ISSN: 0970-938X (Print) | 0976-1683 (Electronic)

An International Journal of Medical Sciences

- Biomedical Research (2016) Volume 27, Issue 3

**JunhaWu ^{1}, Bingfeng Hu^{1}, Xuan Yang^{2*}**

^{1}National High Performance Computing Center at Shenzhen, Shenzhen University, Shenzhen, Guangdong, China

^{2}College of Computer Science and Software Engineering, Shenzhen University, Shenzhen, Guangdong, China

- *Corresponding Author:
- Xuan Yang

College of Computer Science and Software Engineering

Shenzhen University

China

**Accepted date:** February 29, 2016

Landmark based image registration algorithms have been applied to medical image analysis widely. While estimation of the correspondence between landmarks among images are the most important and difficult problem. In this paper, novel landmarks based image registration method combining local tracking and global adjustment of landmarks in 4D-CT lung images is proposed. In order to accurately establish the corresponding landmarks in images at different phases of 4D-CT, we introduce spatial information to structure tensor and use structure tensor to track landmarks in images at different phases. The prior knowledge of respiratory motion is utilized to limit the search area of corresponding landmarks. Further, a cost function measuring similarity of images, bending energy of transformations and spatio-temporal smoothness of trajectories of landmarks along respiratory phases is given. We update the tracked results of landmarks by minimizing a cost function. After the optimization, the correspondence between landmarks is improved and the trajectories of landmarks are enforced to be more smooth, thus, more accurate registration results can be achieved. Experimental results of public lung dataset demonstrate the effectiveness of the proposed method.

Image registration, Structure tensor, Object tracking, Landmark correspondence.

Four Dimensional Computed Tomography (4D-CT), developed for radiotherapy treatment planning, is an imaging technique that allows for the acquisition of time varying 3D image sequences of the thorax through the respiratory cycle [1,2]. 4DCT lung image can be used to estimate respiratory motion model, and then guides radiation therapy in clinical treatment [3,4]. It is a challenge for estimating lung motion model using 4D-CT images, since 4D-CT images contain artefacts such as noisy and lower spatial resolution. Moreover, 4D-CT image contains respiratory induced CT characteristics of the pulmonary parenchyma that will reflect the changes in air content [3]. The respiratory motion is non-uniform from lung expansion to lung contraction, hence, deformable image registration is an effective technique to estimate motion model of lung.

For deformable image registration, there are two main categories. One is intensity based, which aims at aligning dense image intensities of images [4-8]. Intensity based algorithms works better when the register images are very similar (small deformation), but lead to mis-registration for large deformation in images. The other one is landmark based, which attempts to find the correspondence of landmarks between images [9-12], those landmarks mainly including surfaces [13], lines [14] and points [9,15]. Especially, commonly used point landmarks are the unique points in intensity map, such as tip like, saddle like and sphere line structure. Compared to intensity based image registration algorithms, landmark based image registration algorithms avoid solving large system of equations and can handle image registration problem with large deformation. Moreover, landmark based methods are also more flexible because registration features can be customized for the particular study [16]. In order to establish correspondence of landmarks among images, there are manually identifying method [15] and automatically identifying method [17,18]. While manually identifying method is time consuming and introduces errors, which leads to difficult to find true consistent landmarks.

Many researchers proposed methods to automatically establish landmarks correspondence between on one image to another. The automatic landmarks identification algorithms can be divided into two categories: local algorithms and global algorithms. Local algorithms are to establish correspondence of each landmark independently, instead of establishing the correspondence of one landmark based on the correspondence of other landmarks. Tracking and trajectory fitting for landmarks are commonly used methods to find the corresponding positions of landmarks in an image sequence. Most tracking methods are classical template based [19]. Fan et al. [20] employed mean shift algorithm to track landmarks in a 3D image sequence. Xiong et al. [21] detected natural junction structures in the lung and described trajectories of these landmarks by closed B-splines to estimate the lung motion. Castillo et al. [22] included general polynomials into the optical flow equation to calculate a parameterized trajectory of a point by utilizing all available spatial and temporal image information from a local window located at the point.

Global algorithms apply deformable registration to track landmarks positions. Frangi et al. [23] provided a generic framework for automatic selection of corresponding landmarks in 3D shapes, and propagate landmarks in one respiratory phase to the others. Paquin et al. [18] proposed a hybrid multiscale landmark and deformable registration algorithm to make image matching robust with respect to the location of landmarks. Compared with local algorithms, global algorithms establish the correspondence of all landmarks at the same time. Correspondence of each landmark is influenced each other. Global algorithms can acquire more accurate global alignment of landmarks and prevent severe mismatch of isolated landmarks. However, it should be mentioned that global algorithms may compromise the alignment of local structures, because it needs to find a medium between local align accuracy and global align accuracy.

