3d Multi Scale Optic Flow Algorithm English Language Essay

Published: 2021-07-03 06:35:06
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Vimal Chandran
1 Introduction
Cardiac diseases are the leading cause of death and disability in
developed countries and therefore a major public health
challenge. If all forms of cardiac disease were eliminated, the
average life expectancy would increase by 10 years [1]. The
cardiac illness is mainly influenced by the deformation of the
cardiac walls. Assesing the variation in movement of the cardiac
wall may provide a quantitative indication of the health of
cardiac muscle. Montioring and quantification of abnormal cardiac
wall motion is a strong clinical predictor of sudden, arrhythmic
and cardiac death [2].
Estimating the motion field between two consecutive images has
been a heavily investigated field of research for decades. Optic
flow is one of the well known techniques used for motion field
estimation. In this context, the motion field is characterized by
the field of vectors, that shows the displacement of points in
the optic field relative to the observer. Optic flow shows the
velocity field of pixels in the image. In literature [3], there
are several approaches have been proposed for optic flow
estimation and in most of these techniques the assumption is
that, the brightness doesnot change by small displacement and the
motion is estimated by solving the Optic Flow Constrained
Equation.
F_{x}u+F_{y}v+F_{t}=0
where F(x,y,t)\::\mathbb{R}^{2}\times\mathbb{R}^{+}\rightarrow\mathbb{R}
is an image sequence with x, y and t representing the spatial
and temporal coordiantes respectively, F_{x},F_{y},F_{t}
are the
spatiotemporal derivatives, u(x,y,t)\: v(x,y,t),\::\mathbb{R}^{2}\times\mathbb{R}^{+}\rightarrow\mathbb{R}
are the two unkown velocity vectors. It is to be noted that
there are two unknowns with one equation, so a unique solution
cannot be found and this is referred as the "aperture problem" of
the Optic Flow estimation. The unique solution can be obtained if
there are equal number of equation for the number of unknowns.
Inorder to find a probable solution, several methods has been
proposed and they can be grouped into three main categories as
region based, frequency domain based and gradient based
approaches. In the region based approach [3,4,5], velocity is
defined as the shift between regions of subsequent images that
minimizes a sum of squared distance measure for finding the best
match. In frequency domain based approach [6], estimation is
carried out in the frequency domain, where velocity is defined as
energy function and minimization the motion energies is done. In
the gradient based approach [7,8,9], velocity is computed from
the spatiotemporal derivative of the image intensity or filtered
version of the image.
The application of optic flow method to tagged MR images was
introduced by Florack et al [10,11], where a robust differential
technique was developed for scalar and density images. This
generalized framework was adapted to cardiac tagged MR images by
Niessen [12, 13] et al, Suibesuaputra [14]. Van Assen et al [15]
and Florack and Van Asses [16] developed a method based on
multiple independent MR tagging acquition, removing altogether
the aperture problem. Researches in measurement of cardiac motion
are generally carried out in 2D, but for modelling true heart
motion, the 3D model is essential. In 2D optic flow algorithm
captures only the expanison whereas the contraction and rotation
of cardiac tissues are not considered, but with 3D optic flow
algorithm it can be estimated. The latest increase in computation
power made it possible to compute 3D optic flow field from tagged
MRI data.
The 3- dimensional version of the optic flow constrained equation
is given as
F_{x}u+F_{y}v+F_{z}w+F_{t}=0
where F(x,y,z,t)\::\mathbb{R}^{3}\times\mathbb{R}^{+}\rightarrow\mathbb{R}
is an image sequence with x, y, z and t representing the spatial
and temporal coordiantes respectively, F_{x},F_{y},F_{z},F_{t}
are the spatiotemporal derivatives, u(x,y,z,t)\: v(x,y,z,t),\: w(x,y,z,t)\::\mathbb{R}^{3}\times\mathbb{R}^{+}\rightarrow\mathbb{R}
are the three unkown velocity vectors. Similar to 2D, there are
three unknown in this case but only one equation, so the aperture
problem exits. To overcome this problem, Barron [17] extended the
concept developed by Horn and Schrunk, Lucas and Kanade to three
dimension. This method, imposes a constant intensity assumption
which in tagged MR images does not hold due to the tag fading by
T1 relaxation. Inorder to solve the aperature problem and to
model true motion of the heart, we have derived the mutiscale
optic flow technique for 3D case from generalized mutiscale
framework proposed by Florack [11]. Multiple independent MR
tagging acquition are used to solve the aperture problem. In the
experiment, the proposed technique is tested on an artificial 3D
image sequnce, the vectorfield components are calculated and
analysed. The method is further extended to the tagged 3D cardiac
data. In Section 2, generated phantom datasets and real datasets
are discussed. The derived the mutiscale optic flow technique for
3D case is presented in Section 3. Finally in Section 4 and 5 we
describe the experiments, the results and discuss future
directions.
2 Material
2.1 Phantom Data
Image phantom are used to analyse the performance of an
algorithm. In this case, three 3D phantom data is constructed
with a sine function varying in x, y and z direction
respectively. The sequence is constructed by extending in the
temporal direction and with sine function incremented by a
constant t
. The generated phantom sequences are given as:
Fx\left(\sin\left(x+t\right),y,z,t\right),\, Fy\left(x,\sin\left(y+t\right),z,t\right),\, Fz\left(x,y,\sin\left(z+t\right),t\right)\::\mathbb{R}^{3}\times\mathbb{R}^{+}\rightarrow\mathbb{R}
where Fx,\, Fy,\, Fz
represents the 3 dimensional phantom in x,
y, z direction respectively. The generated phantom images has a
resolution of 30\times30
pixels, and an image volume of 30\times30\times30
voxels. There are twenty temporal sequences and the overall
dimension of the generated phanton is 30\times30\times30\times20
representing x\times y\times z\times t
dimensions respectively.
The generated sine phantoms are shown as a varying sine function
in a 3D plot (in Figure 1) and as a 2D sine image (in Figure 2).
[float Figure:

