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Appendix A

LEVENBERG-MARQUARDT ALGORITHM

Given a vector relation y = f(x) where x and y can have different dimen-

sions and an observation y, the objective is to find the vector x which best

satisfies the given relation. More precisely, it is to find the vector x satisfying

y = f(x) + e for which ‖e‖ is minimal.

A.1 Newton iteration

Newton’s approach starts from an initial value x0 and refines this value using

the assumption that f is locally linear. A first order approximation of f(x0+∆)

yields:

f(x0 + ∆) = f(x0) + J∆, (A.1)

where J is the Jacobian matrix and ∆ is a small displacement.

Under these assumptions minimizing e = e0 − J∆ can be solved through

linear least-squares. A simple derivation yields

JTJ∆ = JT e. (A.2)

This equation is called the normal equation. The solution to the problem

is found by starting from an initial solution and refining it based on successive

iterations

xi+1 = xi + ∆i, (A.3)

where ∆i is the solution of the normal equation A.2 evaluated at xi.

One hopes that this algorithm will converge to the desired solution, but it

could also end up in a local minimum or not converge at all. This depends a

lot on the initial value xi.

159

A.2 Levenberg-Marquardt iteration

The Levenberg-Marquardt iteration is a variation on the Newton iteration.

The normal equations N∆ = JTJ∆ = JT e are augmented to N′∆ = JT e

where N ′ij = (1 + δijλ)Nij with δij to be the Kronecker delta.

The value λ is initialized to a small value, e.g. 10−3. If the value obtained

for ∆ reduces the error, the increment is accepted and λ is divided by 10

before the next iteration. On the other hand, if the error increases then λ is

multiplied by 10 and the augmented normal equations are solved again, until

an increment is obtained that reduces the error. This is bound to happen,

since for a large λ the method approaches a steepest descent.

160

Appendix B

SYNTHETIC IMAGE GENERATOR

There exist many algorithms to generate synthetic images for validation

purposes of correlation techniques [e.g. 150, 151]. In this research, synthetic

images are created using the algorithm presented by Zhou and Goodson (2001,

[146]).

A speckle pattern is assumed to be the sum of individual speckles following

Gaussian intensity distribution. First, a number of n speckles are seeded

randomly over an image. Random position of the kth speckle is denoted by

(xk, yk). A speckle is controlled by its diameter a (the same for all speckles)

and its maximum intensity I0 (the value of 255 is used). By integrating a

discretized function of Gaussian distribution, intensity of any pixel (xi, yj) on

the original image can be written as

I1(xi, yj) =1

4πa2I0

n∑k=1

[erf

(xi − xka

)− erf

(xi + 1− xk

a

)]×

[erf

(yj − yka

)− erf

(yj + 1− yk

a

)],

(B.1)

where erf is the error function1.

Deformed images is created by applying the homogenous transformation

up to the first order,

xi = xi + u0 + u,xxi + u,yyj,

yj = yj + v0 + v,xxi + v,yyj,(B.2)

where (xi, yj) is the generated position in the deformed image. The parameters

(u0, v0, u,x, v,x, u,y, v,y) of the transformation have prescribed values, which

1Error function erf(x) is twice the integral of the Gaussian distribution with mean of 0

and variance of 1/2: erf(x) = 2√π

x∫0

e−t2dt.

161

can simulate translation, rotation, stretching and shear effects with sub-pixel

resolution.

Therefore, each pixel on the deformed image has the intensity as given by,

I2(xi, yj) =1

4πa2I0J

n∑k=1

[erf

(xi − xka

)− erf

(xi + 1− xk

a

)]×

[erf

(yj − yka

)− erf

(yj + 1− yk

a

)],

(B.3)

where J is the Jacobian tensor of the deformation, which is computed by

J =

∥∥∥∥∥∥ 1 + u,x u,y

v,x 1 + v,y

∥∥∥∥∥∥ . (B.4)

It should be noted that intensity interpolation of the deformed image has been

done during the discretization (i.e. the integration), and continuity condition

of the intensity is satisfied.

