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Patent US6996288 - Method of calculating shading correction coefficients of imaging systems ... - Google PatentsSearch Images Maps Play YouTube News Gmail Drive More »Sign inPatentsA method of calculating shading correction coefficients for an imaging system from a non-uniform and unknown calibration standard is disclosed. The standard may be of two or more dimensions and divided into several parts. The standard is imaged in an initial position to obtain a set of initial image...http://www.google.com/patents/US6996288?utm_source=gb-gplus-sharePatent US6996288 - Method of calculating shading correction coefficients of imaging systems from non-uniform and unknown calibration standardsAdvanced Patent SearchPublication numberUS6996288 B2Publication typeGrantApplication numberUS 09/964,607Publication dateFeb 7, 2006Filing dateSep 28, 2001Priority dateSep 28, 2000Fee statusLapsedAlso published asUS20020054712Publication number09964607, 964607, US 6996288 B2, US 6996288B2, US-B2-6996288, US6996288 B2, US6996288B2InventorsGang SunOriginal AssigneeGang SunExport CitationBiBTeX, EndNote, RefManPatent Citations (6), Referenced by (10), Classifications (10), Legal Events (4) External Links: USPTO, USPTO Assignment, EspacenetMethod of calculating shading correction coefficients of imaging systems from non-uniform and unknown calibration standards
Typically, an imaging system is used to image and/or to analyze objects in a continuous fashion. That is, every point in the field of view is of interest. In this case, pixel-by-pixel shading corrections should be performed. FIG. 1 illustrates the mapping between object plane and image sensor plane. The rectangular area 200 in the object plane of an imaging system 250 is projected to the image sensor 202 of the imaging system 250. The area 200 is also referred to as the field of view of the system 250. In the example shown in FIG. 1, the image sensor 202 consists of 48 sensor elements 206 (or pixels) in an 8�6 array, denoted as pxl(x, y), x=1, . . . , 8 and y=1, . . . , 6. Correspondingly, there are 8�6 divisions 205, denoted as Rp(x, y), x=1, . . . , 8 and y=1, . . . , 6, in the field of view 200 in the object plane. The region Rp(x, y) is projected and focused to sensor element pxl(x, y) through the lens 201 of the imaging system 250. In this example, there will be 8�6 shading correction coefficients, each of which is an integrated (average) coefficient of the corresponding region.
If the image sensor of an imaging system has NP by MP elements, and ICP(x, y) is the image pixel intensity of pixel pxl(x, y) of an uniform calibration standard, the pixel-by-pixel shading error SEP(x, y) is defined in EQ. 1. Typically, shading errors are normalized so that the average of all errors equals to one. SE P ( x , y ) = IC P ( x , y ) 1 N P � M P ∑ i = 1 N P ∑ j = 1 M P IC P ( i , j ) For x = 1 , … , N P and y = 1 , … , M P EQ . 1 Typically, an imaging system is a linear device. That is, the raw pixel gray levels of the imaging system have a linear relationship with the input signal. For example, when imaging an illuminant object, the input signal is the self-emitted light intensity from the object. In the linear case, image intensity ICP(x, y) is the raw pixel gray-level of the pixel pxl(x, y) minus a stored background value of the corresponding pixel. In the non-linear case, the non-linear relationship first is modeled, and then this non-linear model is used in calculating the imaging intensity ICP(x, y) from the raw gray-levels so that the image intensity ICP(x, y) is linear with the input signal. If the image intensity is not linear to the input signal or the background values are not properly removed, the calculated shading error will not be accurate. In such a system, a skilled person will be capable of adjusting the image intensity values so that they have a linear relationship with the input signal.
In the example shown in FIG. 2, there are 3�2 regions 215, denoted as R(x, y): x=1, 2, 3 and y=1, 2. Each of these regions projects to its corresponding pixel set 216, denoted as P(x, y): x=1, 2, 3 and y=1, 2, in the imaging sensor 202. Each pixel set in this example contains four pixels. For example, P(1, 1), corresponding to R(1, 1), consists of: {pxl(1, 1), pxl(2, 1), pxl(1, 2), pxl(2, 2)}. The generalized shading correction has 3�2 coefficients, and each coefficient corresponds to an area of 4 pixels. Of course, the regions do not have to be in a regular 2-dimensional matrix form. They can be in any form. For the convenience of discussion, the regular 2-dimensional matrix form is used in this document. The calibration standard may be 1-dimensional or 3-dimensional or may have a non-rectangular 2-dimensional form.
