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This study proposes a fast 3D dynamic programming expansion to find a shortest surface in a 3D matrix. This algorithm can detect boundaries in an image sequence. Using phantom image studies with added uniform distributed noise from different SNRs, the unsigned error of this proposed method is investigated. Comparing the automated results to the gold standard, the best averaged relative unsigned error of the proposed method is 0.77% (SNR = 20 dB), and its corresponding parameter values are reported. We further apply this method to detect the boundary of the real superficial femoral artery (SFA) in MRI sequences without a contrast injection. The manual tracings on the SFA boundaries are performed by well-trained experts to be the gold standard. The comparisons between the manual tracings and automated results are made on 16 MRI sequences (800 total images). The average unsigned error rate is 2.4% (SD = 2.0%). The results demonstrate that the proposed method can perform qualitatively better than the 2D dynamic programming for vessel boundary detection on MRI sequences.

Many image edge detection methods have been proposed in the last two decades [

To solve this problem, we propose a 3D-expansion of DP that can seek an optimal surface in a 3D matrix. This method can search succeeding boundaries in an image sequence, as succeeding boundaries form a surface in a 3D matrix. The relationship between two succeeding boundaries is thus considered in the smoothness constraint. To investigate the performance of this method, we design experiments to calculate its accuracy. The significance of this paper is two-fold: (1) we propose a 3D-expansion of DP which can locate an optimal surface among a 3D matrix; (2) we explore the best parameter set of the proposed 3D expansion of DP for round shape objects' boundary detection, and its unsigned relative error is reported.

The rest of this paper is organized as follows. In Sections 2 and 3, we introduce the 2D and 3D-expansion of DP, respectively. In Section 4, we address our experimental designs. In Section 5, we illustrate the experimental results. We then discuss the proposed method and give a conclusion in Sections 6 and 7.

In computer science, DP is a method for solving complex problems, which has similar properties that can be broken down into simpler sub-problems. Therefore, to solve a problem is then transformed to solve different parts of its sub-problems. However, many sub-problems are actually the same. The DP solves each sub-problem once and the result is saved for solving next similar sub-problem. Thus DP reduces the number of computations. A good tutorial on DP can be found in [

Let the feature image saved in a 2D matrix be ^{M×N}. The boundary detection problem is then transformed to an optimization problem that searches for an optimal path. Assume the searching direction is from left to right. The optimization function can be defined to find the optimal path having the following global minimum:
_{x}_{x}_{x+1} are neighborhood. This optimization function can thus be reformulated to implement DP with respect to an iterative cost function:

The index can be stored in the 2D coordinate matrix

Let the 3D matrix R have size _{1} ≥ |_{x}_{x−1}| controls the smoothness on the _{2} ≥ |_{z}_{z−1}| controls the smoothness on the _{x}_{x−1}| ≤ _{1} and |_{z}_{z−1}| ≤ _{2}.

This optimization problem minimizes the iterative cost functions as such:

_{2} are accumulation matrices of size _{2} are thus:

During the accumulation process, the indices must be stored in Y_{1}(_{1} and Y_{2} (_{2} for the backwards tracing [_{2} contains the minimal value in the last column. In the backwards tracing, the y-coordinates of the optimal surface can be obtained sequentially. The minimal value in the last column is found using

The remaining y- coordinates in the same index _{2}(y*(_{1} searches for a node in its previous column (_{i}_{1}, shown in _{1} and weighted by _{1}; both control the smoothness of the surface on _{1}|_{2} searches for the node in its previous column (_{k}_{2}, shown in _{2} and weighted by _{2}; both control the smoothness of the surface on _{2}|_{1}(_{2} matrices thus contain the feature and neighboring information on both

Compared to the traditional DP, the 3D DP has the advantage that it connects curves in two planes (_{1} matrix, it produces the same results as the traditional DP.