Figure 1: 3D Plot of the phantom Fx,\: Fy,\: Fz
at the first
sequence, in x, y, z direction respectively

Figure 2: 2D Image of the phantom Fx,\: Fy,\: Fz
at the first
sequence, in x, y, z direction respectively
]
2.2 Phantom Data with Gaussian Noise
Sensitivity of the algorithm to noise is tested by adding noise
to the phantom sequence, carrying out the same experiments of
noiseless phantom sequence and comparing the results. The
gaussian white noise with zero mean and finite variance is added
to the generated sine phantom sequence. The overall dimension of
the phantom data with noise is 30\times30\times30\times20
representing x\times y\times z\times t
dimensions respectively.
The generated sine phantoms with noise are shown as a varying
sine function with gaussian noise in a 3D plot (in Figure 3) and
as a 2D image (in Figure 4).
[float Figure:

Figure 3: 3D Plot of the phantom with gaussian noise at the first
sequence, in x, y, z direction respectively

Figure 4: 2D Image of the phantom with gaussian noise at the
first sequence, in x, y, z direction respectively
]
2.3 Real Data
Tagged MR imaging is technique that can be used for quantitative
assessment of myocardial contractile function. The tagging
patterns are inherited in the tissue. They move along the tissue
and enables local motion analysis.The 3 dimensional tagged MR
image volume sequence of a patient heart were acquired using a
HARP technique[2] developed at ETH Zurich, Switzerland. It
consists of 23 sequences with a temporal resolution of 30ms.
There are three different views, one short axis and two long axis
views, which are perpendicular with respect to each other.The
images represents a resolution of 112\times112
pixels, and an
image volume of 112\times112\times112
voxels. The overall
dimension of the generated phanton is 112\times112\times112\times23
representing x\times y\times z\times t
dimensions respectively.
The 3D tagged MR image sequence of a patient heart taken one
short axis and two long axis views is shown (in Figure 5 and 6).
[float Figure:

Figure 5: From left to right: short axis view containing
horizontal tags, two long axis views containing both the vertical
and horizantal tags and the combination of the image plane.