In this paper, we propose a novel local global tracking method applied to track landmarks at different phases of 4D-CT images, which combines both the alignment of local structures and alignment of global images. First, landmarks are given in the image at the reference phase, structure tensor based approach is employed to track these given landmarks independently in images at all phases. Since structure tensor has advantages of powerful local patch descriptor, low dimension, multi-scale and anti-noise over other tracking methods, it is capable of tracking landmarks in 4D-CT lung images with complex structures and noise. However, structure tensor describes a local patch by summing all structure tensor matrices inside a local region, which results in an isotropy descriptor. That is, structure tensor cannot be used to identify local structures that are similar to each other in structure but are different in orientation. To tackle this issue, spatial information is introduced to structure tensor in this paper. Besides, we make use of the prior knowledge of respiratory motion to limit the searching area during tracking. The accuracy of landmark tracking has been improved significantly by the improved tracking algorithm based on structure tensor. Although the accuracy of tracking of individual landmark has been improved, there are still isolated landmarks that are mismatched. This is the problem existed in local algorithms inherently because each landmark is tracked independently rather than being constrained by each other. So a spatiotemporal constraint optimization model, which mainly contains the similarity between local regions of images and the spatiotemporal smoothness of deformation function, is proposed in this paper also. An optimization procedure is performed to update the tracking results of landmarks, and the corresponding accuracy of landmarks is improved as a whole and the trajectories of landmarks are constrained to be smooth in temporal dimension.

The 4D-CT image acquired with N phases is denoted as I={I_{t}|
t=0,1,….,N-1}. Supposing that I_{0} is the reference image, and
there are k landmarks in I_{0}. For each landmark p^{o}_{j}, j=1,….,k,
our goal is to track corresponding positions corresponding
positions p^{t}_{j} in target images {I_{t}|t=1,2,…,N-1}. For 4D-CT lung
images, anatomy based landmarks, such as junction structures,
bifurcation points and corners are often preferred, because the
statistics of intensity values and geometrical structures over the
neighbourhood region of these landmarks will not significantly
change during the respiration cycle. The small neighbourhood
region centered at a landmark can be taken as a template, and
then the corresponding region in a successive image I_{t}, t=1,
…,N-1 is found in the sequence according to a similarity
measure.

**Structure tensor**

Structure tensor is served as a powerful analysis tool that can effectively model and estimate the low level feature dependencies, orientation and position of feature pattern. Donoser et al. [24] expanded structure tensor as local patch descriptor to track interested objects in video. We also employed structure tensor to describe characteristics of the local patch centered a landmark, and use this descriptor to track the small neighbourhood region with similar characteristics in an image sequence. The centers of the searched regions are the corresponding landmarks in different phases. The structure tensor of a point x in the image I is defined as follows,

(1)Where V=[ν_{1},….,ν_{L}] is the L dimensional features vector of x
with low level features, such as intensity and derivatives, to be
considered. Since second order derivatives of vessels have
more physical meanings, we define a specified seven
dimensional features vector as,

where I is the intensity of x, and the others are the second
partial derivatives in different directions at x. The seven
dimensional features vector can be used to generate structure
tensor that describes the vessels structures in a local region.
For a local patch P_{x} centered at x in the image I, every point in
P_{x} can be describe by a structure tensor matrix,
correspondingly, the local patch can be described by summing
all the structure tensor matrices of all points in P_{x} as descriptor
SP(x),

The structure tensor of a local patch is a feature descriptor
which can be used to find the local patch with similar features.
However, it cannot be used directly in landmark tracking for
4D-CT lung images. The reason is that the structure tensor of a
local patch contains average information of the local patch,
which is not sensible to the local structures that are similar in
shape but are different in orientation. In a 4D-CT lung image,
there are many similar vessels intersections structures, which
make it difficult to distinguish one from another. As illustrated
in **figure 1**, three synthetic vessel structures in 3D are shown.
The left one P_{R} is the template patch and the right P_{T1} and P_{T2} are two local patches similar to the left one in shape. Based on
the structure tensor, all these three patches are similar to each
other. Indeed, in 4D-CT images, P_{T1} is different from P_{T2}.
Then, it can be guessed that landmarks tracking based on
traditional structure tensor may lead to biased results for 4D-CT
lung images.

**Spatially-extended structure tensor**

We introduce spatial information into the traditional Structure
Tensor (ST) to tackle the isotropic issue, named Spatially
Extended Structure Tensor (SEST). We combine the structure
tensor with spatial information to enhance the identification
ability of structure tensor. Given a 3D local patch P_{x} centered
at x, we uniformly partition the local patch into 8 subpatches
P_{x,i,} i=1,…,8. Let SP_{i} be the structure tensor of the ith sub
patch P_{x,i}, then, the SESP(x) is defined as,

SESP(x)=[SP_{1} SP_{2} … SP_{8}]_{7 × 56} (4)

Where SP_{i}, i=1,…,8 is a 7 × 7 structure tensor matrix.

The spatially extended structure tensor can distinguish the two
patches shown in **figure 1**. As shown in **figure 1**, the reference
patch P_{R} is uniformly divided into 8 local sub patches A_{1}, A_{2},
…, A_{8}. Similarly, the right two patches PT_{1} and PT_{2} are
partitioned as B_{1}, B_{2}, …, B_{8} and C_{1}, C_{2}, …, C_{8} in the same
way.

