image analysis is performed, as shown at block  450 . The image analysis will be discussed in detail in paragraphs that follow. However, once the image analysis has been completed, the user may be presented with an option of zooming in on a particular sub-region, as shown by the decision block  460 . If the user decides to zoom in on a sub-region, the scan resolution is increased as shown at block  455  and a new region of interest within the original region of interest is selected at block  420 . The scan is performed in accordance with the new scan resolution via the method steps previously described. If the user has decided not to zoom in on a sub-region, it is determined if a new region of interest is to be examined as indicated at decision block  465 . If a new region is selected, the associated region of interest is selected at block  420  and the process is repeated to obtain new images of the selected region of interest. If the user decides not to proceed with a new region, the process is exited as shown at block  470 .  
         [0046]     It should be clear to the skilled artisan that even with exercising great care in securing reference device  330   R  and DUT  330   T  to the translation stage, exact duplication of the orientation in space of one object by another object is virtually impossible. Even small differences in the orientation in space of the two devices  330   R ,  330   T  will appear as a corresponding difference in the data obtained during the scan. Thus, the scan for the reference device  330   R  will be performed in one relative orientation of the SQUID detector  313  to the device and the scan of the DUT  330   T  will occur in a different relative orientation between the SQUID detector  313  and the device  330   T . Whereas, differences in the working distance between the SQUID sensor  313  and a device are compensated for during the scanning operation, as described above, relative spatial skew of the parts in the x-y plane add a noise component to the detection of HR defects.  
         [0047]     Simple relative rotation between obtained magnetic field images fails to result in sufficient alignment of the image data to detect the subtle differences in magnetic fields associated with high resistance defects. This shortcoming is depicted in  FIG. 5 , wherein the image of the magnetic field of the reference device  330   R  is indicated at  510  and is composed of N x ×N y  pixels. The reference image  510  includes locations  512   a ,  512   b  which are pixel positions of the known locations of the chosen fiducials.  
         [0048]     The image of the DUT  330   T  is shown at  520  and also consists of N x ×N y  pixels. The DUT image  520  includes locations  522   a ,  522   b  which are pixel positions of the locations of the selected fiducials on the DUT  330   T . As is shown in the Figure, an image processing technique of the prior art attempts to register the image by rotating and translating DUT image  520  so as to align fiducial locations  522   a ,  522   b  respectively with locations  512   a ,  512   b  of reference image  510 .  
         [0049]     As is shown in  FIG. 5 , simple relative rotation of images  510 ,  520  produces significant error to comparative image analysis. Most notably, errors are introduced in the regions where the images do not overlap, such as at region  530  and at region  535 . Thus, any attempt to subtract the data of reference image  510  from DUT image  520  would produce significant error, especially in those regions  530 ,  535 .  
         [0050]     The choice of image resolution may also introduce errors when applying comparative image analysis. For example, if the resolution is chosen too coarse, alignment of the fiducials may prove extremely difficult. This is shown in  FIG. 5  at location  522   b  where the fiducial location of the DUT image  520  occupies two pixels whereas the corresponding fiducial location on reference image  510 , i.e.,  512   b , occupies a single pixel. It should be clear that the actual number of pixels will vary by application, however, the example provided by  FIG. 5  illustrates the nature of the alignment problems in dealing with discrete image data.  
         [0051]     The method of the present invention overcomes the data alignment shortcomings of the prior art by first transforming the image of the magnetic field of the DUT image  520  into a continuous space representation thereof. The data frame of the DUT image  520 , i.e., the region of interest, may then be rotated and translated freely without regard to alignment of specific pixels. Once the data frame has been rotated and/or translated, the continuous representation of the magnetic field of the DUT device is discretized, or pixilated to the same resolution as the reference image  510 . A difference image, i.e., the difference in pixel values of the reference image and the rotated DUT image, is formed and a registration error measure is computed. The process of rotating/translating the data frame, pixilating the continuous-space representation of the magnetic field to the rotated data frame, forming the difference image and computing the registration error measure is repeated until the error measure is minimized. The resulting difference image may then be used to locate any magnetic anomalies.  
         [0052]     In preparation for the comparative image analysis, the image  520  of the magnetic field for the DUT  330   T  is interpolated to produce, 
 
 B   FIT   [x, y]=f   I   [B   T ( i, j )],   (9) 
 
 where B FIT  is a continuous-space representation of the magnetic field in two-dimensional Cartesian coordinates, (x, y), f I  is an interpolating function and B T  is the pixel value of the DUT image  520  at pixel coordinates (i, j). The interpolation function f I  may be any interpolation function suitable for interpolating between discrete image values. In certain embodiments of the present invention, the interpolation function is a cubic or spline surface fitting to produce an interpolating polynomial. The interpolating polynomial is then represented in this discussion as B FIT [x, y]. 
 
