⚡ electrical engineering

Phase shifting digital holography with random phase steps: A novel robust algorithm

This paper presents a novel, robust algorithm for phase-shifting digital holography that eliminates the need for a priori knowledge of reference phase shifts or a calibrated piezo stage, enabling accurate object phase recovery for biological and metrology applications using random phase steps.

Original authors: Damien Peter Kelly, Kristiana Topi, Rainer Wittig

Published 2026-08-27

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CC BY 4.0

Original authors: Damien Peter Kelly, Kristiana Topi, Rainer Wittig

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Light carries two kinds of information about the world it touches: how bright it is, and the precise timing of its waves as they arrive. While cameras and our eyes are excellent at recording brightness, they are blind to that timing, known as phase. This missing piece of information is crucial for seeing transparent objects, such as living cells, which do not block light but do slow it down slightly. Without measuring this delay, a cell looks like a ghostly, invisible smudge. To reveal these hidden details, scientists use a technique called digital holography. This method mixes the light bouncing off an object with a reference beam of light, creating a complex pattern of stripes called a hologram. By analyzing how these stripes shift, researchers can mathematically reconstruct the invisible phase and turn a transparent object into a clear, three-dimensional map of its shape and thickness.

The challenge has always been that to read this map accurately, the reference beam must be shifted in very precise, known steps. Traditionally, this requires expensive, delicate machinery to move a mirror with microscopic precision. If the machine moves the wrong amount, or if the steps are uneven, the resulting image becomes distorted or useless. This reliance on perfect hardware has limited where and how these measurements can be taken. A team of researchers at the University of Ulm has now developed a new way to solve this problem that does not require knowing the exact steps in advance. They created a robust algorithm that can recover the hidden phase information even when the reference light is shifted in completely random, unpredictable ways.

In their work, the researchers demonstrated that they could capture a series of holograms where the reference light was shifted randomly, perhaps by the natural vibrations of a building or a simple, uncalibrated motor. Instead of needing a machine that moves with perfect precision, their new method treats the randomness as a feature rather than a flaw. The process begins by estimating the basic brightness levels of the light involved. From these measurements, the team calculates a preliminary guess of the phase. However, this initial guess comes with a mathematical ambiguity: for every point in the image, there are two possible answers, one real and one that is a mirror image of the truth. To solve this, the algorithm compares different points in the image against a reference point, determining which of the two possibilities is correct for the entire scene.

Once the team has a reliable initial map of the phase, they move to a second stage where they generate a set of "synthetic" holograms. These are computer-generated patterns based on their current best guess of the phase and the light intensities. They then compare these synthetic patterns to the actual holograms they captured in the lab. By finding the synthetic pattern that matches the real data most closely, the algorithm can deduce exactly how much the reference light shifted between each capture, even though that shift was random. With this knowledge of the random shifts, the system performs a final refinement, searching through a wide range of possibilities to lock onto the most accurate version of the phase map. This brute-force search, while computationally intensive, avoids the need for complex mathematical models that often fail when the data is noisy or irregular.

The researchers tested this approach in two very different settings to prove its versatility. In the first experiment, they used a simple setup without a microscope to image a target with a linear slope. Even though they used an uncalibrated motor to create the random shifts, the algorithm successfully reconstructed the slope with high accuracy. In the second experiment, they removed the motor entirely. They built a digital microscope and relied solely on the natural, passive vibrations of the laboratory environment to create the necessary random shifts. They imaged MCF-7 breast cancer cells, which are transparent and difficult to see with standard light. The system successfully mapped the cells' surfaces, revealing their height and shape. By removing the background distortions caused by the microscope lens, they were able to calculate the actual thickness of the cells, measuring them in micrometers.

The results show that the method is both robust and accurate. In computer simulations, the algorithm was able to produce a clear phase map using as few as five holographic frames, and the quality of the image improved as more frames were added. In the physical experiments, the measurements were consistent across multiple runs, showing that the system could handle the noise and imperfections of a real-world environment. The team found that by limiting their search to 256 possible levels for the phase shifts, they could balance the speed of the calculation with the precision of the result. This approach means that high-quality, label-free imaging of biological cells and other transparent objects can be achieved without the need for expensive, perfectly calibrated motion stages. It opens the door to simpler, more portable devices that can measure the invisible properties of light in a wide variety of settings, from biology labs to industrial inspection floors.

Technical Summary: Phase Shifting Digital Holography with Random Phase Steps

Problem Statement
Phase shifting digital holography (PSDH) is a standard interferometric technique for recovering the complex amplitude (magnitude and phase) of an optical wavefield. Traditionally, this requires capturing multiple holograms where the reference field phase is shifted by known, precise amounts, typically achieved using a calibrated piezo motion stage. However, the accuracy of phase recovery in traditional algorithms is heavily dependent on the precision of these phase shifts. If the phase steps are unknown, non-uniform, or random, standard algorithms may fail or stagnate. While previous research has addressed unknown phase shifts using iterative least-squares methods, principal component analysis, or Fourier domain techniques, these often require specific conditions (e.g., a large number of frames, accurate background estimation, or specific distributions of phase shifts) and can be sensitive to noise, particularly when intensity values approach saturation limits.