It can be observed that A_{i} is similar to C_{i} and is different from
B_{i},i=1, …, 8. Correspondingly, the patch P_{R} is similar to P_{T2} instead of P_{T1} based on the similarity of subpatches. That
implies that SEST can be used to distinguish local structure
similar in shape but different in orientation.

**Landmark tracking using spatially extended structure
tensor**

To measure the similarity between local patches, it is required
to define a valid distance measure between structure tensor
matrices. Forserter et al. [25] formulated metric to measure
distance between matrices in connected Riemannian manifold.
But before measuring matrices distance, we should calculate a
time consuming similarity scores. Donoser et al. [24] adapt a
method proposed by Kluckner et al. [26] to approximate the
matrices for more efficient calculation. The approximated
representation relies on repeated random sampling theory and
Monte Carlo simulations. Structure tensor is a symmetric
positive semi definite matrix, the Cholesky decomposition
ST=UU^{T} can be used to decompose structure tensor to obtain a
lower triangular matrix U. The low triangular matrix maps the
standard basis vector b ϵ B, B={e_{i}:1 ≤ i ≤ N}of a L-dimensional
Euclidean space to a vector a, a=U b, a ϵ R^{L}. The
vector represents the characteristics of the structure tensor ST.
The Euclidean distance between vectors is the similarity
measure between different structure tensor. Similar idea can be
used to approximate the spatially extended structure tensor.
Given a spatially extended structure tensor, SESP(x)=[SP_{1} SP_{2} … SP_{8}], SP_{i} can be decomposed as SP_{i}=U_{i}U^{T}_{ i}, i=1,…,8. Each
U_{i} is represented as a vector a_{i} ϵ R^{L}, all ai are cascaded as a
long vector sa=[a^{T}_{1}, a^{T}_{2}, …, a^{T}_{8}]^{T} to represent the spatially
extended structure tensor. The Euclidean distance between
vectors sa is the similarity measure between different spatially
extended structure tensors.

The tracking procedure of each landmark p^{0}_{j}, j=1, …, k, using
SEST is illustrated as Algorithm 1. During this procedure, the
searching area of SEST is different from that of ST.

**Algorithm 1:** tracking algorithm based on spatially extended
structure tensor

Set local patch size l × w × h and the search area size s_{l} × s_{w} ×
s_{h};

Compute SEST (p^{0}_{j}) of the local patch centered at p^{0}_{j};

**for** t=1:N-1 **do**

Let p^{0}_{j} be the initial position of p^{t}_{j};

**for** each p^{t}_{j} in the search area **do**

Compute SEST(p^{t}_{j}) of the local patch centered at p^{t}_{j} in image
I_{t};

Compute D(p^{t}_{j})=||SEST(p^{0}_{j})-SEST(p^{t}_{j})||;.

Update p^{t}_{j} in the search area linebyline;

**end for**

Find the position minimizing D(p^{t}_{j}) as the tracked position of
p^{0}_{j} in I_{t}

**end for**

is different from that of ST. We know ST is generally used to
track objects in video, where the movement and the trajectory
of the interested object is uncertain and unexpected. Then, it is
needed to search the candidate object along different directions
alternatively using ST. However, during respiratory motion, the
vessels intersections move slightly and regularly compared to
objects in video. So the search area can be limited in a small
region. In expiratory period, lung expands downward; in
inspiratory period, lung contracts upward. Hence, compared
with ST tracking algorithm, there is no need to search the
object in both up and down directions for SEST tracking
algorithm. As shown in **figure 2**, the local patch centered at p^{0}_{j} in **figure 2a** is the template, and the cubes in** figures 2b **and** 2c** are the searching areas of the template at two different phases.
What we should do is to search the template upward during
expiratory period (**Figure 2b**), where p^{0}_{j} is at the bottom of the
search area, and downward during inspiratory period** figure 2c**,
where p^{0}_{j} is at the up of the search area. The limited search
area reduces the possibility of mistracking and saves tracking
time.

**Figure 2:** An illustration of the search area for landmark tracking. (a)
is the reference image, where the small cube is the local patch
centered at p^{0}_{j}. The large cube in (b) demonstrates the search area in
expiratory period, and p^{0}_{j} is at the bottom of the search area. The
large cube in (c) demonstrates the inspiratory period, and p^{0}_{j} is at the
up of the each area.

**Cost function**

After tracking landmarks individually, it is inevitable that there
exist tracking errors for several landmarks due to the noise and
artefacts in 4D-CT images. That is, there might be the tracked
positions of several landmarks obviously deviating from their
correct positions. So, it is needed to update the tracking results
of landmarks in images at different phases in a global sense.
An optimization procedure is performed to update the tracking
results of landmarks according to the similarity between
images and the spatiotemporal smoothness of global
deformation function. To update tracking results of landmarks
in a global sense, it is needed to deform the 3D images I_{0} by
the transformation function f_{t} based on the correspondence
between {p^{0}_{j}} and {p^{t}_{j}}, t=1, …, N-1. We employ the Thin
Plate Splines (TPS) [27] to be the transformation by
considering {p^{0}_{j}}as the source landmarks in the I_{0} and {p^{t}_{j}} as the target landmarks in I_{t} have the property that each landmark
has a global influence on the transformation. That is, if the
position of one landmark is updated, then all other voxels in
the image are adjusted as well. This can be an advantage to
update the tracking results of landmarks in a global sense.

