Parallel processing network that corrects for light scattering in image scanners

A signal representative of the total incident flux on each area element of a target plane is generated. Each area element corresponds to a portion of the object from which the flux emanates. Each total incident flux signal is corrected to eliminate therefrom the effects of flux incident on the corresponding area element due to scattering, thereby to generate a signal representative of the flux incident on that area element emanating only from the portion of the object corresponding to that area element. The total incident flux signal is corrected by weighting each total incident flux signal in accordance with a predetermined weighting factor W(x,y), and summing the weighted signal representative of the total flux incident on an area element with a weighted total incident flux signal from each of the other area elements.

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
The present invention relates to an imaging apparatus, and in particular, 
to an imaging apparatus that corrects for the effects of flux scattering. 
2. Description of the Prior Art 
One of the useful properties of photographic film is that it records 
incident radiant flux in a manner that compresses dynamic range. It has 
become standard practice to measure the response of photographic film to a 
given exposure by determining the optical density of the resulting 
transparency. This is accomplished by illuminating the transparency and 
measuring the light flux that is transmitted or reflected. Precise 
measurement of transmitted or reflected flux values when reading 
transparencies is hampered by technical equipment limitations such as 
failure to create even illumination in the plane of the transparency, 
vignetting, optical misalignment, and sensor nonuniformity. Some prior art 
devices have attempted to correct for these limitations, such as the 
device disclosed in Jansson et al., "Implementation and application of a 
method to quantitate 2-D gel electrophoresis patterns", Electrophoresis 4, 
82-91 (1983). 
The measurement of transmitted or reflected flux values may also be 
hampered by other factors such as scattering of flux due to dust on the 
optical elements, scattering of flux due to imperfections in the optical 
system, and scattering of flux from the transparency itself. 
Various attempts have been made to use optical isolation to eliminate or 
reduce the contribution of scattered flux due to these above-listed 
scattering phenomena while measuring the flux transmitted through a 
transparency. An instrument known as a digital scanning microdensitometer 
is used to scan an entire image on a transparency, pixel by pixel, to 
convert it to a series of numbers or to a digitized image. This form of 
the data is useful for subsequent image processing and analysis. Scanning 
can be accomplished by planar motion of a stage containing the 
transparency, or alternately, rotation of a drum having the transparency 
thereon. Such a device, however, requires troublesome mechanical parts 
that demand a high level of precision, limit the speed of the scan, render 
the scanner cost high, and produce wear that limits the life of the 
equipment. As a result, the cost of such a device is excessive. 
Methods and apparatus that correct for resolution distortion in 
spectrometers and other optical devices are summarized in Jansson, 
"Deconvolution: With Applications in Spectroscopy", Academic Press (1984). 
In view of the foregoing it is believed advantageous to obtain the speed 
and convenience of a relatively low cost scanner, yet at the same time 
obtain the precision and accuracy that attends the use of isolation to 
eliminate the effects of scattered flux. 
SUMMARY OF THE INVENTION 
The present invention relates to an apparatus and method for imaging an 
object having flux emanating therefrom, the object having a predetermined 
number of discrete portions thereon. In accordance with the present 
invention, a target plane having a plurality of area elements is defined. 
Each area element corresponds to a portion of the object. A signal 
representative of the total incident flux on each area element is 
generated. Each total incident flux signal is corrected to eliminate 
therefrom the effects of flux incident on the corresponding predetermined 
area element due to scattering, thereby to generate a signal 
representative of the flux incident on that area element emanating only 
from the portion of the object corresponding to that area element. 
The total incident flux signal is corrected by weighting each total 
incident flux signal in accordance with a predetermined weighting factor 
W(x,y), and summing the weighted signal representative of the total flux 
incident on an area element with a weighted total incident flux signal 
from each of the other area elements. Each weighting factor W(x,y) is 
functionally related to a predetermined response matrix R formed of a 
predetermined number of response coefficients R(x,y). A response 
coefficient R(x,y) represents a quantification of the degree to which a 
given one of the area elements, (for example, the x-th area element) is 
affected by flux emanating from a portion of the object corresponding to 
another area element, (e.g., the portion corresponding to the y-th area 
element) that is, due to scattering, incident on the x-th area element. 
Each response coefficient R(x,y) represents a property of the imaging 
apparatus that is not dependent upon an object being observed or upon a 
test object used to calibrate the apparatus.

DETAILED DESCRIPTION OF THE INVENTION 
Throughout the following detailed description similar reference numerals 
refer to similar elements in all Figures of the drawings. 
Although the more detailed mathematical basis for the present invention is 
derived in connection with a more particularized embodiment set forth 
hereinafter, the general nature of the problem to which the present 
invention addresses itself may be understood from the highly stylized 
representation of a generalized imaging apparatus indicated by the 
reference character A in accordance with the present invention shown in 
FIGS. 1A and 1B. An arbitrary object 10 is shown as being disposed in 
generally confrontational relationship with respect to a target plane 12 
forming part of the apparatus A. Although not illustrated it should be 
understood that suitable optical components may be interposed between the 
object 10 and the target plane 12. The entirety of the object 10 that 
confronts the target plane 12 may be understood to be subdivisible into a 
predetermined number N of discrete portions, several of which, viz., the 
portions 10-1, 10-(i-1), 10-i, 10-(j-1), 10-j, and 10-N are illustrated by 
way of example. To insure that only those portions of the object that 
confront the target plane contribute flux it may be necessary or 
desireable in some instances to block certain potential flux paths 
surrounding the object by masks, stops, baffles or similar structures. 