To investigate the accuracy of the proposed method, we design a phantom image sequence with added uniform distributed noise. The image gray level is normalized to be within the range [0,1]. The foreground (artery lumen) is set to 1, and the background is set to 0.4. The image has the size 41 × 41, with simulated artery lumens having radii ranging from 9 to 12 pixels (with step 1) and back to 8 pixels. Each image contains one artery lumen, and there are eight images in a sequence. The noise intensity is randomly given at every pixel, and the SNR (signal-to-noise ratio) is measured as such:
_{level}_{level}_{level}

To extract boundary information, we apply the directional gradient [_{1} = _{2} in _{1} = 1 and _{2} = 2. The factor _{2} is larger because we allow the method to catch larger radius changes in the sequence. The image resize is performed after the image gradient computation, and it is transformed from Cartesian to polar coordinates. A 3D volume containing the gradient images is then used as input for the 3D-expansion of DP.

A mobile MRI (Siemens-type “AvantoTM”, 1.5 T) was used for image acquisition [

A mobile 1.5-T Magnetom (Siemens-Avanto™, Model Mob. MRI 02.05, Siemens Ltd., Erlangen, Germany), with a flexible 6-channel body matrix coil with 6 integrated low-noise preamplifiers (Siemens Ltd.) and bilateral table fixation, was used to acquire SFA MRI sequences from the subjects. All of the participants were athletes. The athletes were fixed in a stretched supine position, head forward on the MR table. To identify the axial perpendicular acquisition location at the right SFA 10 mm beneath the bifurcation of the common femoral artery, a biplane coronal and sagittal localizer (TRUFI: “true fast imaging with steady state precision”; Siemens Ltd.) was used. As in the common carotid artery MR-measurement, changes in the SFA diameter during systole and diastole was assessed with a T2*-weighted gradient-spoiled gradient-echo cine-sequence (FLASH: “fast low angle shot”, Siemens Ltd.) in a two-dimensional cross section view with prospective two-dimensional ECG gating (cardiac triggering). Specific sequence parameters were set thus: flip angle 15°, echo time variable between 4–6 ms (depending on heart rate), repetition time variable 20–40 ms (depending on heart rate), slice thickness 6 mm, field of view 768 cm^{2}, matrix size 512 × 384, pixel width 0.625 mm (pixel area 0.3906 mm^{2}) ISO, pixel bandwidth 250, and 50 images per sequence for one RR-cycle (approximately 300 heart beats per sequence). The total imaging acquisition time was approximately 4 min 30 s to 5 min for each sequence.

Each real MRA image sequence contains 50 images. Our previous study has proposed an automatic SFA position detection method [

The proposed system is applied, and the SFA cross-sectional lumen area of each image is calculated for the following comparison. The comparison is performed by calculating their relative unsigned errors:
_{Automated}_{Manual}

Furthermore, we demonstrate the ability of the proposed method when one image is ruined.

In the real MRA image study, we show the robustness of the proposed method against vagueness.

The Bland-Altman plot [^{2} with SD 2.14 mm^{2}. The 95% confidence intervals are within 4.2 mm^{2}. ^{2} with SD 2.11 mm^{2}. The 95% confidence intervals are within 4.1 mm^{2}. The bias is very limited (0.22 mm^{2}). The system is consistent in its data number (SD = 2.11 mm^{2}). The results demonstrate the system's ability to replace the expert's manual work.

The computer system has an Intel^{®} Core™ 2 CPU T5600, 1.83 GHz, with 2 GB RAM. All programs are designed based on the Matlab platform [

In the proposed algorithm, there are two weighting factors, _{1} and _{2}, and two constraint factors, _{1} and _{2} (_{1} defines the boundary connectivity on the same image. It is thus usually defined to be less than or equal to 2. Distance _{2} defines the boundary connectivity on a different image level and is thus case dependent. If boundaries on different levels have a large distance, then _{2} must be defined larger. In this study, the SFA have 50 images during an R-R interval. The boundary thus does not change much, and _{2} is set to 2.

From the phantom study, we know that the effect of the resize factor is larger than the weighting factors _{1} and _{2}. We therefore set the resize factor to 1.6 (the optimal parameter value in the phantom study) and choose the weighting factors as 0.1 (_{1} = _{2} = s = 0.1) arbitrarily in the real MRI study.