Optimization of tracking results are based on following criteria: 1) the deformed image should be aligned to the target image as a whole, rather than that only the landmarks are corresponding to each other; 2) the spatial bending energy of the spatial transformation should be less; 3) the trajectories of voxel in the 3D image should be smooth also. To update the tracking results of landmarks satisfying the constraints described above, a spatiotemporal cost function is proposed as follows,

(5)Where μ={p^{1}_{j}} U…U {p_{j}^{N-1}}, j=1,2,…,k are all tracked results
of landmarks {p^{0}_{j}} in images at different phases. W is a
neighborhood window of w_{l} × w_{w} × w_{h} voxels. x=(x_{1}, x_{2}, x_{3}) is
the position of a voxel in a 3D image. φ(x, t) is the trajectory of the voxel *x* along temporal dimension by sequentially
connecting the deformed position f_{t}(x), t=1,…,N-1 along
respiratory phases. α and β balance the second and the third
terms. To get better α and β, we do a lot of experiments.
Finally, we set α and β to be le-3.

The value of the cost function is determined by the
transformations {f_{t}, t=1,…,N-1}, which are influenced by
landmarks μ. The goal of optimization is to adjust all tracked
results of landmarks μ to let the value of the cost function E
approach to global minima as much as possible. There are
three energy terms in the spatiotemporal cost function. The
first term E_{sim} measures the similarity between the target image
I_{t} and the image I_{0}(f_{t}) deformed by the transformations f_{t}.
Specifically, the similarity is evaluated in each landmark’s
neighbourhood **W** instead of the whole image, by integrating
all differences between the intensity of every neighbouring
voxel in W(I_{t}, p^{t}_{j}) and the intensity of the corresponding voxel
in W(I_{0}(f_{t}), p^{t}_{j}). The second energy term E_{B} is a bending
energy constraint for the transformation f_{t}. We use Laplacian
operator to impose spatial smoothness. Similar to the first term,
the bending energy is evaluated in each landmark’s
neighbourhood **W**, by integrating all bending energy of every
neighbouring voxel in W(f_{t}). The third energy term E_{S} is a
temporal smoothness constraint for the trajectories of voxels in
the neighbourhood of a landmark. It is known that the
deformation of pulmonary organ from one time point to
another time point is subtle in space, and the motion trajectory of every voxel should be a smooth curve. Similar to the second
term, the temporal smoothness of a trajectory is evaluated by
the Laplacian operator along temporal dimension.

It is noted that the three parts in the cost function is defined on
the transformations f_{t}, and evaluated in each landmark’s
neighbourhood cube. Although the transformations f_{t}, have the
property that each landmark has a global influence on the
transformation when spacing between landmarks is small,
changes of the deformation field is limited to a local
neighbourhood region instead of occurring in the whole image
area when only one landmark is updated. The changes of the
deformation field occur in the local region surrounding the
updated landmark mostly. An example is shown in **figure 3**,
suppose the current landmark are μ={p^{1}_{j}} U…U {p_{j}^{N-1}}, j=1,2,
…,k. Correspondingly, the transformations between the
reference image and the image at other phases are {f_{1},…,f_{t-1},f_{t+1},…,}. Then, the jth landmark p^{t}_{j} at t phase is updated to p^{t}_{j}.
The transformations {f_{1},…,f_{k}} are updated as {f_{1},…,f_{t-1},f_{t+1},
…,} also, where only the tth transformation function is
updated. We subtract f_{t} from f”_{t} and show the subtraction
results in** figure 3**. It can be seen that the changes of the
deformation field after updating p^{t}_{j} are limited in a local region
surrounding p^{t}_{j} instead of occurring in the whole image. This
result is exactly in agreement with the local region cube used
in the cost function. So, it is reasonable to evaluate the changes
locally, as well as the terms used in the cost function.

**Figure 3:** Illustration of locality of deformation changes. Landmark
p^{t}_{j} is slightly moved to p^{t}_{j} and leads to local changes of the
deformation field (marked as the shaded area). p^{t}_{j} is denoted as black
“+” and pi t is denoted as red “+”. The shadow areas illustrate the
changes between f_{t} and f’_{t}.