Given a calibration standard with uniform regions R(x, y): x=1, . . . , N and y=1, . . . , M, the shading error SE(x, y) for the region R(x, y) (or pixel set P(x, y)) is defined in EQ. 2. SE ( x , y ) = IC ( x , y ) 1 N � M ∑ i = 1 N ∑ j = 1 M IC ( i , j ) For x = 1 , … , N and y = 1 , … , M EQ . 2 where: IC(x, y) is the image intensity of region R(x, y). It is calculated by averaging the image pixel intensities of all the pixels corresponding to region R(x, y).
The present invention provides a method for calculating shading correction coefficients for an imaging system, using a calibration standard with N�M regions. Reference is first made to FIG. 3, which illustrates a first calibration standard 10 according to the present invention. The described calibration standard 10 consists of N�M (N columns and M rows) calibration regions 5, which are referred to as R(x, y). Calibration standard 10 is divided into four parts. The variables x and y are the array index. x takes values form 1 to N, denoted as x=1, . . . , N, and y takes values from 1 to M, denoted as y=1, . . . , M. The top-left corner 4 consists of one calibration region 5 at its center; the top-row part 2 consists of N−1 calibration regions 5; the left-column part 3 consists of M−1 calibration regions 5, and bottom-right part 1 consists of (N−1)�(M−1) calibration regions 5. In this exemplary embodiment, N>1 and M>1. This is referred to as the 2D (2-dimensional) case. In the 2D case, both N and M have to be greater than one. If either N or M is equal to one, it becomes a one dimensional case, which is described later later.
W FOV =a�N and H FOV =b�M
IC O(x, y)=G�SE(x, y)�VC(x, y) EQ. 4
Content(R H(x, y))=Content(R(Mod(x+1, N), y)) EQ. 6 for x=1, . . ., N and y=1, . . ., M The function Mod (q, d) in EQ. 6 and subsequent equations is defined as: Mod ( q , d ) = { ( q + m � d - 1 ) % d + 1 if q and d are integers and q < 1 and d ≥ 1 ( q - 1 ) % d + 1 if q and d are integers and q ≥ 1 and d ≥ 1 Invalid Otherwise where: % is a remainder operator (e.g., 1%3=10%3=1).
Given integer q and d, m is the smallest positive integer such that q+m�d≧1.
for x=1, . . . , N and y=1, . . . , M Multiplying 2 to both sides of EQ. 5, multiplying−1 to both sides of EQ. 8 and EQ. 9, and then adding the three new equations together, i.e., 2�(EQ. 5)−(EQ. 8)−(EQ. 9), we have:
ICV(x, y)=2�VC L(x, y)−VC L(Mod(x+1, N), y)−VC L(x, Mod(y+1, M)) EQ. 10
ICV(x, y)=2�IC L O(x, y)−IC L H(x, y)−IC L V(x, y)
ICS(x, y)=2�SE L(x, y)−SE L(Mod(x+1, N), y)−SE L(x, Mod(y+1, M)) EQ. 11
where: A is a (N�M)�(N�M) matrix, i.e., it has N�M rows and N�M columns. It can be represented by a partitioned matrix, as defined below. The matrix E is a N�N matrix as defined below, the matrix I is a N�N identity matrix and matrix 0 is a N�N zero matrix (all coefficients are 0). A = [ E - I 0 . 0 0 0 0 E - I . 0 0 0 0 0 E . 0 0 0 . . . . . . . 0 0 0 . E - I 0 0 0 0 . 0 E - I - I 0 0 . 0 0 E ] NxM x NxM E = [ 2 - 1 . 0 0 0 2 . 0 0 . . . . . 0 0 . 2 - 1 - 1 0 . 0 2 ] NxN I = [ 1 0 . 0 0 0 1 . 0 0 . . . . . 0 0 . 1 0 0 0 . 0 1 ] NxN ICS = ⌊ ICS ( 1 , 1 ) ICS ( 2 , 1 ) . ICS ( N , 1 ) . . ICS ( 1 , M ) ICS ( 2 , M ) . ICS ( N , M ) ⌋ 1 � ( N � M ) SE L = ⌊ SE L ( 1 , 1 ) SE L ( 2 , 1 ) . SE L ( N , 1 ) . . SE L ( 1 , M ) SE L ( 2 , M ) . SE L ( N , M ) ⌋ 1 � ( N � M ) Using standard mathematical techniques, it can be proved that the rank of matrix A is N�M−1. Therefore the solution of SEL(x, y) is not unique. However, since the rank of matrix A is N�M−1, SEL(x, y) can be resolved by arbitrarily specifying only one of the values (e.g., SEL(1, 1) is specified).