In response either to irradiation from incident radiation I.sub.0 from a 
source 14 appropriately positioned with respect to the object 10 or 
because of its self-luminousity, radiant flux emanates from the discrete 
portions of the object 10. It should be apparent by implication from the 
foregoing that, for purposes of this application, the object 10 may be 
understood to be either a self-emissive object, a reflective object, or a 
transparent object. The flux emanating from the object 10 impinges on the 
target plane 12. The target plane 12 contains a predetermined number N of 
area elements. Again, several of the area elements 12-1, 12-(i-1), 12-i, 
12-(j-1), 12-j and 12-N, respectively corresponding to the portions 10-1, 
10-(i-1), 10-i, 10-(j-1), 10-j, and 10-N of the object 10, are shown. In 
FIG. 1A the respective flux paths 16 from only two portions 10-i and 10-j 
of the object 10 are illustrated by the bold dashed lines, referenced by 
respective reference characters 16-i and 16-j, with the flux paths from 
the other portions of the object being suggested. The magnitude of the 
radiant flux from each respective portion is indicated by the characters 
I(i) and I(j). 
In the typical case, as will be more fully developed hereafter, a 
photodetector arrangement generally indicated at 18 is associated with the 
target plane 12. The photodetector arrangement 18 forms part of the 
apparatus A. The photodetector arrangement 18 may be positioned in any 
convenient operative location with respect to the target plane 12. In FIG. 
1A the photodetector 18 is shown as lying generally behind the target 
plane 12. It should also be understood that in various implementations of 
the present invention suitable optical components may also be interposed 
at any appropriate positions between the target plane 12 and the 
photodetector arrangement 18. The term "photodetector" should be 
understood to encompass detectors of energy flux other than 
electromagnetic. 
In the general case, portions of the photodetector arrangement 18 are 
associated with corresponding area elements of the target plane 12. For 
example, the portions 18-1, 18-(i-1), 18-i, 18-(j-1), 18-j and 18-N of the 
photodetector arrangement 18 are respectively associated with the area 
elements 12-1, 12-(i-1), 12-i, 12-(j-1), 12-j and 12-N of the target plane 
12. The total number N of discrete portions into which the object 10 is 
subdivided is generally determined by the number of area elements defined 
on the target plane 12 and by the number of corresponding associated 
portions of the photodetector arrangement 18. It lies within the 
contemplation of the invention that multiple detectors may be configured 
to behave as one detector entity. 
At this point it should be noted that in many implementations of the 
invention the target plane 12 and the photodetector arrangement 18 may be 
physically integrated. For example, the target plane 12 may be defined on 
the surface of a planar photodiode array, on the surface of a light 
sensitive photographic film, or on the surface of a vidicon target. In 
these instances it is apparent that the target plane is inherent in the 
structure of the photodetector arrangement. 
In other instances the target plane and the photodetector arrangement may 
be physically separated from each other. For example, in the case of a 
laser scanner, the focussing and timed positioning of an interrogating 
laser beam is used to illuminate portions of the object in some 
predetermined pattern. The movement of the beam is typically generated 
using a rotatable mirror. Light reflected by (or transmitted through) the 
portion of the object being illuminated at any given instant of time is 
collected by a suitable light collector arrangement. In such an 
arrangement the surface of the collector at that instant defines the area 
element of the target plane corresponding to that illuminated portion of 
the object. The associated photodetector arrangement is usually positioned 
at a location that does not lie in the target plane. 
It should therefore be understood that the depiction in FIG. 1A of separate 
and distinct area elements of the target plane 12 as corresponding to 
separate and distinct portions of the photodetector arrangement 18 should 
not be construed to limit or exclude any apparatus, such as a laser 
scanner, in which area elements of the target plane 12 are identified only 
by their temporal sequence in a timed scan. Although in such case the 
target plane may, in the physical sense, have just one area element, it 
should be appreciated from the foregoing that at any given instant of time 
that target plane corresponds to the portion of the object being 
illuminated. Thus, the timed scan of the beam across the object produces a 
succession of such correspondences and, hence, a succession of area 
elements corresponding to various portions of the object. It is in this 
sense that, in the case of the laser scanner, the target plane may be said 
to exhibit a plurality of area elements, each of which corresponds to a 
portion of an object. 
However the apparatus is configured, each portion of the photodetector 
arrangement 18 is operative to provide a signal that represents the total 
radiant flux impinging on the area element of the target plane 12 
corresponding to that portion of the photodetector arrangement 18. In 
FIGS. 1A and 1B each such signal is carried on an output line 20 
associated with each of the detectors in the photodetector arrangement. It 
should be understood that the signals on the output lines 20 may be 
multiplexed over a single line. Those skilled in the art will understand 
that corrections to the signal on the line 20 output from a given 
photodetector may be necessary to remove nonlinear response of the 
photodetector to flux and to compensate for offset that introduces an 
additive constant to the output of the photdetector. 