         [0054]     With an interpolating function defined, i.e., the data for the magnetic field of the DUT  330   T  being represented in continuous-space form, the image data may be transformed to a new coordinate system which more closely coincides with the coordinate system of the reference image as a originally scanned. Such a transform may take the form of, 
 
 r*   μ   =T   μ [ξ x , ξ y   ]+R[θ]·r   ν ,   (10) 
 
 where exemplary terms are given by 
 
 T[ξ   x , ξ y ]:( x, y )*=( x, y )+(ξ x , ξ y ) 
 
 R[θ]: ( x, y )*=( x, y )· R (θ). 
 
 The transformation of coordinates in accordance with the relationship above is shown in  FIG. 6  where r* μ =(x, y)* is the coordinate of the data originally located at coordinate (x, y). 
 
         [0057]     Once the magnetic image data has been translated and rotated into its new coordinate system, the data may then be re-pixilated, or re-sampled, according to, 
 
 B   i,j   =B   FIT   [x*   i   , y*   j ],   (11) 
 
 where B i,j  is the rotated image data and x* i  and y* j  are the x and y coordinates of the pixel located at (i, j) in the transformed coordinate system. 
 
         [0059]     A difference image is formed via the relationship, 
 
 D   i,j   =R   i,j   −B   i,j ,   (12) 
 
 where D i,j  is the difference image data at pixel location (i, j), R i,j  is the pixel data of the reference image  510  at pixel (i, j) and B i,j  is the transformed image data at pixel location (i, j). However, the difference image D may not be the result of the optimal alignment of reference image  510  and DUT image  520 . To ensure that the image difference is the result of optimum subtraction, a performance estimator, σ k  may be used to provide an indication as to the data alignment in the two images i.e., maximum image registration Examples for estimators σ k  are: 
 
 M =max[| D|]   (13) 
 
d=StdDev]D[  (14) 
 
               s   =     -       ∑     i   ,   j               ⁢           ⁢       D     i   ,   j       ⁢     log   ⁡     [     D     i   ,   j       ]               ,           (   15   )             
 
 where M is the maximum of the norm of D, d is the standard deviation of D and s is the image entropy of image D. It should be clear to the ordinarily skilled artisan that other estimators may be used, such that the minimum value of which will assure that the transformed image is properly registered to the reference image. The minimum value for any of the exemplary estimators σ k  above results in maximum image registration and may be found by iteration over many different transformations, as will be shown below. 
 
         [0062]     Referring now to  FIG. 7 , there is shown a flow chart of the certain key method steps in aligning the data for comparative image analysis, as implemented by certain embodiments of the present invention. The process is entered at start block  710 , from which flow is transferred to block  715 , whereby the image for R of the reference device and the image B T  for the DUT are obtained. The fitting polynomial B FIT (x, y) is determined at block  720 .  
         [0063]     Once the interpolation polynomial has been found, iterations are executed to find the optimal data alignment. At block  725 , initial values for the coordinates of the transform system are initiated and flow is transferred to block  730  where a new set of transformed coordinates are returned. At block  735 , the DUT image data are re-sampled via the interpolating polynomial as described above. Flow is transferred to block  740 , whereby the difference image is formed and a new error estimation is computed at block  745 . It is then determined if the registration error estimation σ[D] is a minimum as indicated at block  750 . If the registration error estimation is not minimum, new values for the transform coordinates are established by adding a small amount thereto (through multipliers k 1 , k 2 , k 3  at block  725 ) and the process is repeated. If it is found at block  750  that σ[D] is at a minimum value, the process is exited at block  760 . At this point, the difference image D is the result of optimum subtraction and can be used for the determination of magnetic anomalies.  
         [0064]     An exemplary image of a magnetic anomaly as determined by the method of the present invention is shown in  FIG. 8 . As shown in the Figure, a signal trace  810  has removed therefrom a “mouse bite” indicated at  815 . As is well known in the art, such a geometric transition presents a significant change in impedance at the location thereof. This defect appears as a Gaussian-shaped peak  820  in the image data of the difference image. The center of the peak indicates the location of the HR defect. It has been found that the method of the present invention may locate such HR defects to within tens of microns.  
         [0065]     Whereas, the illustration of  FIG. 8  depicts a defect having a Gaussian profile in the magnetic signature thereof, other magnetic signatures, such as the magnetic dipole signature, exist and are detectable by the present invention. It should be clear to the skilled artisan that the magnetic field residual due to a particular defect is a function of the geometry thereof and its relationship to the surrounding circuit elements. The location of the defect can be found by knowledge of the magnetic field distribution as a function of the geometry of the emitter. For example, in the case of a magnetic dipole signature, the centroid of the signature region, i.e., the midpoint between dipole extrema, provides the location of the defect. For defects producing peak-shaped signatures, such as the Gaussian peak of  FIG. 8 , the centroid produces the location of the defect. In the case of symmetrical magnetic signature peaks, the centroid coincides with the peak maximum.  
         [0066]     The description of the invention as presented above is illustrative and not restrictive. Many variations, alternatives and modifications will become apparent to those of skill in the art upon reviewing this Disclosure. The scope of the invention described herein should, therefore, be determined not with reference to the description presented above, but instead should be determined with reference to the appended Claims along with their full scope of equivalence.