Taking the exponential transform (inverse of the log transform) and normalization (as in EQ. 2), the corresponding shading error SE(x, y) is defined as: SE ( x , y ) = ⅇ SEO ( x , y , ax , ay , K ) 1 N � M ∑ i = 1 N ∑ j = 1 M ⅇ SEO ( i , j , ax , ay , K ) for x = 1 , … , N and y = 1 , … , M EQ . 13 Using standard mathematical techniques, it can be proved that the difference between any given two solutions SEO(x, y, ax1, ay1, K1) and SEO(x, y, ax2, ay2, K2) of EQ. 12 is constant, for a given set of measured image intensities (ICL O(x, y), ICL H(x, y) and ICL V(x, y)). That is:
This give the following equation: ⅇ SEO ( x , y , ax2 , ay2 , K2 ) 1 N � M ∑ i = 1 N ∑ j = 1 M ⅇ SEO ( i , j , ax2 , ay2 , K2 ) = ⅇ SEO ( x , y , ax1 , ay1 , K1 ) + F 1 N � M ∑ i = 1 N ∑ j = 1 M ⅇ SEO ( i , j , ax1 , ay1 , K1 ) + F = ⅇ SEO ( x , y , ax1 , ay1 , K1 ) 1 N � M ∑ i = 1 N ∑ j = 1 M ⅇ SEO ( i , j , ax1 , ay1 , K1 ) That is, the shading error SE(x, y) defined in EQ. 13 is unique, regardless of the values of ax, ay and K.
As an example, select ax=N, ay=M and K=1 as the solution for calculating the shading error. This solution is referred to as intermediate shading error and denoted as S1(x, y), i.e., S1(x, y)=SEO(x, y, N, M, 1) , whose solution is defined by EQ. 14A and EQ. 14B. S1 ( N , M ) = 1 EQ.��14A ⌊ ICS ( 1 , 1 ) ICS ( 2 , 1 ) . ICS ( N , 1 ) . . ICS ( N , M - 1 ) + 1 ICS ( 1 , M ) ICS ( 2 , M ) . ICS ( N - 1 , M ) + 1 ⌋ 1 � ( N � M - 1 ) = A1 ⌊ S1 ( 1 , 1 ) S1 ( 2 , 1 ) . S1 ( N , 1 ) . . S1 ( 1 , M ) S1 ( 2 , M ) . S1 ( N - 1 , M ) ⌋ 1 � ( N � M - 1 ) EQ.��14B where: Matrix A1 is the top-left (N�M−1)�(N�M−1) sub-matrix of A defined in EQ. 12. That is:
EQ. 14B can be solved by directly calculating the inverse matrix A1 −1, as defined in EQ. 15. It can also be solved by standard iterative methods. ⌊ S1 ( 1 , 1 ) S1 ( 2 , 1 ) . S1 ( N , 1 ) . . S1 ( 1 , M ) S1 ( 2 , M ) . S1 ( N - 1 , M ) ⌋ 1 � ( N � M - 1 ) = A1 - 1 ⌊ ICS ( 1 , 1 ) ICS ( 2 , 1 ) . ICS ( N , 1 ) . . ICS ( N , M - 1 ) + 1 ICS ( 1 , M ) ICS ( 2 , M ) . ICS ( N - 1 , M ) + 1 ⌋ 1 � ( N � M - 1 ) EQ . 15 The shading error SE(x, y) can then be calculated from the intermediate shading error by EQ. 16: SE ( x , y ) = ⅇ S1 ( x , y ) 1 N � M ∑ i = 1 N ∑ j = 1 M ⅇ S1 ( i , j ) for x = 1 , … , N and y = 1 , … , M EQ . 16 Once shading error SE(x, y) is determined, the shading correction coefficients SC(x, y) can be determined from EQ. 3.