**Optimization**

The goal of optimization is to adjust arguments to let the value of cost function approach to global minima as much as possible. Optimization can be defined as follows

(6)where variables μ are all coordinates of current updated
landmarks. Each coordinate of each landmark corresponds to a
variable. 3k(N-1) is the total number of variables in μ.
Optimization of E(μ) is handled by a quasi-Newton approach
in the form of the Limited memory BFGS (LBFGS) method
[28]. LBFGS is an optimization algorithm for unconstrained
optimization, which approximates the Broyden Fletcher
Goldfarb Shanno (BFGS) algorithm using a limited amount of
memory. BFGS is a kind of quasi-Newton methods with high
precision and improved convergence rate compared to simple gradient descent algorithms. In BFGS, a dense 3(L-1)k ×
3(L-1)k matrix is adopted to approximate inverse Hessian
matrix. Since the number of variables in μ is large, it is
unwisely to imply BFGS to perform optimization. Unlike the
original BFGS, LBFGS represents the approximation
implicitly by maintaining a history of the past m updates of μ
and gradient ∇*E*, where generally the history size m can be less
than 10. These updates are used to implicitly do operations
requiring the inverse Hessian product. The LBFGS
optimization requires the gradient of the cost function with
respect to μ, which is approximated by the finite difference.
We denote the gradient of the cost function as ∇*E*= (a1, . . . ,
a3kN ). The ∇*E* is calculated by,

Where e_{i} denotes the ith unit vector, μ_{i} is the ith variable. μ_{i} +he_{i} represents that the ith coordinate is shifted slightly, and
other coordinates are kept fixed. That means only one
landmark is updated slightly and all other landmarks are fixed.
Next, the new transformation fʹ_{t} is estimated based on the
updated landmarks, and the new cost function value E(μ_{i}+he_{i})
is also calculated. It must be pointed out that when only one
landmark is shifted, the deformation field will change locally
upon fixing other landmarks (the fixed landmarks are required
to overlay the image uniformly and densely). That is, the cost
function evaluates the transformation locally, which reflects the
locality of the deformed results when gradient of the cost
function is calculated.

Ten individual patients’ lung CT images provided by DIR-lab dataset are used to evaluate our method. Each case includes the set of 10 phases CT 16-bit integer images from T00 to T90. Images from T00 to T50 are successively acquired in inspiratory period and others from T60 to T90 are successively acquired in expiratory period. For each case, 75 corresponding landmarks from T00 to T50 are provided by experts, and also 300 corresponding landmarks in the maximum inhalation and the maximum exhalation are provided by experts. We first segment lung parenchyma from 4D-CT lung images and map 16-bit data to 8bit data to enhance the intensity of parenchyma. To make it easier to align vessel intersections during the registration, the vessels in parenchyma is enhanced by multiscale hessian filters [29]. However, the vesselness response of Hessian filter is not continue, even more is fragmentary. We use the nonlinear diffusion equation proposed by Perona and Malik [30] to make the enhanced vascular structures continuous and smooth.

**Tracking results**

To demonstrate the tracking performance of our method,
landmarks marked by experts at T00 phase are tracked from
T10 to T50 respectively for the first five cases. In the image at
T00 phase, a local patch centered at a landmark with size of 11
× 11 × 7 is selected as a tracking template. The search area size
is 21 × 21 × 21 for the ST algorithm and is limited to 21 × 21 × 11 for the SEST algorithm. In the optimization process, a
window W in size of 25 × 25 × 25 is used in the cost function. **Figure 4** demonstrates the tracked results for four landmarks at
T50 phase. The first column shows four landmark locations at
T00 phase, and the second column shows the manually
identified landmarks at T50 phase. The third and fourth
columns present the tracked results at T50 phase using the ST
algorithm and the SEST algorithm respectively. We expect that
the tracked position is closer to the manually identified
position as close as possible. The coordinate below each slice
is the coordinate of the point marked by red plus. P11, P13,
P40 and P73 are indices of the four landmarks. From the
coordinates of points, we can see the displacement of these
four landmarks is obvious. Moreover, there are similar
landmarks near these landmarks, which make it difficult to be
tracked. Noted that the tracked results using the SEST
algorithm is more accurate than that using the ST algorithm.
Especially, in the case shown in the last line, the landmark
needed to be tracked is located on a vessel bifurcation, and
there is a similar bifurcation near the bifurcation. The tracking
result by the ST algorithm is misled by the similar structure,
while the tracking result by SEST algorithm is correct exactly.
This example demonstrates the performance of the SEST
algorithm for tracking object in complex background.

To quantitatively evaluate the tracking accuracy of our method,
tracking error is evaluated by computation of the Euclidean
distance between the landmark positions marked by experts
and tracked results. The tracking errors of 75 landmarks for the
first five cases of DIR-lab dataset are listed in **table 1**. From
this table, it can be obviously seen that the performance of the
SEST algorithm is better than the ST algorithm in tracking error significantly. Especially, the respiratory motions in case
3, 4 and 5 are more violent than the other two cases. The
improvement of tracking errors using the SEST algorithm for
these three cases is significant. Moreover, standard deviation of
tracking errors using the ST algorithm is larger than that using
the SEST algorithm, which implies that the tracked results of
some landmarks using the ST algorithm deviate from the
correct positions severely. It also implies that SEST algorithm
is more advantageous than ST algorithm.