In the absence of scattering, radiant fluxes I(i) and I(j) emanating from 
the respective portions 10-i and 10-j of the object 10 along the 
respective flux paths 16-i and 16-j would impinge directly upon 
corresponding area elements 12-i and 12-j of the target plane 12. Those 
direct flux paths from the portions of the object 10 to the corresponding 
area elements of the target plane 12 are indicated in FIG. 1A by the 
collinear bold and fine dashed lines. Absent scattering or absorption the 
flux emanating from a given portion of the object 10 and impinging upon 
the corresponding portion of the target plane 12 would not be attenuated. 
However, in a non-idealized and practically realizable apparatus, 
scattering occurs. In the flux path 16-i, for example, a scattering 
volume, illustrated diagrammatically at 22, will cause a part of the flux 
I(i) to be scattered along one or more alternative flux paths. A 
scattering volume should be understood to potentially contain surfaces 
that specularly reflect flux in either forward and/or backward directions. 
For flux I(i) along the path 16-i two alternative flux paths 24S.sub.1 and 
24S.sub.2 are shown as dotted lines. Similarly, in the case of the flux 
I(j) propagating along the path 16-j a scattering volume 26 would result 
in two alternative flux paths 28S.sub.1 and 28S.sub.2, which are shown as 
dot-dash lines. Some of the flux scattered by the volume 22, for example, 
the flux along the path 24S.sub.2, would be directed toward and impinge 
upon the area element 12-j. This flux is indicated in FIG. 1A by the 
reference character I.sub.S (j,i). Similarly, some of the flux scattered 
by the volume 26, for example, the flux along the path 28S.sub.1, would be 
directed toward and impinge upon the area element 12-i. This flux is 
indicated in FIG. 1A by the reference character I.sub.S (i,j). Some part 
of the flux emanating from each portion of the object 10 would, however, 
impinge on the area element corresponding thereto. This flux is indicated 
in FIG. 1A by the reference characters I.sub.U (i) and I.sub.U (j). It 
should be understood that I.sub.U (i) and I.sub.U (j) may be attenuated by 
absorption losses in the respective flux paths. 
From FIG. 1A it may thus be seen that the total incident flux impinging on 
a given area element of the target plane 12 (and falling upon the 
photodetector arrangement 18 to produce the signal on the line 20 
therefrom) is equal to the flux emanating directly from the portion of the 
object 10 corresponding to the given portion in the target plane 12 plus 
the sum of the flux emanating from other portions of the object and 
scattered onto that given portion of the target plane. Symbolically, for 
the area element 12-i, the following equation holds: 
##EQU1## 
where I.sub.M (i) is the total incident flux impinging on a given area 
element of the target plane 12 as measured by the photodetector 
arrangement 18 associated therewith, 
I.sub.U (i) is the flux emanating directly from a portion of the object 10 
corresponding to the given portion in the target plane 12 that is incident 
on that portion of the target plane, and 
I.sub.S (i, j) is the flux emanating from other portions of the object 10 
and scattered onto the given portion of the target plane 12. 
Similar relationships may be written to define the total measured incident 
flux for each of the other area elements on the target plane 12. 
From the relationship given in Equation (1) it may be seen that the radiant 
flux emanating from a predetermined portion of the object 10 and incident 
on the corresponding area element of the target plane 12 may be 
represented (for the area element 12-i) 
##EQU2## 
In accordance with this invention, the apparatus A includes means generally 
indicated at 30 (FIG. 1B) is provided for correcting a signal 
representative of the total radiant flux incident on a given area element 
to eliminate from that signal the effects of flux incident on the 
corresponding area element due to scattering. The correcting means 30 is 
associated with the photodetector arrangement 18 and is operative to 
correct the signals on the lines 20 to generate a signal on each one of an 
array of output lines 32 that represents the radiant flux incident on a 
given area element on the target plane 12 emanating only from the portion 
of the object 10 corresponding to that area element. The correcting means 
30 may be implemented in a hardwired form comprised of discrete 
components, in an integrated circuit and/or hybrid circuit form, or by an 
appropriately programmed digital computer. 
The correcting means 30 includes an array of summing elements 34-1 to 34-N. 
Each summing element 34 corresponds to one of the area elements 12. As is 
seen in FIG. 1B each summing element 34 is connected to each of the output 
lines 20 from the detectors 18 by an array of lines 22. Each summing 
element 34 has associated therewith an array of N weighting elements 36-1 
through 36-N. Each one of the array of weighting elements 36 associated 
with a given summing element 34 is connected respectively in each one of 
the lines 22 that link that given summing element to the lines 20 from the 
detectors 18. 
Each weighting element applies a predetermined weighting factor W(x,y) to 
the signal on the line 22 in which it is connected. Each weighting factor 
W(x,y) is, in turn, functionally related to a predetermined response 
matrix R formed of a predetermined number of response coefficients R(x,y). 
The functional relationship between the response matrix R and the 
weighting factor W(x,y) may be indicated by the notation W(x,y)=f.sub.x,y 
(R). A response coefficient R(x,y) represents a quantification of the 
degree to which a given one of the area elements, (for example, the x-th 
area element) is affected by flux emanating a portion of the object 
corresponding to another area element, (e.g., the portion corresponding to 
the y-th area element) that is, due to scattering, incident on the x-th 
area element. Each response coefficient R(x,y) is determined in a manner 
to be described using a predetermined test object. Each response 
coefficient R(x,y) is a ratio representing a property of the imaging 
apparatus that is not dependent upon the object being observed or the test 
object used to calibrate the apparatus. 
