The Innovative Brown-Conrady Model

I was recently discussing camera manufacturing with a roboticist who was building a stereo vision system. While most of our conversation covered familiar ground (sensor resolution, baseline distance, synchronization), one question caught my attention:

"Why don't lens manufacturers just tell you which intrinsic model to use during calibration?"

It's a perfectly reasonable question. Camera manufacturers invest enormous resources in optical design and should want their lenses to work seamlessly with computer vision applications. They certainly know their products better than anyone else. So why do we, as users, have to reverse-engineer distortion models for every lens we use?

The answer lies in the history of lens distortion modeling itself, and in the honest limitations that the originators of this field built into their work. To understand why manufacturers stay silent on calibration models, we need only to turn to one of the most widely used frameworks in computer vision: the Brown-Conrady model.

But Why Lens Models?

Everyone who's delved into computer vision has encountered the Brown-Conrady model. Based on publications over 100 years old, it remains the go-to for describing camera lens distortion, largely due to OpenCV’s early adoption of the model as their default camera distortion profile.

The story begins in 1919 with A.E. Conrady's groundbreaking work in his publication "Decentred Lens-Systems". Working at a time when photogrammetry was still a nascent practice, Conrady developed the first rigorous mathematical treatment of lens decentering (aka tangential) distortion. His work, published in the Monthly Notices of the Royal Astronomical Society, laid the theoretical groundwork for understanding how imperfect lens alignment creates systematic errors in photographs. He discussed his analysis in terms of astronomy, but his results were applied to microscopy, photography, and even periscopes.

However, it’s important to note that “applied” here means that his work was used to better understand and improve lens construction techniques, not develop computational correction methods like we use today. Opticians of the era were aware of the aberrations produced by decentered lenses; Conrady’s work was just the first to provide mathematical expressions for those effects.

It was Duane Brown's 1966 publication "Decentering Distortion of Lenses" that transformed Conrady's theoretical work into a practical calibration tool. Brown's contribution was big: he not only refined Conrady's mathematical model, but also demonstrated that decentering distortion could be calibrated analytically rather than requiring perfect physical alignment of lens elements.

The Brown-Conrady Model Explained

The Brown-Conrady model describes lens distortion as a combination of radial and tangential components:

• Radial distortion occurs when the lens elements aren't perfectly spherical, causing straight lines to appear curved (e.g. ”barrel" or "pincushion" effects)

• Tangential distortion (also called decentering distortion) results from imperfect alignment of lens elements, where the optical centers don't lie on a straight line.

The mathematical elegance of the Brown-Conrady model lies in its ability to describe these complex optical phenomena with a relatively simple polynomial expansion. This made it revolutionary for 1960s photogrammetry, where precise measurements from aerial photographs were crucial for mapping applications. MetriCal, Tangram’s calibration suite, has more than one model based on this math directly: OpenCV’s RadTan model and our own Pinhole with BC.

So that’s it. After that, all camera manufacturers used Brown-Conrady for everything. Case closed, no notes.

The Fundamental Problem: Models vs. Reality

Just kidding. It’s still messy.

Brown himself recognized a critical limitation in his work. In the same publication that he presented this unified model, he demonstrated that even the best lens systems exhibit distortion that varies with focus distance, aperture, and even zoom setting. He showed experimentally that distortion isn't constant across a lens's operating range. Instead, it's a complex, multidimensional function that changes with every adjustment you make to your camera.

Consider what this means for a modern zoom lens:

• Distortion varies across the zoom range

• Distortion changes with focus distance

• Distortion shifts with aperture settings

• Even manufacturing tolerances mean identical lens models perform differently

The Impossibility of Perfect Profiles

This is why camera manufacturers don't ship distortion profiles with their lenses: any fixed mathematical model would be, at best, an approximation valid only under specific conditions. A profile accurate at one focal length might be significantly wrong at another, or show just a part of the whole. A correction perfect for infinity focus could introduce errors when focused at minimum distance.

Brown's experimental work showed that residual errors after applying even carefully calibrated corrections could still amount to several microns, enough to cause noticeable artifacts in precision applications. For consumer photography, this might be fine. For autonomous and high-precision applications, they could become a straight-up safety risk.

Camera manufacturers understand that providing "official" distortion profiles would create unrealistic expectations of precision. Instead, they focus on optical design that minimizes distortion mechanically, leaving software corrections to third parties who can iterate and improve their models without being bound to a specific manufacturer's reputation.

The Lesson

Brown and Conrady's work remains foundational because it established the mathematical framework for understanding lens distortion. But their greatest insight might be the recognition that these are just models. They’re useful approximations of reality, not perfect descriptions of it.

The next time you wonder why your lens doesn't come with a factory distortion profile, remember that the very researchers who developed our distortion models understood their limitations. Sometimes the most honest answer is acknowledging what we can't perfectly predict, rather than pretending our models capture every nuance of optical reality.

In the end, the Brown-Conrady model's enduring value isn't its perfection. Instead, it's the honest recognition that lens distortion is more complex than any single mathematical description can capture. That humility, built into the model from its inception, explains why it remains useful more than a century later.

Further Reading

Read Brown's paper for the math; read Conrady's paper for the prose.

• A. E. Conrady, Decentred Lens-Systems, Monthly Notices of the Royal Astronomical Society, Volume 79, Issue 5, March 1919, Pages 384–390, https://doi.org/10.1093/mnras/79.5.384

• Duane C. Brown, Decentering Distortion of Lenses. 1966. https://web.archive.org/web/20180312205006/https://www.asprs.org/wp-content/uploads/pers/1966journal/may/1966_may_444-462.pdf