Doppler Velocity Reconstruction via Streamfunction-Vorticity Formulation
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Solution Overview
Problem
Conventional Doppler image reconstruction methods face challenges in achieving physical consistency and measurement accuracy, particularly in resolving rotational flow features and boundary conditions, leading to underestimation of velocity and inaccurate post-measurement quantities like pressure and energy loss.
Innovation Solution
A novel image processing approach that reconstructs 2D or 3D velocity fields from a single velocity component Doppler image by using a streamfunction-vorticity formulation, applying boundary conditions, and iteratively solving the Poisson Equation to generate an updated streamfunction, thereby improving velocity component reconstruction.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Ease of manufacture
If conventional Doppler image reconstruction methods solve solely the continuity equation with symmetry assumptions, then the formulation is simplified, but measurement accuracy and physical consistency deteriorate
Solution Approach 1:
The patent transforms the velocity field formulation by introducing a streamfunction parameter that automatically satisfies the continuity equation. This parameter change allows the method to move from solving the continuity equation directly to solving a vorticity transport equation, which improves measurement accuracy while maintaining formulation tractability through the streamfunction-vorticity transformation.
Solution Approach 2:
The patent introduces a streamfunction as an intermediary variable that mediates between the continuity equation and the velocity field reconstruction. This streamfunction serves as a bridge that enforces mass conservation automatically while allowing the vorticity equation to be solved, thereby improving physical consistency without requiring direct solution of the continuity equation with symmetry assumptions.
2Reliability
If scalar formulation is used to solve the continuity equation, then a more robust solver is achieved, but boundary conditions become difficult to apply and rotational flow features are under-resolved
Solution Approach 1:
The patent segments the velocity field into two distinct components: a streamfunction that handles the continuity constraint and a vorticity field that captures rotational features. This segmentation allows the robust solver to work with the streamfunction while the vorticity equation separately resolves rotational flow features, eliminating the trade-off between solver robustness and rotational flow resolution.
Solution Approach 2:
The patent transitions from solving a scalar velocity field to solving a two-field system consisting of streamfunction and vorticity. This dimensional expansion in the solution space allows the method to simultaneously satisfy continuity constraints through the streamfunction and resolve rotational features through the vorticity field, overcoming the limitations of scalar formulation.
3Measurement precision
If meteorology methods with momentum equations and time-varying fields are used, then initial estimates are improved, but velocity underestimation occurs and formulations become more complex
Solution Approach 1:
The patent extracts the time-varying and momentum equation components from the meteorology methods, retaining only the essential feature of allowing the field to evolve. By removing the complex momentum equations while keeping the iterative refinement approach, the method achieves good initial estimates without the excessive formulation complexity of full meteorology models.
Applied Scientific Principles
This section explains which scientific principles are used to turn an abstract innovation direction into a practical engineering solution.
Function Achieved in This Case
This method enhances the accuracy of velocity measurements, providing more precise post-measurement quantities such as pressure, circulation, and energy loss, while ensuring physical consistency and minimizing errors in rotational flow features.
Implementation Method 1
Doppler velocity measurement by Doppler image processing is utilized in several fields including meteorology, oceanography, petroleum production, and medical examinations to measure flow velocity of a working fluid
Data Source
AI summary
A non-transitory computer-readable medium encoded with a computer-readable program, which, when executed by a processor, will cause a computer to execute a method of processing an image, wherein the method includes receiving a 2-D color Doppler image. The method additionally includes extracting a single component velocity field of a 2-D plane from the 2-D color Doppler image. Further, the method includes receiving a geometrical boundary of a region of interest within the 2-D color Doppler image. Moreover, the method includes applying a plurality of boundary conditions to the geometrical boundary, an at least one inlet, and an at least one outlet, of the single component velocity field of a 2-D plane. The method additionally includes solving a streamfunction vorticity formulation to reconstruct a transverse velocity component. Further, the method includes outputting a reconstructed 2-D 2-component velocity based on the transverse velocity component.