⚡ electrical engineering

On the Determination of Diameter and 3D Position of Spray Droplets Using Astigmatic Shadowgraphy and Deep Learning

This paper presents a single-camera deep learning framework that combines astigmatic shadowgraphy with YOLO and VGG16 networks to accurately estimate the diameter and 3D position of spray droplets, achieving high precision on experimental data despite being trained exclusively on synthetic images.

Original authors: Mohammad Mehdizadeh Youshanlouei, Alex Ravaioli, Fahrettin Gökhan Ergin, Gabriele Bellani, Massimiliano Rossi

Published 2026-09-21

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CC BY 4.0

Original authors: Mohammad Mehdizadeh Youshanlouei, Alex Ravaioli, Fahrettin Gökhan Ergin, Gabriele Bellani, Massimiliano Rossi

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

In the world of fluid dynamics, understanding how liquid breaks apart into tiny droplets is essential for everything from fueling jet engines to delivering medicine through inhalers. When a spray is released, it is not a uniform mist but a chaotic cloud of particles of varying sizes flying through space. To control these processes, scientists need to know not just how big the droplets are, but exactly where they are located in three-dimensional space. Traditional cameras can easily measure the size of a droplet if it is in sharp focus, but they struggle to tell how far away it is if it is blurry. Usually, solving this puzzle requires complex setups with multiple cameras or lasers, which can be difficult to use in real-world industrial settings.

A team of researchers at the University of Bologna and Dantec Dynamics has developed a simpler way to solve this problem using a single camera and a new kind of artificial intelligence. They took a standard system used for imaging sprays and added a single piece of glass, a cylindrical lens, to the front of the camera. This lens intentionally distorts the light, turning the image of a droplet into an oval shape. The direction and shape of this oval change depending on whether the droplet is in front of or behind the focal point, effectively turning the blur itself into a precise ruler for depth. However, reading these distorted shapes by hand is difficult and slow. To overcome this, the researchers trained a computer program to recognize the patterns in these blurry images and instantly calculate both the droplet's size and its exact position in space.

The challenge was that training a computer to recognize real spray droplets usually requires thousands of photos where the size and location of every single drop are already known. In a real spray, this information is impossible to know beforehand. To get around this, the team created a massive library of fake, or synthetic, images. They used a computer model to simulate how light would pass through their specific lens setup to create thousands of perfect, blurry droplet images with known sizes and positions. They taught two different artificial intelligence networks using these synthetic pictures. The first network, acting like a digital eye, learned to spot individual droplets in a crowded image and cut them out. The second network, acting like a detective, analyzed the shape of each cut-out droplet to predict its depth and diameter. Remarkably, they trained the system entirely on these computer-generated images, never showing it a single real photograph during the learning phase.

When the researchers tested this system on real-world data, the results were surprisingly accurate. They first tested it on a calibration target, a glass plate with dots of known sizes moved to precise depths. The system correctly identified the size and position of these dots, with a depth uncertainty of just 0.15 millimeters over a four-millimeter range, and a size uncertainty of about 2.4 micrometers for droplets between 20 and 100 micrometers. They then applied the same trained system to a real spray of water droplets generated by an aerosol machine. Without any retraining or adjustment, the computer successfully detected droplets in the chaotic spray, even when they overlapped or were partially hidden by others. It mapped out the three-dimensional distribution of the spray, revealing that the droplets were spread relatively evenly through the measurement volume, with sizes clustering around a peak of roughly 35 micrometers.

The study demonstrates that it is possible to turn a simple, single-camera shadowgraphy system into a powerful three-dimensional measurement tool simply by adding a cylindrical lens and using deep learning. The researchers found that the computer's ability to guess the depth based on the shape of the blur was robust, even though it had never seen a real droplet before. While the system struggled slightly at the very edges of the measurement range where the image distortion became extreme, it performed consistently well in the center. This approach offers a straightforward path for industries to measure sprays with high precision without needing expensive, multi-camera setups, proving that a little bit of intentional blur, combined with smart software, can reveal the hidden structure of a spray.

