Junior Research Fellow at Indian Institute of Science

Overview

As a Junior Research Fellow at the Supercomputer Education and Research Centre, Indian Institute of Science, Bangalore, I worked on developing efficient algorithms for image reconstruction in Near Infrared (NIR) Diffuse Optical Tomography. My research focused on optimizing data-collection strategies to identify independent measurements, thereby reducing data acquisition time while maintaining reconstructed image quality. This work resulted in a first-author publication in Medical Physics, a leading journal in medical imaging.

Publication

Data-resolution based optimization of the data-collection strategy for near infrared diffuse optical tomography

• Authors: Deepak Karkala and Phaneendra K. Yalavarthy
• Journal: Medical Physics, Vol. 39, No. 8, pp. 4715-4725 (August 2012)
• DOI: 10.1118/1.4736820

• Published: 18 July 2012

Research Problem

Diffuse Optical Tomography (DOT) is an emerging non-invasive medical imaging modality that uses near-infrared light (600-1000 nm wavelength) to reconstruct internal distributions of optical properties in tissues. It has applications in:

• Breast cancer imaging
• Brain functional imaging
• Small animal imaging studies

Key Challenges:

1. Ill-posed Inverse Problem: Light scattering dominance in soft tissue makes the reconstruction problem nonlinear, ill-posed, and sometimes underdetermined
2. Redundant Measurements: Existing optimization techniques could identify the number of useful measurements but not specific information about which measurements are independent
3. Data Collection Efficiency: No method existed to determine whether a particular measurement is independent or redundant, leading to unnecessary data collection time

Key Contributions

1. Data-Resolution Matrix Based Optimization Method

Developed a novel optimization technique using the data-resolution matrix to identify specific independent measurements among total measurements in a data-collection strategy.

Technical Innovation:

The data-resolution matrix (N) is computed based on the sensitivity/Jacobian matrix (J) and the regularization parameter (λ) used in the reconstruction:

N = JJ^T [JJ^T + λI]^(-1)

Key Properties:

• Diagonal Entries: Indicate the importance/weight of each measurement (values between 0 and 1)
• Off-Diagonal Entries: Show the dependency of measurements on each other
• Model-Based: Depends only on the sensitivity matrix and regularization, independent of noise characteristics

Algorithm for Identifying Independent Measurements:

1. Compute Jacobian (J) using uniform initial guess
2. Set threshold (th) between 0.8 and 1.0 to indicate dependency level
3. Calculate data-resolution matrix: N = JJ^T [JJ^T + λI]^(-1)
4. For each row i of N:
   • If off-diagonal entry N(i,j) > th × N(i,i), mark measurement j as dependent on i
   • Skip analysis for already identified dependent measurements
5. Extract independent measurement indices
6. Construct reduced Jacobian with only independent measurements

2. Diffuse Optical Tomography: Forward and Inverse Problems

Forward Problem:

Modeled continuous wave NIR light propagation in biological tissue using the diffusion equation:

-∇·D(r)∇Φ(r) + μₐ(r)Φ(r) = Q₀(r)

Where:

• D(r): Optical diffusion coefficient
• μₐ(r): Absorption coefficient (imaging parameter to reconstruct)
• Φ(r): Photon fluence density
• Q₀(r): Isotropic light source

Implementation:

• Used Finite Element Method (FEM) to solve the diffusion equation
• Applied Type-III boundary condition for refractive index mismatch
• Generated modeled data under Rytov approximation (natural logarithm of intensity)

Inverse Problem:

Reconstructed absorption coefficients from boundary measurements using iterative least-squares optimization:

Levenberg-Marquardt Optimization:

Minimized objective function: Ω = ||y - G(μₐ)||²

Update equation at iteration i:

Δμₐⁱ = J^T [JJ^T + λI]^(-1) δⁱ⁻¹

Where:

• y: Natural logarithm of experimental amplitude data
• G(μₐ): Modeled data
• δ: Data-model misfit (δ = y - G(μₐ))
• J: Jacobian/sensitivity matrix (dimension: NM × NN)
• λ: Regularization parameter (stabilizes solution)

Reduced Jacobian Approach:

After identifying NK independent measurements, the reduced Jacobian becomes:

Jₙ = J(independent_indices, :)  (dimension: NK × NN)

New update equation:

Δμₐⁱ = Jₙ^T [JₙJₙ^T + λI]^(-1) δⁱ⁻¹

3. Experimental Validation

Numerical Simulations:

Tested on multiple 3D geometries:

1. Cylindrical Domain:

   • Diameter: 86 mm, Height: 100 mm
   • Mesh: 24,161 nodes, 116,757 tetrahedral elements
   • Data collection: 48 fibers in 3 layers → 2,256 measurements
   • Spherical target (20 mm diameter) at 10 mm depth

2. Patient Breast Mesh:

   • Mesh: 18,723 nodes, 94,936 tetrahedral elements
   • Data collection: 16 fibers in circular arrangement → 240 measurements

Optical Properties:

• Background: μₐ = 0.01 mm⁻¹, μₛ' = 1.0 mm⁻¹
• Target: μₐ = 0.03 mm⁻¹, μₛ' = 1.0 mm⁻¹
• Added 1% Gaussian noise to simulate experimental conditions

Experimental Phantom:

Fabricated cylindrical gelatin-based phantom:

• Dimensions: 86 mm diameter × 60 mm height
• Background: μₐ = 0.008 mm⁻¹, μₛ' = 0.9 mm⁻¹
• Cylindrical target (16 mm diameter): μₐ = 0.02 mm⁻¹, μₛ' = 1.0 mm⁻¹
• Data collected at 785 nm wavelength

Results and Impact

1. Increased Independent Measurements

Compared to traditional singular value analysis:

Imaging Domain	Total Measurements	Singular Value Analysis	Data-Resolution Matrix (th=0.9)	Improvement
Cylinder: 1 layer	240	89	111	+25%
Cylinder: 3 layers, in plane	720	231	331	+43%
Cylinder: 3 layers, out of plane	2,256	282	1,093	+288%
Patient breast mesh	240	101	121	+20%
Experimental phantom	240	94	162	+72%

Key Finding: Data-resolution matrix analysis identified 20-288% more independent measurements than traditional singular value analysis.

2. Maintained Image Quality

Threshold Analysis:

• th = 1.0: All measurements used (baseline)
• th = 0.9: 121 independent measurements
  • Relative error: 0.41% compared to baseline
  • Data-model misfit difference: <3% across all iterations
• th = 0.8: 106 independent measurements
  • Relative error: 3.89%
• th = 0.7: 97 independent measurements
  • Relative error increases significantly

Optimal Choice: Threshold of 0.9 maintains image quality (error <1%) while reducing measurements by ~50%.

3. Random Selection Comparison

When 162 randomly selected measurements (same as independent set) were used:

• Poor image quality with major boundary artifacts
• Reduced contrast recovery
• Significantly worse than using 162 independent measurements

Conclusion: Specific selection of independent measurements is critical, not just the count.

4. Computational Efficiency

Computation Time per Iteration (Cylinder: 3 layers):

• All measurements (th=1.0): 476.04 seconds
• Independent only (th=0.9): 462.12 seconds
• Reduction: ~3%
• Overhead for data-resolution analysis: 37.98 seconds (one-time cost)

Memory Benefits:

• Reduced Jacobian dimensions from NM × NN to NK × NN
• Lower storage requirements for sensitivity matrix

5. Effect of Regularization Parameter

Number of independent measurements varies with λ (for patient mesh, th=0.9):

• λ = 0.01: 143 measurements
• λ = 0.1: 138 measurements
• λ = 1: 121 measurements
• λ = 10: 119 measurements
• λ = 100: 116 measurements

Trend: Higher regularization → fewer independent measurements (more measurement dependency).

Conclusion

This research fellowship at IISc provided valuable experience in developing computational algorithms for medical imaging, working on inverse problems, and publishing in high-impact journals. The data-resolution matrix based optimization method I developed addressed a critical gap in diffuse optical tomography - identifying which specific measurements are independent rather than just knowing the count. This work has practical implications for reducing data acquisition time in clinical optical imaging systems while maintaining image quality, contributing to more efficient and economically viable medical imaging protocols.