162

Appendix C

A 3D-DIC METHOD

BASED ON SMTV AND SPIV

C.1 Introduction

In almost all 3D-DIC methods, images of a specimen that has unknown shape

and undergoes unknown deformation, are captured by both cameras and corre-

lated to establish correspondences between image points. The correspondence

task correlating a pair of images captured by the same camera at different

deformation stages, is called temporal-matching ; while the one regarding im-

ages captured by different cameras at the same deformation stage, is known

as stereo-matching. In the task called reconstruction, 3D coordinates and

displacements are calculated from the corresponding coordinates of image

points and camera calibration parameters.

In general, common 3D-DIC methods can be categorized into two main

approaches distinguished by the way that stereo-matching and reconstruction

are combined.

Back-projection: This approach was proposed in [122]. Stereo-matching

is hidden by being combined to reconstruction. A family of candidate 3D

surfaces with unknown coefficients is implicitly reconstructed from image of a

camera, and back-projected into image of the other camera (see Fig. C.1). The

unknown coefficients are defined by correlation between the back-projected

image and the captured image. In this way, inherent systematic errors due to

perspective distortion are reduced, however coordinate and intensity interpo-

lation of candidate surfaces produce additional systematic errors (see [147] for

an assessment).

Matching-reconstruction: This is the approach of recent interests, because

it is straightforward to separate stereo-matching from reconstruction (for de-

163

Figure C.1: Back-projection to combine stereo-matching with reconstruction[122]

tails, see [123, 126, 129, 131]). As the most difficult task, stereo-matching is

imposed with as many constraints as possible, including similarity, epipolar

geometry, ordering, disparity limit, etc. Reconstruction is a simple task of

intersecting two rays starting from camera lenses and going through corre-

sponding image points. Displacement is then computed as the difference in

coordinate of reconstructed shape from original shape. This approach offers

high robustness and versatility, however accumulated systematic errors after

each step is significant.

Both of the approaches are able to reconstruct original shape of the speci-

men, therefore they can be applied for all types of specimen (e.g. plate, shell,

cylinder...). Basically, there is no restraint on arrangement of cameras as long

as images captured at the same time do not differ too much. On the other

hand, since adapted from computer vision where stereo-matching is employed

to obtain high robustness, common 3D DIC methods have unpredictable accu-

racy. Final displacement error usually ranges between 1/30, 000 and 1/5, 000

of field of view, depending on application [127, 136]. Furthermore, extremely

slow image processing is also a problem of these methods (this may not be,

however, a shortcoming because real-time performance is not an objective).

164

The article is organized as following. In the first section, common 3D-DIC

methods applied for experimental solid mechanics are summarized and dis-

cussed. The overview of the proposed approach is given in section 2. Cali-

bration parameters are assumed to be known beforehand, but several matters

regarding calibration are discussed in Section C.5. Detailed description of

displacement calculation is given in Section C.3. Limitation and difficulty of

the approach are also presented.

C.2 The proposed approach

In many experiments, such as buckling of plates, original specimens have a flat

surface. Therefore, both cameras can be arranged to observe the same region

on that surface.

The proposed approach exploits this condition to obtain an obvious cor-

respondence of every pair of points on images of the original shape. For

instance, any pair of point (x01, y

01) on the original image of camera 1 and point

(x02 = x0

1, y02 = y0

1) on that of camera 2, are obviously images of an identical

point on the original surface. By this way, two dense sets of corresponding

image points are obtained without using the intensive stereo-matching.

These sets of points are then tracked by temporal-correlation for each

camera independently, and their in-plane coordinates as observed by each

camera are obtained.

Finally, actual three-dimensional coordinates and displacements are de-

termined by combining those in-plane coordinates in very simple formulae.

Since orientation and position of the original surface with respect to both

cameras are known after calibration, the original shape of specimen is known;

thus consequent undeformed shapes can also be reconstructed as in common

3D-DIC methods.

Generally speaking, this approach can be considered as either an extension

of 2D DIC or a variation of sPIV [152] and sMTV [153] (in sPIV and sMTV,

the flat surface is formed by laser).

165

Figure C.2: Geometrical representation of coordinates and displacements ofimage, object plane and real points. The image planes are placed in front ofthe camera lens only for convenience. Coordinates of points related to originalconfiguration are denoted with the superscript 0, while the ones related todeformed configuration do not have superscript.