Case-1 | Case-2 | Case-3 | Case-4 | Case-5 | |
---|---|---|---|---|---|

Initial | 2.18(2.54) | 3.78(3.69) | 5.05(3.81) | 6.69(4.72) | 5.22(4.61) |

ST | 0.97(1.16) | 0.91(1.07) | 1.21(1.32) | 1.80(2.03) | 2.32(2.45) |

SEST | 0.86(1.13) | 0.76(0.97) | 0.91(1.10) | 1.28(1.29) | 1.29(1.71) |

Improvement | 0.11(0.03) | 0.15(0.10) | 0.30(0.22) | 0.72(0.74) | 1.03(0.73) |

**Table 1:** Mean and standard deviation of tracking errors in mm for 75
landmarks from T00 to T50.

**Optimization Results**

**Tables 2 **and** 3** list the tracking errors of 75 landmarks and 300
landmarks before and after optimization respectively to
compare the performance of our method with optimization and
without optimization. It is observed that the tracking errors
decrease slightly whether for 75 landmarks or 300 landmarks.
Moreover, the standard deviation of tracking errors decrease
also. Especially, the mean and standard deviation of tracking
errors for case 8 is reduced significantly. For case 3, 6 and 9,
the registration results after optimization are not improved. **Figure 5** shows the box plot and the dot graph of 75 and 300
landmarks registration errors for case 3 and case 8. It can be
observed that the median value of the registration results of 75
landmarks is decreased slightly after optimization for case 3, and landmarks with extremely large errors is almost same to
that before optimization; although the median value of the
registration results of 300 landmarks is larger after
optimization, the registration errors of outliers are smaller than
that before optimization for case 3. Hence, in the statistical
sense, the registration errors of case 3 are not improved
significantly. On the contrary, although the median values of
registration errors of both 75 and 300 landmarks for case 8 are
fixed after optimization, the number of landmarks with errors
extremely larger than the median values are less after
optimization, which demonstrate the performance of global
tracking.

Cases | Initial | Tracked results | Optimization results |
---|---|---|---|

1 | 2.18 (2.54) | 0.86 (1.13) | 0.75 (0.97) |

2 | 3.74 (3.69) | 0.76 (0.97) | 0.76 (0.85) |

3 | 5.05 (3.82) | 0.91 (1.10) | 0.91 (0.96) |

4 | 6.69 (4.72) | 1.28 (1.29) | 1.18 (1.28) |

5 | 5.22 (4.62) | 1.29 (1.71) | 1.24 (1.70) |

6 | 7.42 (6.56) | 1.47 (1.87) | 1.46 (1.87) |

7 | 6.66 (6.46) | 1.28 (1.57) | 1.20 (1.26) |

8 | 9.83 (8.31) | 1.67 (1.96) | 1.38 (1.67) |

9 | 7.92 (3.98) | 1.05 (1.05) | 1.10 (1.03) |

10 | 7.30 (6.35) | 1.41 (1.84) | 1.39 (1.75) |

**Table 2:** Comparison of the mean and standard deviation of tracking
errors in mm for the 75 landmarks before and after optimization.

cases | Initial | Tracked results | Optimization results |
---|---|---|---|

1 | 3.89 (2.78) | 0.94 (1.33) | 0.86 (1.27) |

2 | 4.34 (3.90) | 0.78 (0.94) | 0.71 (0.85) |

3 | 6.94 (4.05) | 0.92 (1.11) | 0.92 (1.05) |

4 | 9.83 (4.86) | 1.43 (1.49) | 1.39 (1.44) |

5 | 7.48 (5.51) | 1.49 (1.53) | 1.39 (1.75) |

6 | 10.89 (6.97) | 1.31 (1.41) | 1.25 (1.35) |

7 | 11.03 (7.43) | 1.15 (1.76) | 1.28 (1.72) |

8 | 14.99 (9.01) | 1.68 (1.81) | 1.38 (1.49) |

9 | 7.92 (3.98) | 1.10 (0.99) | 1.10 (0.96) |

10 | 7.30 (6.35) | 1.29 (1.32) | 1.18 (1.40) |

**Table 3:** Comparison of the mean and standard deviation of tracking
errors in mm for the 300 landmarks before and after optimization.

**Figure 5:** Registration errors before and after optimization. (a)
and (b) are results of 75 and 300 landmarks for case 3,
respectively. (c) and (d) are results of 75 and 300 landmarks for
case 8, respectively. The red box plots and the red point show
the registration errors before optimization, and the blue box
plots and the blue point show that after optimization. From left
to right: box plot graphs and dot graphs.

To determine whether our tracking algorithm after optimization
has statistically significant differences between 75 and 300
Landmarks, we compute the p-value for the null hypothesis
that the registration errors of 75 landmarks and that of 300
landmarks come from the same distribution, using a Kruskal-
Wallis test. Kruskal-Wallis test is performed for all ten cases.
The returned value of p are show in **table 4**, which indicates
that Kruskal-Wallis accepts the null hypothesis that almost all
data samples come from the same distribution at a 5%
significance level.

case | p value |
---|---|

1 | 0.0534 |

2 | 0.1216 |

3 | 0.064 |

4 | 0.5331 |

5 | 0.7116 |

6 | 0.0591 |

7 | 0.3163 |

8 | 0.1691 |

9 | 0.223 |

10 | 0.0441 |

**Table 4:** The return value of p using Kruskal-Wallis test to compare
the registration errors of 75 landmarks and that of 300 landmarks.

