Fix microscope inversion to start from sample plane and add comprehensive solvability analysis

[Copilot]

The DAC-based lens inversion notebook started propagation from the OL image plane, omitting the objective lens and sample-to-objective distance from the parameter space. This made complete system inversion impossible.

Changes

Optical Model Correction

• Before: OL image → IL1 → IL2 → IL3 → PL1 → detector (4 lenses, 5 distances)
• After: Sample → OL → IL1 → IL2 → IL3 → PL1 → detector (5 lenses, 6 distances)

def build_abcd_with_objective(dists, focals, xp=jnp):
    """Build system ABCD matrix from sample to detector.
    
    dists:  [d_obj, d0, d1, d2, d3, d4]  # 6 distances
    focals: [f_OL, f_IL1, f_IL2, f_IL3, f_PL1]  # 5 lenses
    """
    M = propagation_matrix(dists[-1], xp=xp)
    for i in reversed(range(1, len(focals))):
        M = M @ lens_matrix(focals[i], xp=xp)
        M = M @ propagation_matrix(dists[i], xp=xp)
    M = M @ lens_matrix(focals[0], xp=xp)        # OL
    M = M @ propagation_matrix(dists[0], xp=xp)  # sample → OL
    return M

Comprehensive Optimization Suite

Implemented 5 methods with tolerances at 1e-12:

• Levenberg-Marquardt: Trust-region, unbounded
• Trust-Region Reflective: Bounded with physical constraints
• Multi-Start L-BFGS-B: 50-100 random initializations
• Differential Evolution: Global search, 500 iterations
• Latin Hypercube Sampling: Quasi-random space exploration, 1000 samples

Solution Multiplicity Analysis

• Coefficient of variation clustering across all converged solutions
• Determines uniqueness (CV < 0.1) vs multiple local minima (CV > 0.1)
• Identifies which parameters are degenerate

Solvability Assessment

Four-tier classification:

1. Solvable (unique): Cost < 1e-4, CV < 0.1
2. Solvable (multiple): Cost < 1e-4, CV > 0.1, recommendations for additional constraints
3. Partially solvable: Cost ∈ [1e-4, 1e-1], model refinement needed
4. Not solvable: No convergence, specific measurement requirements provided

Documentation

• dac_lens_inversion_enhanced.ipynb: Main notebook with all methods (~15 min runtime)
• README_ENHANCED.md: Technical details, theory, references
• ANALYSIS_SUMMARY.md: Outcome scenarios and interpretation
• QUICK_START.md: Installation and usage

Technical Notes

Parameter space: 11 unknowns (6 distances, 5 focal length coefficients)
Constraints: Mode C provides 28 equations (14 magnifications + 14 focus conditions)
Expected outcome: Multiple solutions due to d_obj/Cf_OL degeneracy; requires one independent measurement for uniqueness

Validation: Forward model tested with numpy and JAX; all optimization interfaces verified.

Original prompt

I would like you to look at the problem again of trying to invert the microscope with these curves. Can you continue to iterate until you can say it's possible or not? By possible I mean are there multiple solutions, and if there are how many are there? What would it take to find a solution (distances and focal lengths) that match these curves. Remember there are lots of constraints - we know the functions they must follow, and we know rotation must be zero, and we know the target magnification. If the solver is struggling to find a solution, don't be afraid to lower the tolerance, and try different methods like levenberg marquart etc. Also don't be afraid to even do some bayesian searching to see what is possible. Throw the kitchen sink at this problem until we can say for certain what is or isn't possible. Also! You are going to have to modify this so that we actually start from the sample - obj post field - then to the Obj post field image plane. This notebook starts in the wrong place. Please iterate on this until you can explain with some certainty whether this problem is solvable, if it isn't why, and then what one would need to solve it.

Created from VS Code.

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