Methodology
The authors propose a novel, robust algorithm capable of recovering the object phase $\alpha(x)$ from a series of holograms captured with random, unknown phase steps ( $\theta_n$ ). The method does not require a priori knowledge of the phase shifts or a calibrated piezo actuator. The algorithm operates through four distinct modules:

Measurement Module: The system captures $N$ holograms ( $H_n$ ). The intensities of the object ( $I_o$ ) and reference ( $I_r$ ) waves are either measured directly or estimated from the hologram stack to determine the DC term ( $I_{DC}$ ) and AC term ( $I_{AC}$ ).
Estimator Module:
- The algorithm calculates $\cos[\alpha(x)]$ using the relation $\Gamma_1(x) = [H_1(x) - I_{DC}(x)]/I_{AC}(x)$ .
- Since $\cos[\alpha(x)]$ yields two possible solutions (a real solution $\alpha$ and a conjugate solution $2\pi - \alpha$ ), the algorithm introduces a binary function $T(x)$ to select the correct branch for each pixel.
- To determine $T(x)$ , the algorithm compares a reference pixel ( $x_{ref}$ ) with a comparison pixel ( $x_{comp}$ ). It calculates potential phase steps ( $\theta_n$ ) for both the real and conjugate solutions across multiple frames. By computing a running complex average of the differences between these solutions, the algorithm statistically determines whether the real or conjugate solution is consistent across the frame, thereby resolving the phase ambiguity.
Predictor Module:
- Using the initial phase estimate $\alpha(x)$ , the algorithm generates a set of synthetic holograms ( $Z_p$ ) for a quantized range of phase steps ( $\Theta_P$ ).
- By comparing the actual captured holograms ( $H_n$ ) against these synthetic predictions, the algorithm identifies the specific phase step $\theta_n$ that minimizes the error for each capture. This allows for the precise estimation of the random phase shifts without prior calibration.
Refiner Module:
- With the estimated phase steps $\theta_n$ known, the algorithm performs a final refinement of the phase estimate $\alpha(x)$ .
- It conducts a "brute force" search over a quantized vector of phase values ( $A_Q$ ) to find the value that minimizes the difference between the measured holograms and the model prediction, yielding the final refined phase $\alpha_R(x)$ .

Key Contributions

Random Phase Step Handling: The algorithm successfully recovers phase information without requiring a calibrated piezo stage or known phase steps, relying instead on random phase shifts generated either by an uncalibrated actuator or passive environmental vibrations.
Ambiguity Resolution ( $T(x)$ ): The introduction of the $T(x)$ function provides a robust statistical method to distinguish between the real and conjugate phase solutions, a common source of error in phase retrieval.
Synthetic Hologram Prediction: The use of synthetic holograms to actively estimate the random phase steps and measurement error is a novel concept within this context.
Robustness to Noise: The method demonstrates stability even when $\Gamma_1(x)$ approaches $\pm 1$ , a region where traditional arccos-based estimators typically fail or become noisy.

Results
The authors validated the algorithm through numerical simulations and two distinct experimental setups:

Numerical Simulations: Using 20x1440 pixel holograms with added noise, the algorithm successfully recovered the phase using as few as 5 frames, with accuracy improving as the number of frames increased (up to 35). The refined phase ( $\alpha_R$ ) showed significantly reduced noise compared to the initial estimate, particularly near phase values of $0, \pi,$ and $2\pi$ .
Experimental System 1 (Lens-less Interferometer): An uncalibrated piezo-actuator was used to generate random phase shifts on a linear phase ramp target. The algorithm successfully recovered the phase profile, demonstrating that the structural features of the target were preserved and that the measurement was robust across multiple trials.
Experimental System 2 (Digital Holographic Microscope): A system with no active actuator was used, relying solely on passive laboratory vibrations to create random phase shifts. The system imaged MCF-7 breast cancer cells. The algorithm successfully recovered the phase, which was then unwrapped and corrected for Zernike aberrations (piston, tilt, defocus) to generate a cell height map.

Significance and Claims
The paper claims that this approach offers a robust and flexible alternative to traditional PSDH. By eliminating the need for a calibrated piezo stage, the method simplifies the optical setup and reduces hardware costs. The authors emphasize that the algorithm is effective even when phase shifts are entirely random and unknown, making it suitable for applications where precise mechanical control is difficult or impossible. The results indicate that the method is consistent and accurate for both metrology (linear phase ramps) and biological imaging (label-free cell height measurement), providing a practical solution for environments where passive phase shifting (e.g., via vibration) is the only option. The authors note that while the brute-force search introduces quantization error, the 8-bit resolution (256 levels) strikes an optimal balance between processing time and accuracy for their experimental conditions.

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