Figure 6: From left to right: 2D Image of the short axis view and
two long axis views of the heart.
]
3 Proposed Method
3.1 3D Multi-Scale Optic Flow Algorithm
The 3D image sequence is a real function F(x,y,z,t)\epsilon\mathbb{R}^{3}\times\mathbb{R}^{+}
with x, y, z and t as spatial coordinates and temporal
coordinates respectively. F(x(t),y(t),z(t))
is the trajectory of
the material point projected to the volume plane. The scale space
representation I(x,y,z,t,\sigma,\tau)\epsilon\mathbb{R}^{3}\times\mathbb{R}^{+}
of a 3D image sequence F(x,y,z,t)
is given as the convolution
between the 3D image sequence and the spatiotemporal gaussian
kernel\phi(x,y,z,t,\sigma,\tau)\epsilon\mathbb{R}^{3}\times\mathbb{R}^{+}
.
I\left(x,y,z,t,\sigma,\tau\right)=\left(F\otimes\phi\right)\left(x,y,z,t,\sigma,\tau\right)=\intop_{-\infty}^{\infty}\intop_{-\infty}^{\infty}\intop_{-\infty}^{\infty}\intop_{-\infty}^{\infty}F(x,y,z,t)\phi(x,y,z,t,\sigma,\tau)dxdydzdt
where \phi(x,y,z,t,\sigma,\tau)=\frac{1}{\left(\sqrt{2\pi\sigma}\right)^{3}\left(\sqrt{2\pi\tau^{2}}\right)}\exp\left(-\frac{\left(x^{2}+y^{2}+z^{2}\right)}{2s^{2}}-\frac{t^{2}}{2\tau^{2}}\right)
and x, y and z are the spatial coordinates, whereas \sigma,\tau\epsilon\mathbb{R}^{+}
denotes the spatial and temporal scales of gaussian kernel
respectively The equation gives the blurred version of the image
and strength depends on the choice of the scale. The brightness
constancy assumption states that the pixel intensities F(x(t),y(t),z(t))
remains constant in time. Inorder to have constant intensities
over time, the Lie-derivative of the observation I with respect
to the vectorfield of the motion is zero. The Lie-derivative
captures the variation of space-time quantities along the
integral flow of some vectorfield. The first order Lie-derivative
of the observation L is considered to have a linear model of
optic flow. The Lie-derivative L_{\vec{v}}
of a function I(\phi)
with respect to the vector field\vec{V}
is defined as L_{\vec{V}}I(\phi)
. The Optic Flow Constraint Equation (OFCE) states that the
luminance does not change when we take the derivative along the
vectorfied of motion.
L_{\vec{V}}I(\phi)\equiv0
For scalar images the Lie-derivative of the observed
spatiotemporal image I is defined as
L_{\vec{V}}I(\phi)\equiv I\left(L_{\vec{V}}^{T}\phi\right)
where L_{\vec{V}}^{T}\phi=-\vec{\nabla}.\left(\phi\vec{V}\right)
[11], so we get
-\intop_{-\infty}^{\infty}\intop_{-\infty}^{\infty}\intop_{-\infty}^{\infty}\intop_{-\infty}^{\infty}F(x',y',z',t')\vec{\nabla}\phi(x-x',y-y',z-z',t-t')\vec{V}(x-x',y-y',z-z',t-t')dx'dy'dz'dt'=0
In the equation 7, the optic flow vectorfield \vec{V}
is
unknown, which is computed by deriving a set of equation. The
optic flow vectorfield with \vec{u}(x,y,z,t)
in x direction, \vec{v}(x,y,z,t)
in y direction and \vec{w}(x,y,z,t)
in z direction is
approximated for the first order which gives
I_{x}u+I_{xt}u_{t}+I_{xx}u_{x}+I_{xy}u_{y}+I_{xz}u_{z}+I_{y}v+I_{yt}v_{t}+I_{xy}v_{x}+I_{yy}v_{y}+I_{yz}v_{z}+I_{z}w+I_{zt}w_{t}+I_{xz}w_{x}+I_{yz}w_{y}+I_{zz}w_{z}=-I_{t}