From the response coefficient R(x,y) the value of each of the weighting 
factors W(x,y) is calculated in accordance with any of various methods, 
all to be described. As will be seen the methods differ in their degree of 
precision and in their applicability to the varying scattering 
characteristics that may be present in different apparatus. Thus, the 
weighting factor W(x,y) for the weighting elements will vary, in 
accordance with the method utilized to calculate the same. 
Deriving the weighting factors from the response coefficients, which in 
turn are ratios representative of apparatus properties, will insure that 
the appropriate correction is applied to each signal even though the 
intensities of the fluxes encountered during actual use of the apparatus 
differ from the flux intensities used to calibrate the apparatus and 
produce the response coefficients. 
The signals on all of the lines 22, each weighted by the appropriate 
weighting factor, are applied either additively or subtractively to each 
of the summing elements 34. The cooperative association of the summing 
elements 34 and the weighting elements 36 forms a parallel processing 
network (implemented in any manner as set forth above) that is operative 
to produce on each signal line 32 a signal that is representative of the 
total incident radiant flux on each area element of the target plane 12, 
corrected for the effects of scattering. 
METHOD OF PRODUCING THE RESPONSE COEFFICIENTS 
To produce the response coefficients R(x,y), the apparatus A is calibrated 
using a test object. A preferred test object includes a plurality of flux 
sources, the number of flux sources corresponding to the number of 
detectors in the photodetector arrangement 18. Preferably, the test object 
is selected to generally cover the entire field that confronts or would 
confront an imaging apparatus in accordance with the present invention 
when the apparatus is in use. The flux sources are illuminated singly, in 
some predetermined order, and the response signal on each line 20 from 
each detector 18 is monitored. Following such a procedure, a table may be 
produced. The table may conveniently take a matrix form. The matrix 
contains the response of each detector 18 to an individual flux source. 
The response coefficients are functionally related to the response signals. 
In accordance with the preferred technique for producing the response 
coefficients, the values of the detector responses for a given active flux 
source are divided by a quantity representative of the true flux emanating 
from the given active flux source. These quotients, one for each detector 
and all related to a single flux source, appear in a single column of a 
response coefficient matrix. Repeating this procedure for each other 
sequentially activated flux source yields the other columns of the 
response coefficient matrix. 
The quantity representative of the true flux emanating from the given 
active flux source may be established in any convenient manner. For 
example, the quantity representative of the true flux emanating from the 
given active flux source may be defined to be proportional to the sum of 
the values of the detector responses for a given active flux source. 
Alternatively, the quantity representative of the true flux emanating from 
the given active flux source may be determined by measuring the actual 
flux output from the source, assuming that such a measurement can be 
conveniently made. Other alternative methods for defining the quantity 
representative of the true flux emanating from the given active flux 
source may also be used. 
It should be appreciated that other forms of test object may be found 
useful or the response coefficients may be calculated based upon the 
design of the apparatus. However formulated (whether empirically or from 
first principles) a matrix of response coefficients representing the 
properties of the apparatus is generated. 
THE WEIGHTING FACTORS 
As noted earlier, each of the weighting factors W(x,y) is functionally 
related to at least its corresponding response coefficient R(x,y). Each of 
the weighting factors W(x,y) may also be functionally related to some 
others of the response coefficients. Depending upon the degree of 
precision required and the nature of scattering in a given apparatus 
different formulations may be specified to calculate the weighting factors 
W(x,y) to be employed in the correcting means 30. The weighting factors 
are also preferably expressed in matrix form, with the dimension of the 
weighting factor matrix corresponding to the dimension of the response 
coefficient matrix. The weighting factor matrix is shown in FIG. 1A. The 
weighting factor matrix W is an N.times.N square matrix that includes a 
main diagonal indicated in FIG. 1A by the reference character M. All of 
the other diagonals of the weighting factor matrix extend parallel to and 
in the same direction as the main diagonal M. 
In accordance with a first method, each weighting factor W(x,y) is directly 
proportional to at least its corresponding response coefficient R(x,y), 
although, as noted, it may be functionally related to others of the 
response coefficients. 
In accordance with a second method, assuming that the appropriate timing, 
switching and storing elements are provided, an even more precise 
approximation may be achieved. 
To implement the second method, as is seen in FIG. 1B, the correction means 
30 includes a plurality of storage arrangements 40-1 to 40-N. Each storage 
arrangement 40 is respectively connected to one of the summing elements 34 
by a line 41. The output of each storage arrangement 40 is carried over a 
line 42. Each storage arrangement 40 may be implemented in any convenient 
fashion, with only the storage arrangement 40-i being illustrated in 
detail, it being understood that all of the other storage arrangements 
being functionally identical to it. As seen in FIG. 1B, each storage 
arrangement 40 includes a first and a second storing element 43A, 43B that 
are alternatively connectible between the line 41 and the line 42 by a 
respective input switch 44 and an output switch 45. The switches 44 and 45 
are connected in opposition to each other so that at any given time one of 
the storing elements is connected to the line 41 and the other of the 
storing elements is connected to the line 42. Each storage arrangement 40 
may thus be seen to define storing means connected to the output of each 
of the summing elements for storing the signal therefrom. It is, of 
course, understood, that the storing means may be implemented in any of a 
variety of ways, so long as the storing function performed thereby allows 
the correcting means to provide, in temporal succession, increasing more 
accurate corrections that reduce the effects of scattered flux on the 
total radiant flux signal, until sufficient accuracy is obtained. 