**Registration results and analysis**

Furthermore, we compared the tracking error of our method with spatial temporal image registration [31].

B-Spline based 4D registration algorithm [32], 4D optical flow
method based on trajectory modeling [22], and trajectories
tracking algorithm [21]. The registration errors by spatialtemporal
image registration, B-splines based 4D registration,
trajectories tracking algorithm and our method on 75 landmarks between T00 and T50 phases are listed in **table 5**.
The registration errors by 4D optical flow method based on
trajectory modeling, B-splines based 4D registration,
trajectories tracking algorithm and our method on 300
landmarks are listed in** table 6**. It is observed from **table 5 **that
mean tracking errors of SEST algorithm for 75 landmarks are
smaller than trajectories tracking algorithm and B-Spline based
4D registration algorithm, and closer to spatial-temporal image
registration results for the first five cases. Unfortunately, Metz
et al. did not published registration results for case 6 to case 10.
For case 6 to case 10, the mean tracking errors by SEST
algorithm are smaller than that by B-Spline based 4D
registration algorithm and closer to that by trajectories tracking
algorithm. By comparing the registration results for 300
landmarks, listed in **table 6**, it can be clearly seen that the
tracking errors of our experimental results are almost the
smallest among these four methods.

label | Initial | Spatial-temporal | BSpline | Trajectories tracking | SEST |
---|---|---|---|---|---|

case 1 | 2.18 (2.54) | 0.79 (0.46) | 0.97 (0.71) | 0.97 (0.65) | 0.75 (0.97) |

case 2 | 3.74 (3.69) | 0.86 (0.52) | 0.98 (0.61) | 0.93 (0.54) | 0.76 (0.85) |

case 3 | 5.05 (3.82) | 0.92 (0.53) | 1.17 (0.66) | 1.09 (0.59) | 0.91 (1.10) |

case 4 | 6.69 (4.72) | 1.07 (0.75) | 1.37 (0.97) | 1.41 (0.81) | 1.18 (1.28) |

case 5 | 5.22 (4.62) | 1.13 (0.86) | 1.46 (1.40) | 1.35 (0.87) | 1.24 (1.70) |

case 6 | 7.42 (6.56) | 2.26 (1.96) | - | 1.36 (0.92) | 1.46 (1.87) |

case 7 | 6.66 (6.46) | 1.85 (1.96) | - | 1.26 (0.82) | 1.20 (1.26) |

case 8 | 9.83 (8.31) | 2.40 (1.95) | - | 1.40 (0.98) | 1.38 (1.67) |

case 9 | 7.92 (3.98) | 2.05 (1.95) | - | 1.21 (0.70) | 1.10 (1.03) |

case 10 | 7.30 (6.35) | 1.69 (1.52) | - | 1.27 (0.86) | 1.39 (1.75) |

**Table 5:** Mean and standard deviation of registration errors in mm for
the 75 landmarks from T00 to T50.

label | Initial | 4D optical flow | BSpline | Trajectories tracking | SEST |
---|---|---|---|---|---|