The region signal intensity VC(x, y) can also be determined from EQ. 4, if the gain of the imaging system is known. That is: VC ( x , y ) = IC 0 ( x , y ) G � SE ( x , y ) EQ . 17 If only the relative region signal intensity (VCR(x, y)) is required, it can be calculated without knowledge of the system gain. That is: VC R ( x , y ) = IC 0 ( x , y ) / SE ( x , y ) 1 N � M ∑ i = 1 N ∑ j = 1 M IC 0 ( i , j ) / SE ( i , j ) EQ . 18 FIG. 3 shows calibration standard 10 which has a top-left configuration. There are many other configurations that can be used to calculate shading errors and the region signal intensities. For example, FIG. 8 shows a calibration standard 60 with a bottom-right configuration. As with the top-left calibration standard 10 described in relation to FIG. 3, this bottom-right calibration standard 60 has four detachable parts with N�M uniformly arranged calibration regions 5. The bottom-right corner 24 consists of one calibration region 5 at the center; the bottom-row part 22 consists of N−1 calibration regions 5; the right-column part 23 consists of M−1 calibration regions 5, and the part 21 consists of (N−1)�(M−1) calibration regions 5. The calibration regions 5 are referred to as RBR(x, y).
ICS BR(x, y)=2�SE L(x, y)−SE L(Mod(x−1, N), y)−SE L(x, Mod(y−1, M)) EQ. 21
The corner part (e.g., part 4 in FIG. 3) of calibration standard 10 (FIG. 3) consists of only one calibration region 5. Referring to FIG. 12, a calibration standard may alternatively have more than one region in the corner part. FIG. 12 shows a calibration standard 30 with a top-left configuration and with 2 calibration regions in the corner part 34. The top-row part 32 consists of N-2 calibration regions; the left-column part 33 consists of 2�(M−1) calibration regions, and bottom-right part 31 consists of (N-2)�(M−1) calibration regions. The horizontally shifted-and-rotated arrangement corresponding to calibration standard 30 will require shifting TWO columns. Similar to the deduction of EQ. 11, we can also derive an equation to solve the shading error SE(x, y) from the image intensities of the initial calibration standard, a horizontally shifted-and-rotated arrangement, and a vertically shifted-and-rotated arrangement. However, unlike EQ. 11, solving this equation requires specifying TWO values of SE(x, y), e.g., SE(1, 1) and SE(2, 1) are given. Therefore, more than one initial value of the shading error SE(x, y) needs to be defined if the corner parts of a calibration standard consists of more than one calibration region. In order to obtain the correct shading correction coefficients for an imaging system using a calibration standard 30, the ratio of the shading errors for the two specified regions must be known a priori.
In certain applications, fabrication of a physically detachable calibration standard may not be convenient or possible. In this case, the shift-and-rotation can be achieved by performing two shifts of a calibration standard. FIG. 13 shows calibration standard 70 with N�M calibration regions RS(x, y) with an imaginary top-left configuration. The dotted lines 81 and 82 in FIG. 13 represent the imaginary divisions to the standard 70 into a top-left configuration, and imaginary parts 71, 72, 73 and 74 correspond to part 1, 2, 3 and 4 in FIG. 3. It is to be noted that the use of the word “imaginary” here indicates only that calibration standard 70 cannot be conveniently detached to separate parts 71, 72, 73 and 74.
IC O(x, y)={overscore (IC)} O(x, y)�(1+e P O(x, y))+e A O(x, y)
IC H(x, y)={overscore (IC)} H(x, y)�(1+e P H(x, y))+e A H(x, y)
IC V(x, y)={overscore (IC)} V(x, y)�(1+e P V(x, y))+e A V(x, y) EQ. 22
SE(x, y)={overscore (SE)}(x, y)�(1+e SE(x, y)) EQ. 23
As an example of averaging multiple calculation from the same calibration standard, SE(1)(x, y) and SE(2)(x, y) may be defined as the results of two acquisitions and calculations of shading errors of an imaging system from a calibration standard with a top-left configuration. The averaged shading error can be defined as: SE ( x , y ) = SE ( 1 ) ( x , y ) + SE ( 2 ) ( x , y ) 2 EQ . 24 As an example of averaging different calibration standards, SETLL(x, y) may be defined as the results from a top-left calibration standard and SEBR(x, y) may be defined as the results from a bottom-right calibration standard. A simple averaged shading error can be defined as: SE ( x , y ) = SE TL ( x , y ) + SE BR ( x , y ) 2 EQ . 25 The statistics (such as mean and variance) of calculation error eSE(x, y) is position-dependent, even if such statistics of measurement error is position-independent (e.g., all eP O(x, y), eP H(x, y), eP V(x, y), eA O(x, y), ea H(x, y) and eA V(x, y) are μ=0 and σ=0.1). By using standard statistical methods and knowledge of measurement error, the error mean μESE(x, y) and error standard deviation σESE(x, y) of eSE(x, y) can be estimated for given measurement errors (e.g., Gaussian noise with μ=0 and σ=1 for the additive measurement errors and Gaussian noise with μ=0 and σ=0.05 for the proportional measurement errors). Using the error statistics μESE(x, y) and σESE(x, y), weighting factors can be derived. Instead of using a simple averaging, a weighted averaging can be performed. SE ( x , y ) = w TL ( x , y ) � SE TL ( x , y ) + w BR ( x , y ) � SE BR ( x , y ) w TL ( x , y ) + w BR ( x , y ) EQ . 26 where: wTL(x, y) and wBR(x, y) are the weighting factors for the top-left configuration and bottom-right configuration respectively.