In accordance with the second method the correction means 30 further 
includes a set of feedback weighting elements 46-1 to 46-N. The output 
line 42 from each storage arrangement 40 is connected to one of the 
feedback weighting elements 42. Each feedback weighting element 46 is 
operative to weight a signal on the output line 42 of its associated 
storage arrangement 40 by a predetermined feedback weighting factor Z(x). 
In the preferred instance each feedback weighting factor Z(x) is 
proportional to a corresponding response coefficient R(x,x). 
The correcting means 30 is further modified in accordance with the second 
method to further include a set of switches 48 associated with each of the 
summing elements 34. Each set of switches 48 contains (N-1) switching 
elements. Each of the switch elements 48 associated with a given one of 
the summing elements 34 is connected in the appropriate line 22 
intermediate the feedback weighting element 46 associated with that 
summing element and one of the weighting elements 36 associated with that 
summing element. The switches 48 are connected in a manner such that each 
of the lines 22 associated with a summing element 34 has a switch 48 
therein, except there is no switch 48 provided in the subscripted line 22 
corresponding to the subscript of the summing element. For example, in the 
case of the summing element 34-i, there is a switch 48 connected in each 
of the lines 22 associated with the summing element 34-i, with the 
exception that there is no switch 48 connected in the line 22-i, that is 
the line having the corresponding subscript. Thus, a given summing element 
remains connected to the total flux signal generating means to which it 
corresponds through the weighting element 36. The switch elements 48 are 
operative to disconnect a given weighting element from all of the other 
total flux signal generating means with which it is associated and to 
connect that weighting element to a predetermined one of the feedback 
weighting elements. Each switch 48 may be implemented in any convenient 
fashion, with only one of the switches 48 being illustrated in detail, it 
being understood that all of the other switches 48 are functionally 
identical to it. 
In operation, when the correcting means 30 is initialized the lines 22 are 
not opened by the switches 48 and the first storing element 43A is 
connected to the output of the summing element 34. The first order 
corrected flux signals thus appear at the output of the summing elements 
34 and are tracked and stored by the first storing element 43A in the 
storage arrangement 40 associated with that summing element 34. The 
switches 48 are asserted to disconnect selected ones of the weighting 
elements 34 from the total flux signal generating means 18 with which they 
are associated and to connect each of those weighting elements to a 
predetermined one of the feedback weighting elements 46. Once asserted a 
switch 48 remains in this position throughout subsequent steps of this 
method. Simultaneously with the opening of the lines 22 by the switches 
48, the switches 44 and 45 are asserted to permit the stored first order 
flux signals (stored by the storage element 43A) to be applied to the 
weighting elements 46 while the outputs of the summing elements 34 are 
simultaneously applied to the second storing elements 43B. The outputs of 
the summing elements 34 yield the second order corrected flux signals, 
which are tracked and stored by the second storing elements 43B. A 
reversal of the positions of the switches 44 and 45 applies the stored 
second order flux signals (stored in the second storage element 43B) to 
the feedback weighting elements 46. The outputs of the summing elements 34 
then yield the third order corrected flux signals. In this fashion the 
connections of the storing elements are cycled by repeated operation of 
the switches 44 and 45 until the outputs of the summing elements 34 
provide a sufficiently accurate representation of the radiant flux 
incident on a given area element in the target plane 12 emanating only 
from the portion of the object 10 corresponding to that area element. 
As with the implementation of all of the methods described in this 
application, the switching, timing and storing and feedback elements may 
be implemented by an appropriately programmed digital computer. 
In accordance with a third method, in addition to each element in the 
matrix of weighting factors being functionally related to the 
corresponding response coefficient, the values of the weighting factors 
along the main diagonal M of the weighting factor matrix are equal, and 
the values along each of the other parallel diagonals are also equal, so 
that each diagonal contains a sequence of identical values that might (or 
might not) be equal to the values in one of the other diagonals. Another 
way of visualizing it is that all rows are right-shifted replicas of the 
first row, each successive row one step to the right, with the exception 
that new values must be added at the left to fill the space made available 
as the rows are shifted to the right. Values on the right disappear as the 
shifting pushes them beyond the places available. A matrix such as this is 
said to be "Toeplitz". A Toeplitz weight matrix might be most useful to 
correct for flux scattering in an apparatus in which the amount of flux 
scattered from each portion of the object to a given target plane area 
element depends on the distance and direction from the given target plane 
element to the target plane element corresponding to the portion of the 
object. 
In accordance with a fourth method, the value of a given element on the 
main diagonal M of the weighting factor matrix is equal to or 
substantially equal to a first value, and, in addition, the values of all 
the other elements in the column of the weighting factor matrix containing 
that element are equal or substantially equal to a second value. Each of 
the columns may be different from all of the others having unique first 
and second values. The matrix is thus not Toeplitz. 
Finally in a fifth method, the weighting factor matrix is again Toeplitz, 
with the values of the elements on the main diagonal being equal or 
substantially equal to a first value. In addition, the values of all of 
the other elements in all of the columns of the weighting factor matrix 
are equal or substantially equal to a second value. 
In accordance with a sixth method, weighting factors are the values of a 
matrix that is the inverse of the response coefficient matrix. This may be 
cumbersome to implement, or inverting the response matrix may involve 
numerical difficulties. Sometimes, when numerical difficulties arise, 
pseudoinverse techniques may be employed. Pseudoinverse techniques are 
disclosed in Abbott, "Regression and the Moore-Penrose Pseudoinverse", 
Academic Press, New York (1972). 
