Digital image correlation using Newton-Raphson ... - Springer Link

Digital Image Correlation Using Newton-Raphson Method of Partial Differential Correction by H.A. Bruck, S.R. McNeill, M.A. Sutton and W.H. Peters III

ABSTRACT--Digital image correlation is finding wider use in the field of mechanics. One area of weakness in the current technique is the lack of available displacement gradient terms, This technique, based on a coarse-fine search method, is capable of calculating the gradients. However the speed at which it does so has prevented widespread use. Presented in this paper is the development and limited experimental verification of a method which can determine displacements and gradients using the Newton-Raphson method of partial corrections. It will be shown that this method is accurate in determining displacements and certain gradients, while using significantly less CPU time than the current coarse-fine search method.

Introduction Digital image processing is becoming increasingly important as an experimental tool. In the field of experimental mechanics, researchers are investigating the use of digital image processing for photoelasticity and moire fringe analysis, as well as direct measurements of displacements, velocities, and strains. The use of digital image processing to measure displacements of objects under loading to sub-pixel accuracy has been developed and used at the University of South Carolina. ~-3 The technique, which uses coarse-fine search to correlate images, has been found to be accurate and reliable if used properly. The technique is computationally intensive when used for measurement of displacements and displacement gradients. A new digital-image-correlation algorithm has been developed which will allow for the computation of displacements and displacement gradients using far less computation time. The algorithm is based on the Newton-Raphson method of partial differential correction. This paper presents the theory of the method and experimental verification of the theory.

Digital Image Correlation A technique has been developed for the determination of displacements and displacement gradients from video images based on digital image correlation. In this application digital image correlation is used to compare two digitized images so as to determine the deformation between images. The experimental equipment used to acquire digital images for this work is shown in Fig. 1. The system uses a standard video camera attached to a video digitizer. The digitizer transforms the image to a 512 • 512 set of numbers representing the image. Each number represents the intensity of light impinging on a small area of the camera sensor, which is referred to as a pixel. The value of each pixel is typically an eight-bit number (i.e., ranges in value from 0 to 255) with the lowest value representing black, highest value white, and values in between representing different shades of gray. Digital image correlation then becomes a job of comparing subsets of numbers between the two digital images. A typical correlation function which measures how well subsets match is S'(x,y,u,v,

1 -

Ou

au

Ox"

Oy

Ov '

Ox

~v ) _ '

Oy

v~ I F ( x , y ) * G ( x * , y*)] [I7, ( F ( x , y ) 2) * E ( G ( x * , y * ) ' ) ] ''2

(1)

where F ( x , y ) is the gray level value at coordinate (x, y ) for one image and G ( x * , y * ) is the gray level value at point ( x * , y * ) of the second image. The coordinates (x, y ) and (x*, y*) are related by the deformation which occurred between acquisition of the two images. If the motion of the object relative to the camera is parallel to the image plane, then they are related by Ou

Ou

x* = x + u + --~--x A x + --~-y A y Ov Ov y * = y + v + --~--x A X + - - - ~ H.A. Bruck is Graduate Student, S.R, McNeill (SEAl Member) is Assistant Professor, M.A. Sutton (SEAl Member) is Associate Professor, and W.H. Peters III (SEM Member) is Professor, University of South Carolina, Department of Mechanical Engineering, Columbia, SC 29208. Paper was presented at the 1988 SEAl Spring Conference on Experimental Mechanics held in Portland, OR on June 5-10. Original manuscript submitted: August 12, 1987. Final manuscript received: February 8, 1989.

(2) Ay

where u and v are the displacements for the subset centers in the x and y directions respectively. The terms AX and Ay are the distances from the subset center to point (x, y). Image correlation is performed by determining values for u , v , O u / O x , O u / O y , O v / O x , and O v / O y which

Experimental Mechanics * 261

minimize the correlation coefficient (S). In the past, the coarse-fine search technique for determining this minimum tried many possible combinations of the variables within a given range, and compared the correlation factor for each set. Because of the large number of calculations this technique required, it was usually used for determining u and v only. The Newton-Raphson method, which is being presented here, gives corrections for initial guesses of the six deformation parameters and will converge to the solution with substantially fewer calculations.

polation. The functional form for the gray level at (x*, y*), which is between pixels ( i , j ) , (i + l , j ) , ( i , j + 1), and (i + 1, j + 1), using a bicubic spline is G(x*,y*)

= aoo + a,o(X') + a2o(x') 2 + a3o(X') 3 +

+ aol(y ') + ao2(y ,)2 + ao3(Y ,)3 + a , ( x ' )

(y') +

a21(x')2(y ') + a ~ i ( x ' ) a ( y ') + aoz(y ')" + a12(x') ( y ,)2 + + a22(x')Z(y')2 + a,3';(x')'(y ')" + a 3 , ( x ' ) ' ( y ,)a

Sub-pixel Interpolation

(4)

Because of the operational characteristics of video cameras and digitization circuits, the information obtained is discrete in nature. This means that no gray level information is available between pixels. If only pixel center coordinates are used for x, y , x*, and y* in eq (1), the displacements and gradient terms would not be independent. If this were the case, the deformation parameter would always have to be chosen such that x* and y* be a digitized pixel location. The nature of most. problems to be studied is such that the displacement and gradient terms are independent. Therefore, an approximation of gray level values between pixels is needed. For the coarse-fine search method, bilinear interpolation is used. Bilinear interpolation approximates the gray level value at a point ( x * , y * ) , which is between the pixels ( i , j ) , (i + 1 , j ) , ( i , j + 1) and (i + 1 , j + 1), by