C.3 Recovery of 3D coodinate and displacement

For convenience, object frame is placed on surface of the original specimen so

that its Z-axis is perpendicular to the surface and directs to the cameras. The

plane XY is thus referred to as object plane (see Fig. C.2).

Cameras are specified by their lens positions with respect to the object

frame, which are (−c1, 0, l1) for camera 1 and (c2, 0, l2) for camera 2 (where

c1, c2, l1, and l2 are all positive). Since the both cameras are arranged so that

their optical axes intersect at origin of the object frame, orientation of camera

i is defined by an angle αi. For each camera, another frame is attached to the

center of image plane, to define coordinate of an image point to be (xi, yi).

166

The recovery of 3D coordinate and displacement consists of four steps:

• In the first step, image displacement (dxi, dyi) is calculated by cross-

correlating deformed image with original image. Then, deformed image

coordinate (xi, yi) can be defined as (xi = x0i + dxi, yi = y0

i + dyi), where

(x0i , y

0i ) denotes the original image coordinate of camera i.

• The second step is to determine coordinate (Xi, Yi) of the point on object

plane to which a point (xi, yi) on image plane is projected.

• In the third step, coordinate (X, Y, Z) in 3D object frame is recovered

from (Xi, Yi). Deformed shape of the specimen is reconstructed.

• In the final step, 3D displacement (dX, dY, dZ) is calculated by sub-

tracting the coordinate (X,Y, Z) of deformed surface by coordinate of

the original surface (X0, Y 0, Z0).

Given as follows are the formulae used in the last three steps (detailed

derivations are given in Appendices).

An image point (xi, yi) given in pixel units is projected to the object plane

to be (Xi, Yi) given in mm unit, by the following equations [153].

For camera 1,

X1 =h1m1x1

h1 cosα1 −m1x1 sinα1

(C.1a)

Y1 = m1y1

(1 +

x1 sinα1

h1

)(C.1b)

For camera 2,

X2 =h2m2x2

h2 cosα2 +m2x2 sinα2

(C.2a)

Y2 = m2y2

(1− x2 sinα2

h2

)(C.2b)

where h1 and h2 (in mm) are distance from the origin of object frame to lens

center of camera 1 and camera 2, respectively; and m1 and m2 are manification

factors expressed in mm/pixel.

167

Real coordinate (X, Y, Z) is reconstructed from the above in-plane coordi-

nates (X1, Y1) and (X2, Y2) by:

X = X1 −d1l1 (X1 −X2)

d1l2 + d2l1(C.3a)

Y =Y1Y2 (l2−l1)

Y1l2 − Y2l1(C.3b)

Z =l1l2(X1 −X2)

d1l2 + d2l1(C.3c)

where d1 ≡ c1 +X1 and d2 ≡ c2 −X2 (refer to Fig. C.2 for the definitions of

c1, c2, l1, and l2).

Displacement (dX, dY, dZ) is calculated from real coordinate of the de-

formed surface point (X, Y, Z) and the corresponding original one (X0, Y 0, Z0)

by:

dX = X −X0 (C.4a)

dY = Y − Y 0 (C.4b)

dZ = Z − Z0 (C.4c)

The above equations are true for any point, thus repeating the procedure

for all image points yields a dense and three-dimensional displacement field.

The six parameters m1,m2, c1, c2, l1, and l2 are defined in the calibration

task. The angle αi is either measured directly or calculated from ci and li.

The distance hi between camera lens and object frame origin is difficult to be

measured directly, thus calculated from ci and li by Pythagorean theorem.

C.4 Practical limitation and difficulty

It is interesting that all of the above equations, if slightly modified, would

also be valid for arbitrary shapes, since the object plane is not necessarily the

original surface (the original surface is only required to contain the intersection

point of optical axes, but not to align with any particular surface). That means

the proposed approach should be as general as common 3D-DIC methods.

168

Unfortunately, the required cameras alignment such that the original shape

must appear at the same position and scale on captured images, can only be

obtained by using a planar alignment grid (at least with non-arbitrary shape).

Therefore, the proposed approach is limited to applications where specimens

have originally flat surface.