The first order vectorfield gives an equation with 15 unknown
components of vector field
\left\{ u,u_{t},u_{x},u_{y},u_{z},v,v_{t},v_{x},v_{y},v_{z},w,w_{t},w_{x},w_{y},w_{z}\right\}
. This has to solved but with one equation unique solution is not
possible. However, because the Lie-derivative of the image
vanishes identically so do all their partial derivatives. So the
equations may be added for the vanishing Lie-derivatives with
respect to x, y, z and t since it represents the first order
vectorfield. This gives four extra equations. In total there are
five linearly independent equations
\begin{array}{c}
I_{x}u+I_{xt}u_{t}+I_{xx}u_{x}+I_{xy}u_{y}+I_{xz}u_{z}+I_{y}v+I_{yt}v_{t}+I_{xy}v_{x}+I_{yy}v_{y}+I_{yz}v_{z}+I_{z}w+I_{zt}w_{t}+I_{xz}w_{x}+I_{yz}w_{y}+I_{zz}w_{z}=-I_{t}\\
I_{xx}u+I_{xxt}u_{t}+\left(I_{x}+I_{xxx}\right)u_{x}+I_{xxy}u_{y}+I_{xxz}u_{z}+I_{xy}v+I_{xyt}v_{t}+I_{xxy}v_{x}+I_{xyy}v_{y}+I_{xyz}v_{z}+I_{xz}w+I_{xzt}w_{t}+I_{xxz}w_{x}+I_{xyz}w_{y}+I_{xzz}w_{z}=-I_{xt}\\
I_{xy}u+I_{xyt}u_{t}+I_{xyx}u_{x}+I_{xyy}u_{y}+I_{xyz}u_{z}+I_{yy}v+I_{yyt}v_{t}+I_{xyy}v_{x}+\left(I_{y}+I_{yyy}\right)v_{y}+I_{yyz}v_{z}+I_{yz}w+I_{yzt}w_{t}+I_{xyz}w_{x}+I_{yyz}w_{y}+I_{yzz}w_{z}=-I_{yt}\\
I_{xz}u+I_{xzt}u_{t}+I_{xxz}u_{x}+I_{xyz}u_{y}+I_{xzz}u_{z}+I_{yz}v+I_{yzt}v_{t}+I_{xyz}v_{x}+I_{yyz}v_{y}+I_{yzz}v_{z}+I_{zz}w+I_{zzt}w_{t}+I_{xzz}w_{x}+I_{yzz}w_{y}+\left(I_{z}+I_{zzz}\right)w_{z}=-I_{zt}\\
I_{xt}u+\left(I_{x}+I_{xtt}\right)u_{t}+I_{xxt}u_{x}+I_{xyt}u_{y}+I_{xzt}u_{z}+I_{yt}v+\left(I_{y}+I_{ytt}\right)v_{t}+I_{xyt}v_{x}+I_{yyt}v_{y}+I_{yzt}v_{z}+I_{zt}w+\left(I_{z}+I_{ztt}\right)w_{t}+I_{xzt}w_{x}+I_{yzt}w_{y}+I_{zzt}w_{z}=-I_{tt}
\end{array}
There are five linearly independent equation and fifteen
unknowns, so the unique solution is not possible. This is
refereed as the aperture problem, where the number of vectorfield
to be estimated is larger than the number of linearly independent
equations. In other words, there is not enough information to
recover the optic flow at one point by looking at first order
derivative of image intensity.
Inorder to solve this problem, multiple independent MR tagging
acquition are used, in this case, the same 3D image sequence is
taken in all the three directions x, y and z as F^{X}(x,y,z,t)\epsilon\mathbb{R}^{3}\times\mathbb{R}^{+}
, F^{Y}(x,y,z,t)\epsilon\mathbb{R}^{3}\times\mathbb{R}^{+}
and F^{Z}(x,y,z,t)\epsilon\mathbb{R}^{3}\times\mathbb{R}^{+}
respectively. When their equations are solved with the scale
space representation in directions x, y and z as I^{X}(x,y,z,t,\sigma,\tau)\epsilon\mathbb{R}^{3}\times\mathbb{R}^{+}
,I^{Y}(x,y,z,t,\sigma,\tau)\epsilon\mathbb{R}^{3}\times\mathbb{R}^{+}
and I^{Z}(x,y,z,t,\sigma,\tau)\epsilon\mathbb{R}^{3}\times\mathbb{R}^{+}
respectively. Each direction provide five equation, when added
they provide fifteen equation in total. The first order
vectorfield provides fifteen unknowns and there are fifteen
equation, so unique solution is obtained. This is given in the
form of matrix notation as
AV=-a
