Cylindrical Warping

Cylindrical warping re-projects a photograph onto the surface of a cylinder centred on the camera. It is used to build panoramas: once every photo is on the same cylinder, neighbouring photos differ only by a horizontal shift, which is much simpler to align than a rotation.

A planar image of a grid, and the same image warped onto a cylinder with two focal lengths. Vertical lines stay vertical and straight, while horizontal lines bow. A smaller focal length means a stronger effect.

Why Use a Cylinder?

A normal photo is a planar projection (a flat sensor). When a camera pans around, the projection of the same world onto several planes is related by a homography (Homographies and Perspective Warping), and a wide panorama on a single plane stretches badly at the edges.

On a cylinder, equal turns of the camera correspond to equal horizontal distances along the surface. So for a camera that rotates about its vertical axis, each new photo is a pure translation of the previous one in cylindrical coordinates. Alignment then needs only a shift (found by feature matching and robust fitting, see RANSAC) instead of a full homography per pair.

flowchart LR
    A[Photos taken while panning] --> B["Warp each onto a cylinder (needs focal length f)"]
    B --> C["Estimate the horizontal shift between neighbours"]
    C --> D[Blend overlapping regions]
    D --> E[Panorama]

The Mapping

Let $f$ be the focal length in pixels, and measure image coordinates $(x,y)$ from the image centre. A point on the image plane lies at angle $\theta$ around the cylinder axis and height $h$ up the cylinder:

$\theta =\arctan \left(\frac{x}{f}\right),h=\frac{y}{\sqrt{x^2+f^2}}$

The cylindrical image coordinates (unrolling the cylinder, with radius $f$) are

$x^{\prime }=f\theta =f\arctan \left(\frac{x}{f}\right),y^{\prime }=fh=\frac{fy}{\sqrt{x^2+f^2}}$

For warping, use the inverse mapping: for each output pixel $(x^{\prime },y^{\prime })$ find its source $(x,y)$ and interpolate there (Image Warping):

$x=f\tan \left(\frac{x^{\prime }}{f}\right),y=\frac{y^{\prime }}{\cos (x^{\prime }/f)}=y^{\prime }\frac{\sqrt{x^2+f^2}}{f}$

Behaviour of the mapping:

• Near the centre ($x\ll f$), $x^{\prime }\approx x$ and $y^{\prime }\approx y$: the centre is almost unchanged.
• Vertical lines stay vertical, since $x^{\prime }$ depends only on $x$. Horizontal lines curve, because $y^{\prime }$ shrinks with $|x|$.
• Edge pixels are pulled in, so the result is not rectangular and the border has to be cropped or filled.
• The mapping is non-linear, so it cannot be written as a $3\times 3$ matrix like the transformations in Image Warping.

Choosing the Focal Length

$f$ must be in pixels and should match the real camera. It can come from the camera’s calibration, from the lens’s field of view,

$f=\frac{W/2}{\tan (\text{FOV}/2)}$

or from image alignment. A wrong $f$ leaves curved seams and misalignment. Wide-angle lenses with strong lens distortion should be undistorted first.

Assumptions and Limits

• The camera rotates about (nearly) its optical centre. Translating the camera adds parallax, which no warp can remove.
• The rotation axis is vertical and the camera is level, otherwise the shifts between frames are no longer pure horizontal translations.
• Covers a 360-degree horizontal sweep but limited vertical range. A spherical warp is the analogue when the camera also tilts a lot.

Code: Image Warping Snippets.

Source: Szeliski, Computer Vision: Algorithms and Applications, chapter on image stitching (cylindrical and spherical coordinates). The formulas were verified with a round-trip test in my own implementation.

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Related: Computer Vision MOC, Image Warping, Homographies and Perspective Warping, Image Interpolation, Image Warping Snippets