IC 1 O(x)=G 1 �SE 1(x)�VC 1(x) for x=1, . . . , N EQ. 29
Similar to the 2D case, SE1(x) is defined as the solution (intermediate shading error) of EQ. 32 when SE1 L(N)=1, and the shading errors, defined in EQ. 33, are uniquely defined. SE1 ( x ) = ⅇ S11 ( x ) 1 N ∑ i = 1 N ⅇ S11 ( i ) for x = 1 , … , N EQ . 33 Once the shading error is calculated, the shading correction coefficients SC1(x), signal intensity VC1(x), and relative signal intensity VC1 R(x) can be calculated. SC1 ( x ) = 1 SE1 ( x ) for x = 1 , … , N EQ . 34 VC1 ( x ) = IC1 0 ( x ) G1 � SE1 ( x ) for x = 1 , … , N EQ . 35 VC1 R ( x ) = IC1 0 ( x ) / SE1 ( x ) 1 N ∑ i = 1 N IC1 0 ( i ) / SE1 ( i ) for x = 1 , … , N EQ . 36 As with the 2D case (FIG. 14), the shading errors can also be calculated from a right configuration as well. FIG. 16 and FIG. 18 show mat the right configuration is symmetrical to the left configuration. That is, the initial setup of a right calibration standard is the shifted-and-rotated arrangement 110 of a left calibration standard and vice versa. Therefore, there is no difference between a left configuration and a right configuration, except in the order they are given to an imaging system.
We have presented both 1D and 2D cases. The same shift-and-rotation principle can be extended to the 3D (3-dimensional) domain as well. In the 3D domain, the field of view is 3 dimensional, and a calibration region becomes a calibration cube. The extension of a 2D calibration standard with a top-left configuration is a 3D calibration standard with a top-left-front configuration as shown in FIG. 19. The calibration standard 400 has N�M�L calibration cubes and has 8 detachable parts. Part 401 consists of (N−1)�(M−1)�(L−1) calibration cubes; part 402 has (N−1)�(L−1) calibration cubes; part 403, which is not visible (blocked by parts 406 and 408) in this figure, consists of (M−1)�(L−1) calibration cubes; part 404 consists of 1 calibration cube; part 405 consists of (N−1)�(M−1) calibration cubes; part 406 consists of (L−1) calibration cubes; part 407 consists of (N−1) calibration cubes; and part 408 consists of (M−1) calibration cubes. The configuration is named as top-left-front because of the position of part 404 is at top-left-front. A horizontally shifted-and-rotated arrangement of the calibration standard 400 will be done by shifting the entire standard 400 to the left by one column and then moving parts 403, 404, 406 and 408 to the right side of the parts 401, 407, 402 and 405 respectively. Similarly, the vertically shifted-and-rotated arrangement and the front-to-back shifted-and-rotated arrangement will be form in a similar manner.
Reference is again made to FIG. 3. The method of the present invention allows the log-transformed shading error (SEL in EQ. 11 or EQ. 12) of each region 5 to be represented as a factor of the log-transformed shading errors of other regions 5. An intermediate shading error for each region may be calculated using standard matrix techniques, if the shading error for one region is specified (assuming that corner part 4 has a single region so that matrix A in EQ. 12 has a rank of N�M−1). The shading error may then be calculated as the normalized value of the intermediate shading error (EQ. 13 or EQ. 16). The effect of the selection in the intermediate shading errors is eliminated by the normalization.
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