EQU -o-0-o- 
The present invention is believed to find particular utility when 
implemented in the environment of a densitometer apparatus generally 
indicated by reference character 50 in the stylized schematic 
representation of FIG. 2. The densitometer 50 generally corresponds to the 
apparatus A of FIG. 1. It is the function of this densitometer apparatus 
50 to measure the optical density at a plurality of specifically localized 
areas or pixel locations of a transparency F that is inserted into the 
apparatus 50. 
The densitometer 50 includes a light box 52 in which is disposed a light 
source 14', analogous to the source 14 in FIG. 1. The box 52 is vented and 
cooled as appropriate. The upper surface of the light box 52 is defined by 
a transparent diffuser plate 58. Disposed over the plate 58 is a mask 60 
having a slit 60S therein, provided for a purpose to be made clearer 
herein. 
A light-tight enclosing shroud 62 is mounted to the light box 52. Access to 
the interior of the shroud 62 is afforded through a suitable door 64. A 
support member for the transparency, for example, in the form of a 
transparent glass platen 68, is mounted on suitable rollers 70 within the 
shroud 62. The platen 68 defines the support surface on which the 
transparency F, corresponding to the object 10 in FIG. 1, may be mounted. 
The platen 68 is mounted for reciprocal rectilinear movement in the 
directions indicated by the arrows 74. A suitable roller drive arrangement 
is, of course, provided to effect relative movement between the 
transparency F on the platen 68 and the mask 60. It should be understood 
that the term support member as used herein is meant to encompass any 
other arrangement for supporting the transparency, including a rotating 
drum with a window or a frame-like member supporting the transparency only 
at its edges. 
A lens 76 is supported on a suitable lens bracket 78 that is secured in any 
convenient fashion in the upper portion of the shroud 62. Disposed a 
predetermined focal distance above the lens 76 is an array of detectors 
18' that extend in a linear fashion normal to the plane of FIG. 2. It 
should be apparent that the array of detectors 18' corresponds to the 
photodetector arrangement 18 in FIG. 1A. It should also be readily 
apparent from the foregoing that the receptor surface 12' of the detectors 
18' in the array corresponds to the target surface 12 of FIG. 1A. 
Throughout the remainder of the discussion the area elements of the target 
plane 12 and the photodetectors 18 associated with those area elements may 
be viewed as merged into the detectors 18'. The output lines 20' from the 
detectors 18' are connected to the correcting means 30, which may be 
physically housed in the shroud 62 or in any other convenient location. It 
should be understood that the signals on the output lines 20' may be 
multiplexed over a single line such that at a given instant of time the 
single line may correspond to one of the lines depicted as 20'. 
As may be understood from FIGS. 2 and 3, as relative motion occurs between 
the platen 68 with the transparency F thereon with respect to the slit 60S 
in the mask 60 an entire scan line of the transparency F is exposed to 
illumination from the source 14' and is imaged by the lens 76 onto the 
linear array of photodetectors 18'. The exposed line of the transparency 
may be subdivided into a predetermined number of pixels 1 to N, which 
corresponds to the number of the detectors 18'. 
FIG. 4 is a highly stylized schematic drawing showing the disposition of 
flux falling upon a single pixel i and upon a single neighboring sample 
pixel j in the densitometer shown in connection with FIGS. 2 and 3. 
Density at the i-th pixel location of the transparency is defined as 
EQU D(i)=log.sub.10 [I.sub.0 (i)/I(i)] (3) 
where I.sub.0 (i) is the flux incident on the film at position i and I(i) 
is the flux transmitted at that position. The transparency transmittance, 
T(i), at location i is given by 
EQU T(i)=I(i)/I.sub.0 (i). (4) 
Thus the density D(i) may be expressed as 
EQU D(i)=log.sub.10 [1/T(i)]. (5) 
In the subsequent description, the determination of transmittance T(i) is 
detailed. The conversion of these measurements to density D(i) is readily 
understood as being accomplished via Equation (5). 
As can be seen from FIG. 4, the disposition of a spatially varying flux 
I.sub.0 (i) incident upon the i-th pixel in the exposed line of the 
transparency F causes this pixel i to transmit flux 
EQU I(i)=T(i)I.sub.0 (i). (6) 
Some of the transmitted flux, the unscattered flux I.sub.U (i), is passed 
directly to the detector at detector location i. The remainder of the 
transmitted flux, 
EQU I(i)-I.sub.U (i)=I.sub.0 (i)T(i)-I.sub.U (i), (7) 
is scattered to other locations within the densitometer. Some of it 
ultimately falls upon other detectors. 
That portion which falls upon detector element j may be defined as I.sub.S 
(j, i). FIG. 4 also shows the disposition of flux I.sub.0 (j) falling on a 
neighboring pixel location j of the transparency. 