G(x*,y*)

where aoo is the gray level of pixel ( i , j ) . Using the first

= aoo + a~o(X') + a o ~ ( y ' ) + a H ( x ' ) ( y ' ) (3)

where aoo = gray level of pixel ( i , j )

a,o = gray level of pixel (i + l , j ) - aoo ao~ = gray level pf pixel ( i , j + I) - aoo a . = gray level of pixel (i + 1, j + 1) - aoo - aw - ao, The x ' and y ' are the x and y distance from pixel ( i , j ) , Figure 2 shows a gray level surface before interpolation is used. A representation of the gray level surface obtained from using the bilinear interpolation on an eight x eight set of pixels is shown in Fig. 3. In the work for this paper it was found that higher accuracy could be obtained by using bicubic spline inter-

!, , I

CkNERA

9[

.9

"

I

ISH PC/AT

Fig. 2--8 • 8 s u b s e t of unintel )olated intensity data

I

I

VAX

11/780

Fig. 1--Schematic diagram o'f digital-image-processing system

262 9 September 1989

Fig. 37--8 x 8 subset of intensity data using bilinear interpolation

order and cross derivatives at nodal locations, the remaining coefficients are determined so that all first order and cross derivatives are continuous between adjacent surfaces. A bicubic representation of an eight x eight set of pixels is shown in Fig. 4.

U V

0u

p~ =

Newton.Raphson Method of Partial Differential Corrections

Ov

As stated earlier, the goal of digital image correlation is to find the six deformation parameters, u, v, a u / O x , a u / a y , a v / S x , and O v / a y , that minimize the correlation function [eq (1)]. The technique, which uses the Newton-Raphson method of partial differential corrections, allows for the determination of these parameters using less computation time than in the previously used coarse-fine search method. The Newton-Raphson method is based on the calculation of correction terms which improve initial guesses. The correction for guess i is given by

Ov V(Pi) is the Jacobian matrix. Each term in the Jacobian is the derivative of the correlation function evaluated at guess L The Jacobian matrix is OS Ou OS Ov

(5)

APt = - H - ' ( P I ) * V (Pi)

OS Ou

where V(P,) =

TABLE 1--AVERAGE CPU TIME* (IN SECONDS) per subset Bicubic

Test Performed Translation Rotation

aS Ou

(7)

as av

Column

50 x 50 Subsets

20 x 20 Subsets

50 x 50 Subsets

20 x 20 Subsets

20.72 55.36

5.95 6.63

54.21 n/a

8.28 n/a

* F o r a Vax 11 / 780 m i n i c o m p u t e r

(6)

Ou

as av

a (-T7) H(Pi) is the Hessian matrix which are the second derivatives of the correlation function

Fig. 4 - - 8 x 8 subset of intensity data using bicubic spline interpolation

Experimental Mechanics 9 263

H(Pl) = O2S OuOu

O'S OuOv

O~S OvOu

O~S OvOv

02S OU

O'S OV

o(-~-x)OU

a~S OU +,+ (-~;)

OU

a2S

OU ) +(-~r Ov

Ou

a~S OV ) ~ (-~;

Ov

OU

02S OU ) +.~ (--~-y

+u+(~;)

O2S

o (-~r) o .

o

02S au

a~S

o (-~-) o (-~r)

Ov

a~S Ou

Ou

Ou

a~S OV

Ov

Ou

Ov

(~-~/) a,, +(-~; ) Ov o (-~;)o ( 9 a~S

o (--~-~) au

)

a2S

Ov o(-~-) o (-~--~) ~(-~-) o (-~--~) Ov

Ou

Ov

Ov

a~S Ov

a2S

au

(-~-~)

ov

O (9 Ou a y - ~0 Ov

a~S

02S Ov

Ov ~(-~-~) Ov o(-~-~)

Ov

Ov

on~S

a2S Ov

a2S

a (-~y) av

o2s OV

Ou

a2S

O2S On

a~S 01] a2S

Ov

a2S Ov Ov a (--~-~)+ (-~-~)

Ou

o2s OV aua (-~y

~+ (~-~)

a2S Ou

Ou

02S [Jy

Ov (8)

Start

{

>. . . . .

Read data: (1) arrays [A] and [B] (2) size and center of selected subset of [A] (3) estimate displacements U and V (4) displacement check range

I

Smin=Sinitial

I

Calculate (x~) and B - l ( x l ) "

{

{

/IX

I v xi=-H-l(xi)* (xi) {

I

{ Use coarse-fine search to find optimum i n i t i a l integer displacements in displacement range.

v

Xi+l=Xi+ x i

I

/1\

I IYes

Calculate xi+ 1 Correlation Coefficient,Si+ 1

{

I

/I\

I

{No

lNo V

Si+l-S i < oscillation limit

{ ICheck to see I ~

---~

lif No{ V

x I and x 2

Fig. 5-- Flow chart for Newton-Raphson program

/{\

IYes

I

>--->

x I or x21

{inverted

I Yes{ #

V ~

l Nol

'{

No

..... > counter less than 10?

> divergence[ limit {

increment ................ { i n i t i a l

264 9 September 1989

Is the oscillation

l v

I Yes V

/IX

[

I v

C a l c u l a t e bicubic s p l i n e coefficients.

I n i t i a l i z e displacement gradients to 0 in the parameter vector,p(xi).

increment the oscillation counter

No{

lYes

~->lSi+l/Si > 1/divergence limit{

lYes > ...... >

lstopl< ........

Output optimized parameter vector,

P(Xoptimlzed)