Confidance limits

When dealing with thousands of images featuring dynamically behaving shocks, achieving accurate tracking can be tough, especially if the shock contrast is low or it’s not perfectly aligned. That’s where the SliceListGenerator.GenerateSlicesArray function comes in. (A detailed methodology for this process will be available in an upcoming publication.)

The core challenge is that relying on a single pixel row for shock tracking can lead to lost contrast. To combat this, the SliceListGenerator.GenerateSlicesArray function employs a multi-faceted approach:

Vertical Averaging for Contrast: To boost the shock’s visibility, the function applies vertical averaging of pixel rows. This feature is activated through the attribute slice_thickness. This process combines information from several rows around the shock area, making it stand out more effectively.
Precise Angle Estimation: Shocks are not always conveniently perpendicular to the image’s horizontal axis. To ensure accurate averaging, images need to be rotated so the shock is perpendicular to the image frame. Getting this angle right is critical; errors can cause the shock to blend in, thereby reducing tracking accuracy. This angle estimation is automatically activated when slice_thickness is defined. However, the tracking range can be separately defined using the attribute sat_vr, which stands for shock angle tracking vertical range.
Robust Angle Refinement with RANSAC: To prevent misalignment from incorrect angle estimates, SliceListGenerator.GenerateSlicesArray, employs a statistical technique called Random Sample Consensus (RANSAC).
- It begins by taking random samples of tracked shock points.
- Then, it fits a line to these points using the vertical least squares method.
- This process is repeated, iteratively maximizing the number of “inliers” (points that fit the line well), ensuring a highly robust and accurate shock angle estimation. The maximum number of RANSAC iterations ($itr_{max}$) is determined probabilistically.
more info. can be found inc_tracking_support.ransac
Post-Processing Evaluation: After fitting the line, its accuracy is evaluated using linear regression relations, which are activated with the attribute conf_interval. This helps assess the scatter of the points and determines the confidence ($z_j$) of the estimated line. The shock’s standard deviation also provides valuable insights into its dynamics. more info. can be found tracking_accuracy.error_analysis
Outlier Elimination: For more accurate angle estimation, Tukey’s fences approach [1, 2] is utilized. This method is automatically activated when conf_interval is defined and is used to remove outliers, especially if they exhibit high leverage and uncertainty. more info. can be found tracking_accuracy.IQR
Averaging Shock Angles: Finally, for the entire set of images, the average shock angle is calculated. This can be done using either a simple arithmetic average ($m$) or a weighted average ($m_w$). The weighted average is particularly useful as it accounts for variations, using the inverse standard deviation of the slope at each snapshot to give more reliable estimates greater weight. more info. can be found tracking_accuracy.compute_weighted_average

Example

Run the following code:

from ShockOscillationAnalysis import SliceListGenerator

if __name__ == '__main__':
    # Define the snapshots path with glob[note the extention of imported files]
    imgPath = r'test_files\raw_images\*.png'

    f = 2000    # images sampling rate
    D = 80      # distance in mm

    output_directory = r'results\Slicelist_confidance-results'

    # iniate the SliceListGenerator module
    SA = SliceListGenerator(f, D)

    # use GenerateSlicesArray function
    ShockwaveRegion ,n ,WR, Scale = SA.GenerateSlicesArray(imgPath,
                                                        # Define the reference line locations
                                                        Ref_x0=[109, 726], Ref_y0=617,
                                                        # Define the range of images to be only imported
                                                        within_range = [50,200],
                                                        scale_pixels=True,
                                                        # as scaled tracking reference values in mm
                                                        slice_loc=10,
                                                        # to crop the slices by vertical reference line
                                                        full_img_width=False,
                                                        # in mm
                                                        slice_thickness=[5, 'mm'],
                                                        # Define shock angle tracking vertical range in mm
                                                        sat_vr=[-5, 8, 'mm'],
                                                        # Define confidence interval with 98% accuracy
                                                        conf_interval=0.98,
                                                        # number of samples to determine the average inclination
                                                        shock_angle_samples=33,
                                                        # Define the number of points to track for angle estimation
                                                        nPnts = 15,
                                                        # to preview the tracked points during angle determination
                                                        angle_samples_review=3,
                                                        # information for angle determination
                                                        inclination_est_info=[110, (474, 591), (463, 482)],
                                                        # to preview the final setup before proceeding
                                                        preview=True,
                                                        # display properties
                                                        avg_preview_mode='avg_ang',
                                                        points_opacity=0.5,
                                                        points_size=5,
                                                        # the directory where the slice list will be stored
                                                        output_directory=output_directory,
                                                        # additional comments to the stored slice list file name
                                                        comment='-SliceList',
                                                        )

Review the spacified domain and reference lines, then press Esc to abort or any other key to continue:
$_static\img\T5\T5-3c.png$

The software will estimate the shock angle, store the angle_samples_review and preview the rotated image, press Esc to abort or any other key to continue:

Img Shape is: (900, 960, 3)
Image scale: 0.12965964343598055 mm/px
Slice center is located at:
    - 540px in absolute reference
    - 9.98mm (77px) from reference `Ref_y0`
Shock angle tracking vertical range above the reference `Ref_y0` is:
    - In (mm)s from 18.02mm to 4.93mm
    - In pixels from 139px to 38px
Shock inclination test and setup ... ✓
Warning: Requested files are out of range; Only available files will be imported from 50 to 100
Import 33 images for inclination Check ...
[====================] 100%
Shock inclination estimation ...
Shock tracking started ... ✓
Calculating confidance limits ... ✓
weighted average shock angle: 93.79±0.109 deg,   σ = 0.00083 deg
Angle range variation: [79.95, 104.28], σ = 5.10 deg
Average shock loc.: 469.75±6.47 px
Average shock angle: 95.03±1.90 deg
Plotting tracked data ...
[====================] 100%
Processing time: 5 Sec

$_static\img\T5\T5-61.png$ $_static\img\T5\T5-70.png$ $_static\img\T5\T5-83.png$ $_static\img\T5\T5-4.png$ $_static\img\T5\T5-5.png$

Note

In the first and third images, red uncertain points significantly deviate from the actual shock location due to the complexity of the shock structure. However, the algorithm successfully identified and excluded these outliers, focusing instead on the correct shock angle.
These uncertain points can influence the arithmetic average of the shock angle. However, in the weighted average approach, images like the first and third—which exhibit high standard deviation—are assigned lower weights compared to the second image. This results in a more reliable average estimation.
The standard deviation for arithmetic average are notably higher than the weighted average, indicating greater variability and lower confidence. This difference is also reflected in the overall confidence level of the result.
A log.txt file is generated in the results directory. It contains details about the tracking process and the operations performed.

The software will generate the slice list and store the data:

RotatedImage: stored ✓
DomainImage: stored ✓
working range is: {'Ref_x0': [109, 726], 'Ref_y1': 540, 'avg_shock_angle': array([95.03081317,  1.90169604, 93.79126163,  0.10905212,  5.10271056]), 'avg_shock_loc': array([469.7515693 ,   6.47399096,  17.37128401])}
Importing 50 images ...
[=================== ] 98%
ImageList write: Image list was stored at: results\Slicelist_confidance-results\2.0kHz_10mm_0.12965964343598055mm-px_tk_38px_-SliceList.png

$_static\img\T5\T5-6.png$