The flux scattered from pixel location j in the transparency F to 
corresponding pixel location i in the detector plane can be expressed as a 
product: 
EQU I.sub.S (i, j)=S(i, j)T(j)I.sub.o (j), for j.noteq.i. (8) 
Here the quantity S(i, j) is the ratio of the flux that emanated from the 
sample at location j that ultimately falls upon detector location i, to 
the flux transmitted at sample location j. For i varying from 1 to N 
define the quantity 
EQU S'(i,i)=I.sub.U (i)/[T(i)I.sub.o (i)] (9) 
and, for both i and j varying from 1 to N, i.noteq.j, define 
EQU S'(i, j)=S(i, j) (10) 
so that the total flux incident upon the detector element i is 
##EQU3## 
The quantity S'(i, i) is the ratio of the unscattered flux transmitted at 
sample location i to the total flux transmitted at sample location i. 
The spatial variation of the incident flux may be incorporated with S'(i, 
j) into a new quantity 
EQU R(i, j)=S'(i, j)I.sub.o (j) (12) 
so that Equation (11) becomes 
##EQU4## 
Equation (13) may be expressed as a matrix equation 
EQU I.sub.M =RT, (14) 
in which I.sub.M is a column vector whose elements are the measured fluxes 
I.sub.M (i), R is the scanner response matrix whose elements are the R(i, 
j) defined above, and T is a column vector whose elements are the sample 
transmittances T(i). 
EQU -o-0-o- 
The method of estimating the true sample transmittances of a transparency 
is now described. 
It is the principal function of the densitometer apparatus in accordance 
with the teachings of the present invention to determine the true 
transmittances, hence the true densities, of a transparency. Generally 
speaking, transparencies having known and carefully specified 
transmittances are first introduced into the densitometer. Resulting 
fluxes are measured by the detectors 18'. From the known transmittances 
and measured fluxes the densitometer response matrix and its inverse, or 
other quantities needed for subsequent calculations, are determined. A 
transparency having unknown transmittances is then introduced into the 
densitometer. The resulting fluxes are measured. Quantities derived from 
the response matrix are then applied to the measurements of the unknown 
transparency to obtain estimates of the true transparency transmittances. 
The general method is summarized in FIG. 5. 
All of the methods to be described for use in conjunction with the 
densitometer apparatus contain an implicit or explicit correction for 
errors introduced by the detection of scattered flux. The corrections are 
described in the context of the response matrix R. This matrix compactly 
summarizes the optical characteristics of the densitometer essential to 
this description. It expresses in a manner appropriate to the description 
of a densitometer the scattering characteristics that are more generally 
described by the scattering matrices S(i, j) and S'(i, j). Following a 
brief statement of the general principle, and of the techniques used to 
determine R, six methods will be detailed that provide estimates of the 
values T(i), which may in turn be used to obtain estimates of D(i), the 
true sample densities, via Equation (5). 
In order to determine the response matrix R, a calibration transparency 
that transmits light from only one pixel at location j is introduced into 
the densitometer. Resulting fluxes are measured at all locations along the 
line of the scan. From Equation (13) we see that matrix elements R(i, j) 
are determined by 
EQU R(i, j)=I.sub.C (i)/T.sub.C (j), (15) 
where I.sub.C (i) is the value of measured flux I.sub.M (i) with the 
calibration transparency in place, and T.sub.C (j) is the transmittance 
T(j) of the calibration transparency. In order to determine all of the 
values of the matrix R, a succession of calibration transparencies must be 
introduced, each transmitting flux at just one pixel location j, the 
succession of calibration transparencies covering all possible locations 
j. 
COMPUTATION OF WEIGHTING FACTOR MATRIX 
In the first method of this invention, after determination of the values of 
matrix R, a sample transparency having unknown transmittances is 
introduced into the densitometer. Resulting fluxes I.sub.M are measured. 
Useful approximations to the true transmittances T(i) are obtained in one 
step via the equation: 
##EQU5## 
The transmittances are thus obtained from the measurements in one step by 
a process that involves a single matrix product. 
The process may be implemented in a parallel processing network, shown 
generally in FIG. 1B, and a portion of which is again shown in detail in 
FIG. 6. In this figure the weighted sum and the weighting factors W 
required to correct a single pixel i are explicitly shown as quotients of 
various R values in the response matrix. All the weighted sums and the 
entire set of weighting factors needed to correct for scattered light in a 
very small three-detector-element densitometer are shown in FIG. 7. 
Although probably too limited for practical usefulness, it is included 
here since its simplicity lends itself to a complete illustration of the 
various values of the weighting factors W in the correcting means 30. 
Of course, the process may also be implemented in serial fashion in a 
digital computer or other type of signal processor. The general idea of 
determining and storing response matrix values, and subsequently employing 
them to correct for scattering is captured in the flow diagram of FIG. 5. 
This one-step method is simple, fast, and convenient, yet performs with 
good accuracy. It represents a good compromise between accuracy and 
complexity. The one-step method is therefore the preferred method. 
If, however, additional accuracy is required, a second method may be 
employed that requires repeated application of Equation (16) of the first 
method. In the second method, the values of R and I.sub.M (i) are 
determined as in the first method. As in the first method, Equation (16) 
is applied. In this second method, however, the first resulting values 
T(i) are considered to be merely first approximations to the true 
transparency transmittances. They are used to form the products R(j, 
j)T(j) and substituted for the I.sub.M (j) in the quantity governed by the 
summation sign of Equation (16). Thus, I.sub.M (j) is replaced by a value 
that is proportional to the result T(j) of a previous application of the 
expression for T(i). (The I.sub.M (i) in the left-hand term is not 
replaced.) Applied again in this fashion, Equation (16) now yields a new 
set of T(i), a second approximation to the true transmittances. The 
weighting factors employed in the weighted sum are exactly the same ones 
as those employed in the first method described above. This iterative 
process is repeated until the differences between successive sets of T(i) 
are negligibly small. 