A =\left(\begin{array}{ccccccccccccccc}
I_{x}^{X} & I_{y}^{X} & I_{z}^{X} & I_{xt}^{X}\tau^{2} & I_{xx}^{X}\sigma^{2} & I_{xy}^{X}\sigma^{2} & I_{xz}^{X}\sigma^{2} & I_{yt}^{X}\tau^{2} & I_{xy}^{X}\sigma^{2} & I_{yy}^{X}\sigma^{2} & I_{yz}^{X}\sigma^{2} & I_{zt}^{X}\tau^{2} & I_{xz}^{X}\sigma^{2} & I_{yz}^{X}\sigma^{2} & I_{zz}^{X}\sigma^{2}\\
I_{xx}^{X} & I_{xy}^{X} & I_{xz}^{X} & I_{xxt}^{X}\tau^{2} & I_{x}^{X}+I_{xxx}^{X}\sigma^{2} & I_{xxy}^{X}\sigma^{2} & I_{xxz}^{X}\sigma^{2} & I_{xyt}^{X}\tau^{2} & I_{y}^{X}+I_{xxy}^{X}\sigma^{2} & I_{xyy}^{X}\sigma^{2} & I_{xyz}^{X}\sigma^{2} & I_{xzt}^{X}\tau^{2} & I_{z}^{X}+I_{xxz}^{X}\sigma^{2} & I_{xyz}^{X}\sigma^{2} & I_{xzz}^{X}\sigma^{2}\\
I_{xy}^{X} & I_{yy}^{X} & I_{yz}^{X} & I_{xyt}^{X}\tau^{2} & I_{xxy}^{X}\sigma^{2} & I_{x}^{X}+I_{xyy}^{X}\sigma^{2} & I_{xyz}^{X}\sigma^{2} & I_{yyt}^{X}\tau^{2} & I_{xyy}^{X}\sigma^{2} & I_{y}^{X}+I_{yyy}^{X}\sigma^{2} & I_{yyz}^{X}\sigma^{2} & I_{yzt}^{X}\tau^{2} & I_{xyz}^{X}\sigma^{2} & I_{z}^{X}+I_{yyz}^{X}\sigma^{2} & I_{yzz}^{X}\sigma^{2}\\
I_{xz}^{X} & I_{yz}^{X} & I_{zz}^{X} & I_{xzt}^{X}\tau^{2} & I_{xxz}^{X}\sigma^{2} & I_{xyz}^{X}\sigma^{2} & I_{x}^{X}+I_{xzz}^{X}\sigma^{2} & I_{yzt}^{X}\tau^{2} & I_{xyz}^{X}\sigma^{2} & I_{yyz}^{X}\sigma^{2} & I_{y}^{X}+I_{yzz}^{X}\sigma^{2} & I_{zzt}^{X}\tau^{2} & I_{xzz}^{X}\sigma^{2} & I_{yzz}^{X}\sigma^{2} & I_{z}^{X}+I_{zzz}^{X}\sigma^{2}\\
I_{xt}^{X} & I_{yt}^{X} & I_{zt}^{X} & I_{x}^{X}+I_{xtt}^{X}\tau^{2} & I_{xxt}^{X}\sigma^{2} & I_{xyt}^{X}\sigma^{2} & I_{xzt}^{X}\sigma^{2} & I_{y}^{X}+I_{ytt}^{X}\tau^{2} & I_{xyt}^{X}\sigma^{2} & I_{yyt}^{X}\sigma^{2} & I_{yzt}^{X}\sigma^{2} & I_{z}^{X}+I_{ztt}^{X}\tau^{2} & I_{xzt}^{X}\sigma^{2} & I_{yzt}^{X}\sigma^{2} & I_{zzt}^{X}\sigma^{2}\\
I_{x}^{Y} & I_{y}^{Y} & I_{z}^{Y} & I_{xt}^{Y}\tau^{2} & I_{xx}^{Y}\sigma^{2} & I_{xy}^{Y}\sigma^{2} & I_{xz}^{Y}\sigma^{2} & I_{yt}^{Y}\tau^{2} & I_{xy}^{Y}\sigma^{2} & I_{yy}^{Y}\sigma^{2} & I_{yz}^{Y}\sigma^{2} & I_{zt}^{Y}\tau^{2} & I_{xz}^{Y}\sigma^{2} & I_{yz}^{Y}\sigma^{2} & I_{zz}^{Y}\sigma^{2}\\
I_{xx}^{Y} & I_{xy}^{Y} & I_{xz}^{Y} & I_{xxt}^{Y}\tau^{2} & I_{x}^{Y}+I_{xxx}^{Y}\sigma^{2} & I_{xxy}^{Y}\sigma^{2} & I_{xxz}^{Y}\sigma^{2} & I_{xyt}^{Y}\tau^{2} & I_{y}^{Y}+I_{xxy}^{Y}\sigma^{2} & I_{xyy}^{Y}\sigma^{2} & I_{xyz}^{Y}\sigma^{2} & I_{xzt}^{Y}\tau^{2} & I_{z}^{Y}+I_{xxz}^{Y}\sigma^{2} & I_{xyz}^{Y}\sigma^{2} & I_{xzz}^{Y}\sigma^{2}\\
I_{xy}^{Y} & I_{yy}^{Y} & I_{yz}^{Y} & I_{xyt}^{Y}\tau^{2} & I_{xxy}^{Y}\sigma^{2} & I_{x}^{Y}+I_{xyy}^{Y}\sigma^{2} & I_{xyz}^{Y}\sigma^{2} & I_{yyt}^{Y}\tau^{2} & I_{xyy}^{Y}\sigma^{2} & I_{y}^{Y}+I_{yyy}^{Y}\sigma^{2} & I_{yyz}^{Y}\sigma^{2} & I_{yzt}^{Y}\tau^{2} & I_{xyz}^{Y}\sigma^{2} & I_{z}^{Y}+I_{yyz}^{Y}\sigma^{2} & I_{yzz}^{Y}\sigma^{2}\\
I_{xz}^{Y} & I_{yz}^{Y} & I_{zz}^{Y} & I_{xzt}^{Y}\tau^{2} & I_{xxz}^{Y}\sigma^{2} & I_{xyz}^{Y}\sigma^{2} & I_{x}^{Y}+I_{xzz}^{Y}\sigma^{2} & I_{yzt}^{Y}\tau^{2} & I_{xyz}^{Y}\sigma^{2} & I_{yyz}^{Y}\sigma^{2} & I_{y}^{Y}+I_{yzz}^{Y}\sigma^{2} & I_{zzt}^{Y}\tau^{2} & I_{xzz}^{Y}\sigma^{2} & I_{yzz}^{Y}\sigma^{2} & I_{z}^{Y}+I_{zzz}^{Y}\sigma^{2}\\