We have proposed a three dimensional DP, which can find an optimal surface in a 3D matrix where features are normally saved. This method has robustness against strong noise, especially when one or two images are absent in the image sequence. The proposed property is useful, especially for detecting an artery's boundary in an MRI when one or two images in the artery's boundary are vague. The connectivity can help rebuild the artery's boundary. This method can be extended to track an object's boundary in image sequences, especially when partial occlusion exists. Furthermore, it is possible to embed another model into the DP and let it extract a special shape, including a round [

The applications of the proposed algorithm are not restricted to vessel segmentation. It is also practical for bone segmentation in 3D CT slices [

Our future aim is to test this system on sonographic image sequences to detect the intima on both sides of the CCA inner wall to extract the inner wall lumen diameter changing with time. Combined with the blood pressure information, the lumen diameter change can compute the arterial elasticity, which is an important marker in clinics. As a final remark, we must note that this method might be extended to the dual surface detection in a 3D volume, such as the dual dynamic programming in a 2D case [

We have developed a novel algorithm that can detect the optimal surface in a 3D volume matrix. This algorithm is applied to detect SFA boundaries in MRA image sequences. According to our phantom study, we conclude that the optimal parameter values for using the 3D DP are thus: image resize factor = 1.6, and weighting factor s = 0.02. In the MRA real image study, the average relative unsigned error is 2.4% ± 2.0% in 16 MRI sequences (800 images) compared to the manual tracings. The algorithm proposed in this study is reliable with repeatable results that can replace the experts' manual work.

This work was supported by the National Science Council (NSC), Taiwan, under Grant NSC 100-2221-E-039-001. The author acknowledges Uwe Schütz (University Hospital of Ulm, Germany) for providing the MRI sequences and their manual tracings to the gold standard.

The 3D matrix R contains the feature of an image sequence. The structure of the accumulation matrices (_{1} and (_{2} is organized in different orientations.

The accumulation matrices C_{1} and C_{2}. The dimensions of (_{1} and (_{2} is same as those of R.

Two raw MRA images in one sequence. Each sequence contains 50 images. (

The phantom images with added noise from different SNRs.

The unsigned error plots of different phantom images. SNR is (

The contour detection results. SNR = 14 dB, s = 0.01, resize factor = 1.6. Averaged unsigned error = 1.5%.

One image is ruined. The proposed method can still hold the contour because of the continuity of the 3D DP method (_{2} = 1).

The traditional DP fails to hold the contour, as there is no information between image slices.

Boundary detection results. Nine sequential resultant images are shown. The vessel boundaries are vague in the first five images.Their exact boundaries are difficult to determine. Based on the continuity consideration in three dimensions, the proposed method can detect the correct vessel boundaries.

Bland-Altman plots. The solid line indicates the mean, and dash lines are ± 1.96 SD. (^{2}); (^{2}).

The cross-sectional SFA area changing in time (sequence no. 9). The unit in the y-axis is mm^{2}. The x-axis unit is image number.

The software system GUI. The system can read DICOM and other image formats supported by MatLab. However, only the DICOM format offers pixel size information.

The average unsigned relative errors on 16 MRI sequences (800 total images).

1 | 2.4 | 1.8 | 9 | 2.1 | 2.6 |

2 | 3.2 | 2.1 | 10 | 2.9 | 2.5 |

3 | 1.4 | 1.3 | 11 | 1.4 | 1.0 |

4 | 3.2 | 2.7 | 12 | 2.5 | 1.7 |

5 | 3.6 | 2.6 | 13 | 2.4 | 1.2 |

6 | 3.1 | 2.4 | 14 | 3.2 | 2.0 |

7 | 1.9 | 1.6 | 15 | 1.6 | 1.7 |

8 | 2.7 | 2.8 | 16 | 1.4 | 1.1 |

Each sequence contains 50 images. Resize factor = 1.6, s = 0.1.