Difficulty would arise when aligning both cameras so that their optical

axes intersect in space and at a point on specimen surface. Even when this

requirement is met, there would be another difficulty in adjusting the cameras

so that they observe the same region on specimen surface. Hopefully, this kind

of alignment is popular in sPIV [152] and sMTV [153], so it should be obtained

without too much difficulty.

C.5 Calibration

Calibration is the determination of the necessary parameters mi, ci, and li (as

well as αi or hi, if it has not been related to ci and li).

There are three possible approaches to calibration. The first one is calibra-

tion by pure computation of all related parameters (including effective focal

length, pixel size, pixel aspect ratio, pixel skewness ratio, two components

of image center displacement, lens distortion coefficients, three components

of lens center coordinate, and three components of lens plane orientation),

where the only information needed is relative distances between some object

points on a planar calibration target (consult [90] for details). The second

approach is to use the Eq. C.1–Eq. C.4 in reverse. Considered as known

parameters, the real displacement (dX, dY, dZ) is prescribed precisely and the

corresponding image displacement (dxi, dyi) is calculated by cross-correlation.

Reversely, the desired parameters are then considered as unknowns and solved

by optimization technique. This approach is less computational than the first

one, because only necessary parameters are computed. The last approach is to

directly measure as many physical parameters as possible, which is discussed

in the following paragraphs.

169

The optical lens of the type Canon EF28-135mm F3.5-5.6 IS USM used in

the present work has variable effective focal length and unknown effective lens

position with respect to any fixed point on the lens. Therefore, the distance

ci and li, thus hi, cannot be defined physically. To tackle this problem, one

might use the image plane (i.e. the plane of CMOS sensor) as a fixed plane

with positioning parameters c′i and l′i (see Fig. C.3).

Once the magnification factor mi is determined and the pixel size si is

known (si is usually given by camera manufacturer; for these cameras it is

about 6µm), the ratio hi/fi can be calculated from:

mi =hifisi. (C.5)

Then, the desired parameters ci and li are calculated from the parameters

c′i and l′i by using the following relations.

l′ili

= 1 +fihi

(C.6)

c′ici

= 1 +fihi

(C.7)

The problem now becomes to determine the positions of image planes which

are not measurable either. In the Canon EOS 20D and Canon EOS 30D

cameras, a fixed point that can be defined physically is the center of mounting

screw-hole (denoted as the unfilled circle in Fig. C.3), by the distance Ci and

Li. The distance δ from this point to the image plane, which is the same for

both cameras, should be given by camera manufacturer (approximately, this

distance is 5mm). Using this information, one can define the position of image

plane by:

l′i = Li + δ cosαi (C.8)

c′i = l′i tanαi (C.9)

where the angle αi = arctan(Ci/Li).

170

Figure C.3: Practical geometry of a camera. The mounting screw-hole whichcan be measured physically is marked with the unfilled circle. The distancefrom that point to the image plane is δ ≈ 5mm.

The only remaining problem is to determine the magnification factor mi.

Commonly, it can be done by placing a flat pattern with some known real

distance R (in mm) and parallel with the image plane, measuring the corre-

sponding image distance ri (in pixel), and then calculating mi = R/ri.

In this calibration approach, sensitivity of the measurands (such as δ, si, Ci, Li)

to resulted parameters that are used as inputs of Eq. C.1–Eq. C.4, must be

investigated carefully. It should be also to conduct simulation on artificial

object and images to study effects of the uncertainties in those parameters to

the final displacement.

C.6 Remarks

The approach to measurement of full-field and three-dimensional displacement

using the idea of sPIV and sMTV, is presented. Three calibration approaches

are introduced, including pure-computation commonly used in 3D-DIC meth-

ods, reverse-computation of the given equations, and direct-measurement. In

the author’s judgement, priority is given to the latter two approaches because

171

they are easy to implement and control. Of the two approaches, the reverse-

computation is preferable for the particular cameras and lenses used in this

work.

Alignment of the cameras must meet two requirements. The first one is

that optical axes of both cameras must intersect at a point on original specimen

surface. The second requirement is that both cameras must observe the same

region on the original specimen surface. This alignment offers the advantage of

avoiding stereo-matching, but causes the limitation to originally flat specimens.