I_{xt}^{Y} & I_{yt}^{Y} & I_{zt}^{Y} & I_{x}^{Y}+I_{xtt}^{Y}\tau^{2} & I_{xxt}^{Y}\sigma^{2} & I_{xyt}^{Y}\sigma^{2} & I_{xzt}^{Y}\sigma^{2} & I_{y}^{Y}+I_{ytt}^{Y}\tau^{2} & I_{xyt}^{Y}\sigma^{2} & I_{yyt}^{Y}\sigma^{2} & I_{yzt}^{Y}\sigma^{2} & I_{z}^{Y}+I_{ztt}^{Y}\tau^{2} & I_{xzt}^{Y}\sigma^{2} & I_{yzt}^{Y}\sigma^{2} & I_{zzt}^{Y}\sigma^{2}\\
I_{x}^{Z} & I_{y}^{Z} & I_{z}^{Z} & I_{xt}^{Z}\tau^{2} & I_{xx}^{Z}\sigma^{2} & I_{xy}^{Z}\sigma^{2} & I_{xz}^{Z}\sigma^{2} & I_{yt}^{Z}\tau^{2} & I_{xy}^{Z}\sigma^{2} & I_{yy}^{Z}\sigma^{2} & I_{yz}^{Z}\sigma^{2} & I_{zt}^{Z}\tau^{2} & I_{xz}^{Z}\sigma^{2} & I_{yz}^{Z}\sigma^{2} & I_{zz}^{Z}\sigma^{2}\\
I_{xx}^{Z} & I_{xy}^{Z} & I_{xz}^{Z} & I_{xxt}^{Z}\tau^{2} & I_{x}^{Z}+I_{xxx}^{Z}\sigma^{2} & I_{xxy}^{Z}\sigma^{2} & I_{xxz}^{Z}\sigma^{2} & I_{xyt}^{Z}\tau^{2} & I_{y}^{Z}+I_{xxy}^{Z}\sigma^{2} & I_{xyy}^{Z}\sigma^{2} & I_{xyz}^{Z}\sigma^{2} & I_{xzt}^{Z}\tau^{2} & I_{z}^{Z}+I_{xxz}^{Z}\sigma^{2} & I_{xyz}^{Z}\sigma^{2} & I_{xzz}^{Z}\sigma^{2}\\
I_{xy}^{Z} & I_{yy}^{Z} & I_{yz}^{Z} & I_{xyt}^{Z}\tau^{2} & I_{xxy}^{Z}\sigma^{2} & I_{x}^{Z}+I_{xyy}^{Z}\sigma^{2} & I_{xyz}^{Z}\sigma^{2} & I_{yyt}^{Z}\tau^{2} & I_{xyy}^{Z}\sigma^{2} & I_{y}^{Z}+I_{yyy}^{Z}\sigma^{2} & I_{yyz}^{Z}\sigma^{2} & I_{yzt}^{Z}\tau^{2} & I_{xyz}^{Z}\sigma^{2} & I_{z}^{Z}+I_{yyz}^{Z}\sigma^{2} & I_{yzz}^{Z}\sigma^{2}\\
I_{xz}^{Z} & I_{yz}^{Z} & I_{zz}^{Z} & I_{xzt}^{Z}\tau^{2} & I_{xxz}^{Z}\sigma^{2} & I_{xyz}^{Z}\sigma^{2} & I_{x}^{Z}+I_{xzz}^{Z}\sigma^{2} & I_{yzt}^{Z}\tau^{2} & I_{xyz}^{Z}\sigma^{2} & I_{yyz}^{Z}\sigma^{2} & I_{y}^{Z}+I_{yzz}^{Z}\sigma^{2} & I_{zzt}^{Z}\tau^{2} & I_{xzz}^{Z}\sigma^{2} & I_{yzz}^{Z}\sigma^{2} & I_{z}^{Z}+I_{zzz}^{Z}\sigma^{2}\\
I_{xt}^{Z} & I_{yt}^{Z} & I_{zt}^{Z} & I_{x}^{Z}+I_{xtt}^{Z}\tau^{2} & I_{xxt}^{Z}\sigma^{2} & I_{xyt}^{Z}\sigma^{2} & I_{xzt}^{Z}\sigma^{2} & I_{y}^{Z}+I_{ytt}^{Z}\tau^{2} & I_{xyt}^{Z}\sigma^{2} & I_{yyt}^{Z}\sigma^{2} & I_{yzt}^{Z}\sigma^{2} & I_{z}^{Z}+I_{ztt}^{Z}\tau^{2} & I_{xzt}^{Z}\sigma^{2} & I_{yzt}^{Z}\sigma^{2} & I_{zzt}^{Z}\sigma^{2}
\end{array}\right)
V=(u,v,w,u_{t},v_{t},w_{t},u_{x},v_{x},w_{x},u_{y},v_{y},w_{y},u_{z},v_{z},w_{z})^{T}
a=(I_{t}^{X},I_{xt}^{X},I_{yt}^{X},I_{zt}^{X},I_{tt}^{X},I_{t}^{Y},I_{xt}^{Y},I_{yt}^{Y},I_{zt}^{Y},I_{tt}^{Y},I_{t}^{Z},I_{xt}^{Z},I_{yt}^{Z},I_{zt}^{Z},I_{tt}^{Z})^{T}
4 Results
4.1 Phantom Data
4.1.1 Experiment 1
Inorder to assess the accuracy of the proposed 3D mutiscale optic
flow algorithm, an experiment was performed on a more realistic
sine phase grid phantom with non regid motion such as contraction
and expansion. The generated sine phantom has the dimension of 30\times30\times30\times20
representing x\times y\times z\times t
respectively. Isotropic
spatial scale of 2 and temporal scale of 1.5 is used for testing
the algorithm. The fifteen vectorfield components are computed
and their vectorfields taken for the fifth time sequence in the
middle of the image volume are shown in Figure 7.
[float Figure:

\;


\;


Figure 7: Fifteen vectorfield components computed by the proposed
algorithm on the generated phantom image.
]
The values of the all vectorfield components at specific location
such as I(1,1,1), I(10,10,10), I(20,20,20) and I(30,30,30) are
analysed. Figure 8 shows the results obtained for different
velocity components at different locations in the 3D image. The x
axis represents the image sequence (time), and yaxis represents
the optic flow velocity.
[float Figure:





Figure 8: Fifteen vectorfield components at different locations
in the 3D image and in different sequences. (x-axis : image
sequences, y-axis : optic flow velocity)
]
[float Figure:





Figure 9: Fifteen vectorfield components at I(20,20,20) in the 3D
image with different isotropical scale setting. (x-axis : image
sequences, y-axis : optic flow velocity)
]
From the Figure 8, it can seen that the optic flow velocities are
continuously changing for different time sequences. The major
variation are found for the extreme voxels located at I(1,1,1)
and I(30,30,30), this is because of the image boundary
conditions. The minor variation are found for the voxels located
at I(10,10,10) and I(20,20,20), which is of interest. The
generated sine phantoms is incremented with a constant (1, in
this case) in the temporal direction for every iteration, so the
expected optic flow velocity is 1 in {u, v, w} vectorfield and
zero for the rest, because their derivatives vanishes. As
expected, the vectorfields {u, v, w}located at I(10,10,10) and
I(20,20,20) gives satisfactory results, except some variations in
the starting and ending part of the sequence due to temporal
boundary condition and the values remains constant from 3-17
sequence.
4.1.2 Experiment 2
The scale selection plays a key role in multiscale optic flow
estimation because the strength of blurring depends on it. In
order to evaluate the proposed method for different scale
setting, an experiments is performed and the results are
analysed. The experiment involves the usage of same sine phantom
sequence as described before, with isotropic scales {1,2,3}and
non isotropic scale{1,1.5,2} setting in both spatial and temporal
coordintates and the results are compared. Figure 9 and 10 shows
the results obtained for different velocity components at the
location I(20,20,20) in the 3D image. The x axis represents the
image sequence (time), and yaxis represents the optic flow
velocity. s(x,y,z,t) represents the spatial scales in x, y, z
direction and the temporal scale, that are used for testing the
algorithm.
[float Figure:





Figure 10: Fifteen vectorfield components at I(20,20,20) in the
3D image with different non-isotropical scale setting. (x-axis :
image sequences, y-axis : optic flow velocity)
]
From the Figure 9 and 10, the optic flow vectorfields {u, v, w}
gives the values 1 and rest 0, as expected. There are some
variations in the vectorfields in the begining and the end of the
sequences, which is due to the temporal boundary condition before
a constant value is achieved,. The faster the constant value is
achieved the better the scale would be. Isotropic scale results
looks convincing than the non isotrpic results, which is as
expected because of uniform deformation in all the axis.
4.2 Phantom Data with noise
4.2.1 Experiment 3
The sensitivity of the proposed 3D mutiscale optic flow algorithm
is tested with an experiment where the gaussian white noise with
zero mean and finite variance, is added to the generated image
phantom sequences. The gaussian white noise is uncorrelated
between pixels and it resembles tagged cardiac MR, because
different regions show different deformations.
An experiment to test the sensitivity of the proposed algorithm
is performed on generated image phantom sequences with gaussian
noise with different scale settings as before. The optic flow
vectorfields are computed and the results are shown figure 11 and
12, for a particular location I(20,20,20) in the 3D image. The x
axis represents the image sequence (time), and yaxis represents
the optic flow velocity. s(x,y,z,t) represents the spatial scales
in x, y, z direction and the temporal scale.
[float Figure:





Figure 11: Fifteen vectorfield components at I(20,20,20) in the
3D noisy image, with different isotropical scale setting .
(x-axis : image sequences, y-axis : optic flow velocity)
]
[float Figure:





Figure 12: Fifteen vectorfield components at I(20,20,20) in the
3D noisy image with different non-isotropical scale setting.
(x-axis : image sequences, y-axis : optic flow velocity)
]
From the Figure 11 and 12, the optic flow vectorfields {u, v, w}
should have the values 1 and rest 0, as before. All the
vectorfield components shows major variation, this is because the
algorithm depends entirely on the voxel brightness and no
critical point features are selected. So even small change in the
voxel intensity value can bring about major change in optic flow
velocity values. So, proper preprocessing has to be carried out.
The blurring process by increasing the scale can give better
prediction, which is as expected. The higher the scale the better
the prominent information is captured and better the optic flow
velocity values. In the experiment, It is evident that the
blurring has reduced the gaussian noise and the constant value
with minor variations is achieved faster for higher scale. The
better results are obtained for the isotrpic scale (3,3,3,3) when
compared to the rest. Figure 13, shows the amount of change
between the optic flow velocities obtained for without noise and
ground truth (1 in this case), for the isotropic scale (3,3,3,3).
[float Figure:

Figure 13: Variation in Optic flow velocity calculated for
isotropic scale of (3,3,3,3) at I(20,20,20) of Phantom data with
noise
]
The variations in the optic flow velocities seems to be in the
order of 0.2 for the region of interest excluding the pixels near
the boundary. So the proper scale selection is also depend on the
noise. Similar to the above results the isotropic scaling gives
better results that non isotropic because of uniform deformtion
in all the direction. However in real time appropriate scale at
different locations in the cardiac muscle may be different, since
the heart exhibits different deformations in different regions
such as streching and compression.
4.3 Tagged Cardiac MR Data (Not yet Completed this part)
The MR tags are chessboard like patterns constructed from stripes
move along with the moving tissue. The tag fading is an MR
property and occurs due to finite relaxation time T_{1}
and this
property however does not affect the vanishing image gradient.
The proposed 3D mutiscale optic flow algorithm, is tested on the
tagged cardiac MR data from multiple acquisitions. The dimension
of the data used for testing the algorithm is 30\times30\times30\times20
representing x\times y\times z\times t
respectively. The
isotropic spatial scale of 2 and temporal scale of 1.5 is used.
The results are shown in figure.
The appropriate scale at different locations of the heart is
essential since different regions exhibit different deformations.
The optic flow velocities are retrieved at a certain scale
without considering the size of the basis gaussian function. The
choice of scales greater than zero may provide good results with
respect to noise due to the smoothing related to the increase in
scale. Based on the locatation also, the scale has to be chosen.
5 Discussion
We have proposed a new method for estimating the cardiac motion,
by means of 3D mutiscale optic flow. Initially there are five
independent equation computed for the 3D image sequence in one
direction, and the first order OFCE has fifteen unknown vector
field components. To get a unique solution the same 3D image
sequence is obtained from other two directions, which gave
another ten independent equation. So there is a total of fifteen
independent equations and fifteen unknowns, so the aperatre
problem is removed and unique solution is obtained. In real time,
the multiple independent MR tagging acquitions are used for
solving the aperature problem.
The performance of the algorithm is analysed by performing the
experiment 1. The sine phantom is generated with displacement of
1 unit over frame. The optic flow method, finds the instant
velocity in pixels per frame, which is approximately the same as
the displacement over the frame. The vectorfield gave the correct
results for the region of interest excluding the pixels near the
boundary. The vectorfield derivatives vanishes as expected.
The sensitivity of the algorithm with respect to scale is assesed
in experiment 2. The same phantom setting is used, but with
different isotropic and non-isotropic scale. The scale induces
the blurring effect into the image and strength of blurring
depends on scale selection. The same results in above experiment
is expected, and computed vectorfields with isotropic scale gave
the good results compared to non-isotropic scale. Since the image
has uniform deformation in all the axis, the same the scale in
all directions better the results.
The sensitivity of the algorithm with respect to scale is
examined in experiment 3. The sine phantom with gaussian noise
with displacement of 1 unit over frame is used with different
isotropic and non-isotropic scale. The vectorfield components
computed in this method depends entirely on the voxel intensity
values. Small changes in voxel intensity can alter the
vectorfield. The vectorfield obtained from the experiment showes
large variation when compared to previous results. The scale
selection played a key role in removing the noise information.
Higher scale gave better results than the lower scales.
(*The algorithm is tested on the tagged cardiac MR data sequence.
The tagged sequence is obtained from three simultaneous
acquisition is used. *, Not yet Completed this part)
6 Future Work
The Optic Flow Constrained Equation (OFCE) is approximated to the
first order in the proposed algorithm, but the approximation is
feasible for infinite order. The infinite order of approximation
gives the true instant velocity. The first order approximation
assumes the local pattern as linear, and correct solution is
obtained if the linearity condition is satisfied. If the local
pattern is non-linear, the solution depends on the scale to
linearize the pattern, eventhough the scale change doesn't
influence the derivatives. The zeroth order OFCE gives the
velocity components independent of scale and only the derivatives
depends them. So the optic flow velocity error will be higher
than first order and it decreases with increase in order.
The order of OFCE has to altered based on the motion pattern,
because for linear pattern the first order approximation gives
better results, and for non-linear pattern higher OFCE has to
chosen. The higher order derivative have propagation of noise, to
avoid that better scale selection has to be done. Hence in case
of tagged cardiac MR sequences, different regions exhibit
different motion, so in future the algorithm has to been tuned to
different order and different scale depending on the motion
pattern in a particular region.

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