In the third method of this invention, the one-step method with 
shift-invarient scattering, we have noted that for some scanners the 
matrix R(i, j) may be Toeplitz or approximately Toeplitz. That is, all 
rows are shifted replicas of the first row so that Equation (16) may be 
expressed in a form containing a partial discrete convolution. 
In order to implement the third method, the matrix R and the fluxes I.sub.M 
(i) are first determined as in the first and second methods. A useful 
approximation to the true transmittances is then obtain by applying 
##EQU6## 
In this equation we have defined a new singly subscripted variable 
R(p)=R(q, r), where p=q-r. This definition is made possible by the 
Toeplitz nature of R. Since all the rows of R are similar in this case, 
any row may be used to obtain the values of the R(i). If the rows are only 
approximately identical, or if they are uncertain due to noise, average 
values of the R(i) may be used which may be computed from the shifted 
values of the R(i) obtained from all the rows. This method has an 
advantage over the previous two methods in that a full matrix of values 
for R need not be stored. Furthermore, the partial convolution in Equation 
(17) may be computed with the aid of electronic transversal filters or 
tapped delay lines. 
The fourth method of this invention, the non-shifting response method, 
makes use of the observation that some scanners detect the same amount of 
scattered flux, or approximately the same amount, at each pixel along the 
detector, and that this flux is a weighted sum of the illuminating flux 
and sample transmittances along the scan line in the sample plane. In this 
case, all the rows of the scattering matrix S are identical. 
In order to implement the fourth method of this invention, the matrix R and 
the fluxes I.sub.M (i) are first determined as in the first three methods. 
New quantities R'(j) are then computed from the matrix R via 
##EQU7## 
A useful approximation to the true transmittance is then obtained by 
applying 
##EQU8## 
For each sample scan having a new set of transmittances T(i), the 
right-hand term above is the same for all pixels i. This method is 
therefore easier to apply because it only involves subtracting a constant 
value from all the measurements I.sub.M (i)/R'(O). The constant needs to 
be redetermined for each scan line, however. 
In the fifth method of this invention, the product-free method, we note 
that the product R'(j)I.sub.M (j) in the fourth method requires 
computation that would be desirable to avoid. Accordingly, it is possible 
to make the approximation that the I.sub.M (j) need not be weighted in 
computing the sum. That is, all the weights in the weighted sum are one. 
In order to implement the fifth method of this invention, the quantities 
R'(j) and the fluxes I.sub.M (i) are obtained as in the fourth method. A 
constant a is then computed: 
##EQU9## 
Useful approximations to the true transmittances are then obtained by 
applying 
##EQU10## 
The constant a is seen to be independent of measurements I.sub.M (i) and 
therefore does not need to be re-determined for each scan line. 
Note also that Equation (14) may be solved for unknown T by finding the 
inverse of matrix R: 
EQU T=R.sup.-1 I.sub.M. (23) 
This equation suggests a sixth method. It shows that it is possible to 
determine the true sample transmittances T(i), hence the true densities 
D(i), from the measured fluxes I.sub.M (i), provided that the densitometer 
response matrix is known. Numerical methods such as Gaussian elimination 
may be used to compute R.sup.-1. Such a computation may be effected, for 
example, using a digital computer operating in accordance with any 
suitable commercially available program for computing an inverse of 
matrix, such as available from IMSL, Houston, Tex. As noted previously, 
inverting the response matrix may involve numerical difficulties and, in 
such instances a seventh method, employing pseudo-inverse techniques, as 
referenced previously herein, may be used. 
EQU -o-0-o- 
Computing the quantities to be subtracted in the above methods may involve 
excessive time and circuit complexity. Because the response matrix R 
varies slowly with position and because the correction term may be small, 
it is not always necessary to compute every element of the summations 
required. Sparse sampling techniques can thus be employed. More 
specifically, the contribution of each and every element does not need to 
be included. Samples might be either evenly spaced or random. It could be 
useful to sample most densely where the contributions are the largest. It 
is even possible to make weights implicit in the sample spacing. In such a 
method, the samples are simply added together, yet the sum computed is 
actually the required weighted sum. Interpolation methods are also useful 
with sparse sampling. 
EQU -o-0-o- 
While a densitometer has been hereinbefore disclosed as the principal 
application of the apparatus A (and method) in accordance with the present 
invention, it will be understood by anyone skilled in the art that the 
present invention applies equally as well to a light microscope and other 
light imaging systems such as a camera or a telescope where the object or 
sample is remote from the apparatus containing the invention or to imaging 
systems that employ fluxes of particles such as the electron microscope 
and scanning electron microscope. In the case of a light camera or a 
telescope, a calibration setup would be employed to determine the values 
of a matrix characteristic of the scattering. Such a setup would employ 
single point-like sources of light flux in a manner analogous to the use 
of samples having known transmittance. The electron microscope would 
likewise require use of special samples of known electron transmission or 
scattering properties in order to determine the values of the scattering 
matrix. 
Those skilled in the art, having the benefit of the teachings of the 
present invention as hereinbefore set forth, will appreciate numerous 
modifications which may be imparted thereto. It should be understood that 
such modifications are to be construed as lying within the contemplation 
of the present invention, as defined by the appended claims.