In-plane displacements, as measured independently by each camera by

cross-correlation, are combined to calculate three-dimensional displacement

and reconstruct three-dimensional shape in the much simpler equations than

those used in common 3D-DIC methods.

Derivation of object plane coordinate (Eq. C.1–Eq. C.2)

A point (xi, yi) (in pixel) on image plane of the camera i is first projected onto

a parallel plane containing the object frame origin to be an intermediate point

(X i, Y i) (in mm) by the relation of a pin-hole camera model (see Fig. C.4).

X i = mixi (C.10a)

Y i = miyi (C.10b)

In the above equation, the magnification factor, expressed in unit of mm/pixel,

is mi = (hi/fi)si. The parameters si is the actual size of a pixel in image sensor

expressed in unit of mm/pixel. The parameter hi is the distance between the

origin of object frame and the center of camera lens, given in mm. The focal

length fi is defined as the distance between the center of camera lens and the

image plane, with the unit of mm.

The intermediate point (X i, Y i) is then mapped to a point (Xi, Yi) on the

object plane as follows.

172

Figure C.4: Projection of an image point (xi, yi) to a point (Xi, Yi) on objectplane. The intermediate point is (X i, Y i) which lies on a virtual plane parallelwith the image plane.

173

For camera 1, geometry relation of similar triangles gives:

X1 sinα1

h1

=X1 cosα1 −X1

X1

(C.11)

Substituting Eq. C.10a into Eq. C.11 and doing some manipulation result in

the Eq. C.1a.

For camera 2, since the angle α2 has opposite sign, the geometry relation

becomes:−X2 sinα2

h2

=X1 cosα2 −X2

X2

(C.12)

Then, the Eq. C.2a can be derived in similar way.

For the Y -coordinates, similar triangles give the following relations:

Y1 − Y 1

Y 1

=X1 sinα1

h1

(C.13a)

Y2 − Y 2

Y 2

=−X2 sinα2

h2

(C.13b)

Substituting Eq. C.10b into Eq. C.13a and Eq. C.13b respectively results

in Eq. C.1b and Eq. C.2b.

Derivation of real coordinate (Eq. C.3)

As shown in Fig. C.5, the distance from X1 to X2 on the X-axis is devided into

two parts of δ1 and δ2 by a perpendicular line (whose length is magnitude of

the real coordinate Z). With the definitions of d1 = c1 +X1 and d2 = c2−X2,

it is not difficult to obtain the following geometry relations.

δ1d1

=Z

l1(C.14a)

δ2d2

=Z

l2(C.14b)

δ1 + δ2 = X1 −X2 (C.14c)

After substitution of Eq. C.14a and Eq. C.14b into Eq. C.14c and suitable

174

Figure C.5: Recovery of real coordinate (X, Y, Z) from two in-planecoordinates (X1, Y1) and (X2, Y2).

175

arrangement, one should have the Eq. C.3c given again as

Z =l1l2(X1 −X2)

d1l2 + d2l1. (C.15)

Furthermore, it can be seen that X = X1 − δ1. Thus, as combined with

the resulting Eq. C.3c and Eq. C.14a, that relation leads to Eq. C.3a, which

given as

X = X1 −d1l1 (X1 −X2)

d1l2 + d2l1(C.16)

The Y -coordinate can be derived by making use of the following geometry

relations.

Z

l1=Y − Y1

Y1

(C.17a)

Z

l2=Y − Y2

Y2

. (C.17b)

Instead of substituting the Eq. C.3c of Z directly into either Eq. C.17a or

Eq. C.17b, it is more convenient to manipulate both of them to obtain the

Eq. C.3b which is

Y =Y1Y2 (l2−l1)

Y1l2 − Y2l1. (C.18)

176

CURRICULUM VITAE

Nguyễn Trần Nam is a Master student at the Institut

Teknologi Bandung (Indonesia) from 2005 to 2007. He

received his Bachelor Degree from the Ho Chi Minh

City University of Technology (Vietnam) in 2005. His

research interest includes 3D Digital Image Correlation

and its applications to experimental mechanics, optical

methods for 3D shape reconstructions, and structural

mechanics. From 2007 to 2010, he involves in a doctoral research in optical

shape measurement technology at the Loughborough University (UK).

177