# Trackerless 3D Freehand Ultrasound Reconstruction: A Review
## 2 Materials and Methods
### Data Extraction
#### 2.2.2 Quantitative Outcome Measures of 3D Reconstruction Methods  
For each study, relevant quantitative outcome measures were extracted for the proposed reconstruction method and important ablation experiments. These ablation experiments involve systematically removing or modifying components of a model to evaluate their impact on performance, thereby elucidating the contribution of individual elements within the model. Evaluation of the reconstruction methods entailed extraction of trajectory reconstruction errors, volume reconstruction errors, and the computation time. If none of the following described outcome measures were reported within a study, the main outcome measure deemed relevant was extracted.  
Trajectory Reconstruction Errors  
Evaluation of trajectory errors involved two principal groups of metrics:  
**(1) Error between successive frames:** The reconstruction of the trajectory relies on the prediction of transformation parameters between successive frames. Consequently, the most direct metric to gauge model performance is the mean absolute error (MAE) between predicted and ground truth transformation parameters across a sequence of N frames, as outlined in Equation (1):  
\[\text{MAE}(t_{x},t_{y},t_{z},\theta_{x},\theta_{y},\theta_{z})=\frac{1}{N} \sum_{k=1}^{n}\left|\vec{p}\left(T_{k}(i\to j)^{\text{pred}},T_{k}(i\to j)^{ \text{GT}}\right)\right| \tag{1}\]  
Here, \(|\vec{p}(A)|\) computes the element-wise absolute difference between each corresponding parameter of the predicted \(T_{k}(i\to j)^{\text{pred}}\) and ground truth \(T_{k}(i\to j)^{\text{GT}}\) transformation matrices for each pair (k) of successive US frames (i \(\rightarrow\) j). This matrix consists of the translation (\(t_{x}\), \(t_{y}\), \(t_{z}\)) and rotation (\(\theta_{x}\), \(\theta_{y}\), \(\theta_{z}\)) parameters. The MAE was considered parameter-wise for all transformation parameters or in subsets denoting only translation, orientation, or out-of-plane parameters.  
In addition to element-wise parameter analysis, error measurement was conducted based on the spatial location of frames, denoted as the frame error. This method calculates the Euclidean distance between the predicted and ground truth frame locations, averaged over the four corners of each frame. This approach offers a broader perspective on the positional accuracy of the frame sequence without focusing solely on individual transformation parameters.  
**(2) Drift measures:** Another key metric in trajectory reconstruction is the final drift (FD), defined by the Euclidean distance between the positions of the center of the last frame of the estimated trajectory and the last frame of the ground truth trajectory. Since the prediction of the trajectory of the sweep is based on a transformation between successive frames, errors accumulate over the length of the sweep. This accumulated error is reflected in the metric FD. Additionally, the final drift rate (FDR) (%) was extracted, which is the FD normalized by the ground truth length of the sweep. Important to note is that a minimal FD does not indicate a satisfactory reconstruction for complex scan strategies such as a loop scan, which requires additional evaluation metrics.  
Volume Reconstruction Errors  
For volume reconstruction errors, quantitative metrics such as Dice overlap and the absolute difference in volume, in milliliters, were extracted. These measures assess the degree of overlap and volumetric accuracy between the reconstructed volume and the ground truth, providing a robust framework for evaluating the performance of 3D reconstruction algorithms.