Technical Summary: On the Determination of Diameter and 3D Position of Spray Droplets Using Astigmatic Shadowgraphy and Deep Learning

Problem Statement
Characterizing spray droplets is critical across automotive, aerospace, pharmaceutical, and agricultural sectors, where drop size and spatial distribution govern processes like combustion and coating. While established optical techniques exist, they face significant limitations. Phase Doppler methods are limited to single-point measurements; interferometric techniques (ILIDS/IPI) and digital holography often require coherent illumination, complex setups, or suffer from computational intensity and limited depth accuracy. Conventional shadowgraphy is a mature, robust technique for non-spherical and opaque droplets but is inherently two-dimensional; it cannot resolve the out-of-plane ( $z$ ) coordinate, forcing the rejection or correction of defocused particles. While Defocus Particle Tracking (DPT) and Astigmatism Particle Tracking Velocimetry (APTV) offer 3D localization, applying astigmatism to polydisperse spray droplets in a single-camera shadowgraphy setup presents challenges: the ambiguity of the defocus function relative to the focal plane and the lack of established theoretical models for generating reliable synthetic images of polydisperse particles.

Methodology
The authors propose a single-camera approach that combines astigmatic shadowgraphy with a two-stage Deep Learning (DL) pipeline to simultaneously estimate droplet diameter ( $d$ ) and 3D position ( $x, y, z$ ).

Optical Setup: A cylindrical lens ( $f=500$ mm) is placed in front of the objective lens of a commercial micro-scale shadowgraphy system (Dantec Dynamics ShadowSizer). This introduces controlled astigmatism, causing defocused droplets to appear as ellipses with orientations that vary depending on whether the droplet is above or below the focal plane, thereby resolving depth ambiguity.
Synthetic Image Generation: Since obtaining ground-truth experimental data for training is impractical, the authors generated a large dataset of synthetic images using two models:
- Synth1 (Training/Validation): Based on a computationally efficient Gaussian-based Point Spread Function (PSF) model (Rossi and Kähler) calibrated to experimental images. Overlaps are handled via intensity summation.
- Synth2 (Independent Test): Based on a physically accurate, ray-tracing generator (MicroSIG) that treats droplets as opaque objects and handles occlusion more realistically.
Deep Learning Architecture:
- Detection & Localization: A YOLO v12 (You Only Look Once) network detects droplets and determines their in-plane ( $x, y$ ) coordinates, handling overlapping particles.
- Regression: A modified VGG16 Convolutional Neural Network (CNN) predicts the depth ( $z$ ) and diameter ( $d$ ) from cropped droplet images. The architecture includes flattened and dense layers with ReLU and Linear activations.
Training Strategy: Both networks were trained exclusively on calibrated synthetic images (Synth1) with data augmentation (noise, lighting variations) to generalize to real-world conditions. No experimental images were used for training the regression network.

Key Results
The methodology was validated through three complementary tests:

Synthetic Generalization: When tested on the independent ray-tracing dataset (Synth2), the model achieved median errors of 0.174 mm for depth (over a 4 mm range) and 2.172 µm for diameter (20–100 µm range). Discrepancies in depth prediction were observed only near the focal boundary ( $z < 1$ mm), likely due to spherical aberrations not fully captured in the training model.
Calibration Target: Applied to experimental images of a calibration target with known ground truth, the system achieved a median error of 0.149 mm in depth and 2.41 µm in diameter. The model struggled slightly with 20 µm droplets at the measurement volume boundaries.
Experimental Spray: The system successfully analyzed water droplets from an aerosol generator. The YOLO detector achieved a mean Average Precision of 98% (IoU 0.5). The statistical distribution of detected droplet diameters (peaking around 35 µm) was consistent with the manufacturer's nominal value (25 µm) and typical spray distributions. The depth distribution was uniform across the measurement volume, except for deficits at the boundaries attributed to low signal-to-noise ratios or model inaccuracies.

Significance and Claims
The paper claims to present a robust, single-camera methodology for the simultaneous 3D sizing and localization of polydisperse spray droplets. Its primary significance lies in:

Simplicity and Integration: The method introduces astigmatism via a single cylindrical lens placed before the objective, making it straightforward to integrate into commercially available shadow-sizing systems without complex optical reconfigurations.
Synthetic-to-Real Transfer: It demonstrates that deep learning models trained solely on computationally inexpensive, calibrated synthetic images can generalize effectively to unseen synthetic data (generated via ray-tracing) and real experimental spray data without retraining.
Performance: The approach provides consistent diameter determination with uncertainties around 2.1–2.4 µm and depth uncertainties of 0.15 mm over a 4 mm measurement range, successfully handling overlapping droplets in dense sprays.

The authors conclude that while limitations exist near the measurement volume boundaries due to optical aberrations and low signal-to-noise ratios, the proposed framework offers a viable alternative to complex multi-camera or coherent illumination techniques for industrial spray characterization. Future work is suggested to focus on refining optical models or exploring alternative arrangements, such as birefringent crystals.

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