case 1 | 3.89 (2.78) | 0.97 (1.02) | 1.15 (0.60) | 1.05 (0.52) | 0.86 (1.27) |

case 2 | 4.34 (3.90) | 0.86 (1.08) | 1.06 (0.62) | 1.00 (0.41) | 0.71 (0.85) |

case 3 | 6.94 (4.05) | 1.01 (1.17) | 1.27 (0.68) | 1.15 (0.62) | 0.92 (1.05) |

case 4 | 9.83 (4.86) | 1.40 (1.57) | 1.55 (1.17) | 1.39 (0.72) | 1.39 (1.44) |

case 5 | 7.48 (5.51) | 1.67 (1.79) | 1.82 (1.59) | 1.43 (0.89) | 1.39 (1.75) |

case 6 | 10.89 (6.97) | 1.58 (1.65) | - | 1.32 (0.80) | 1.25 (1.35) |

case 7 | 11.03 (7.43) | 1.46 (1.29) | - | 1.29 (0.70) | 1.28 (1.72) |

case 8 | 14.99 (9.01) | 1.77 (2.12) | - | 1.39 (0.82) | 1.38 (1.49) |

case 9 | 7.92 (3.98) | 1.19 (1.12) | - | 1.32 (0.68) | 1.10 (0.96) |

case 10 | 7.30 (6.35) | 1.59 (1.87) | - | 1.24 (0.68) | 1.18 (1.40) |

**Table 6:** Mean and standard deviation of registration errors in mm for
the 300 landmarks in T00 and T50 phases.

**Figure 6** shows the isosurface rendering of the T00 lung
images of case 5 and case 8. The trajectories of 75 landmarks
from their T00 to T50 positions are displayed in the lung
images also. The displacement vectors of 75 landmarks
identified by experts for case 5 and case 8 are shown in **figures
6a **and** 6c** respectively, and the corresponding trajectories
estimated by our method are shown in **figures 6b **and** 6d** respectively. Visually, most landmarks are tracked correctly.
Noted that isolated landmarks are mistracked for case 5, **figure
7** shows the displacement vectors and residual error vectors of
300 landmarks for case 5 and case 8. **Figures 7a **and** 7c** show
the displacement vectors of 300 landmarks at T50 phase
provided by experts in anterior projection for case 5 and case 8 respectively. **Figures 7b **and** 7d **show the corresponding
residual error vectors. The error vectors starts from the
manually marked position to that tracked by SEST in the
image. Noted that these residual error vectors are short and
distribute uniformly approximately. Several large error vectors
can be observed which correspond with mistracking.

**Figure 6:** Landmark points and tracked trajectories. 75 landmarks
identified by experts are utilized to demonstrate the tracking accuracy
of our method. (a) and (c) show the displacement vectors of 75
landmarks provided by experts in anterior projection for case 5 and
case 8. Landmarks are in red and the trajectories of landmarks are
shown in blue. (b) and (d) show the tracked trajectories of 75
landmarks for two cases.

**Figure 7:** Landmark points and tracked errors. 300 landmarks
identified by experts are utilized to demonstrate the tracking accuracy
of our method. (a) and (c) show the displacement vectors of 300
landmarks provided by experts in anterior projection for case 5 and
case 8. Landmarks are in red and the trajectories of landmarks are
shown in blue. Residual error vectors are shown in (b) and (d). The
error vectors starts from the manually marked position to that tracked
by SEST in the image at T50 phase.

The box plots and the dot graph of the 75 and 300 landmarks
registration errors of each case in the DIR-lab dataset are
shown in **figures 8 **and** 9** respectively. The number of outliers
of the 75 landmarks target registration errors in all phases of
each case is between 5 and 24, as listed in **table 7**. The average
ratio of the 75 landmark points’ target registration error outliers is 2.8%. The number of the 300 landmark points’
target registration error outliers in the ME phase of each case is
between 1 and 34. The average ratio of the 300 landmark
points’ target registration error outliers is 2.7%. The outliers in **Figure 8 **and** Figure 9** are caused by two reasons. The first one
is that some of the landmarks are difficult to be tracked, which
may create relative large tracking errors. The second one is that
several landmarks marked by experts are not correct, which
leads to deviation of the mean tracking error.

cases | 75 landmarks | 300 landmarks | ||||
---|---|---|---|---|---|---|

number | mean | percent | number | mean | percent | |

1 | 24 | 3.1 | 6.4% | 34 | 3.5 | 11% |

2 | 7 | 3.5 | 1.8% | 1 | 2.9 | 0.3% |

3 | 5 | 3.7 | 1.3% | 4 | 4.4 | 1.3% |

4 | 8 | 6.3 | 2.1% | 6 | 8.6 | 2.0% |

5 | 5 | 6.6 | 1.3% | 5 | 11 | 1.7% |

6 | 12 | 9.7 | 3.2% | 5 | 7.9 | 1.7% |

7 | 7 | 6.3 | 1.8% | 4 | 5.8 | 1.3% |

8 | 15 | 7.5 | 4.0% | 7 | 8.3 | 2.3% |

9 | 5 | 4.2 | 1.3% | 11 | 3.5 | 3.7% |

10 | 19 | 8.9 | 2.4% | 6 | 8.1 | 2.0% |

**Table 7:** Comparison of the mean and standard deviation of tracking
errors for the 75 and 300 landmarks before and after optimization.

**Temporal smoothness**

Furthermore, we will evaluate the temporal smoothness of the
trajectories of landmarks after optimization. Case 8 and case 9
are selected as examples, where the tracking errors are
improved greatly after optimization for case 8, and are almost fixed after optimization for case 9, especially in phase 4. The
velocity magnitudes between consecutive phases for some
trajectories in case 8 before and after optimization are
illustrated in **figures 10a **and** 10b** respectively. **Figure 10c **and** 10d** show the results for case 9.

Based on the gradient of the velocity magnitudes, we select landmarks that change the velocity magnitudes significantly after optimization to plot. It can be observed that the velocity magnitudes are more stable after optimization for case 8 and case 9, which implies that smooth trajectories of landmarks can be achieved using our method.

4D-CT image registration algorithm based on local-global landmarks tracking is proposed in this paper. Spatial information is introduced into structure tensor to improve its description ability about image objects. The tracking algorithm based on the spatially extended structure tensor is capable of tracking landmarks position locally at all phases in the 4D-CT image. During the tracking procedure, the prior knowledge of respiratory motion can be used to limit the searching area. Furthermore, the spatial-temporal optimization model combining the image similarity, bending energy of transformations, and smoothness of landmark trajectories is a global tracking model to update the locally tracked results of landmarks. After optimization, the tracking errors of landmarks are decreased and the trajectories of landmarks are smoother. Experimental results of open source data from DIR-lab show that the local-global landmark tracking is feasible and robust.

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