Cylindrical Projection for Panoramas

Cylindrical projection lets you map a perspective camera image onto the surface of a virtual cylinder around the camera, then “unroll” that cylinder into a flat image. In this space, pure yaw rotations of the camera turn into horizontal translations, which is ideal for stitching many views into a 360° panorama. Compared to planar homographies, cylindrical (or spherical) projection reduces perspective distortions for wide fields of view and makes global alignment simpler for rotational panoramas.

Geometrically, you can think of each pixel as a ray from the camera center through the image plane. Cylindrical projection re-parameterizes that ray by its azimuth angle (mapped to horizontal coordinate) and elevation (mapped to vertical coordinate), using the focal length $f$ and image center $(c_x,c_y)$. The given formulas express exactly that mapping. In practice, stitching pipelines first warp each image into a cylindrical coordinate system, then align and blend them in this common domain.

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1. Background Knowledge (key concepts)

• Pinhole camera model A pixel at $(x,y)$ corresponds to a 3D ray $(X,Y,Z) \propto (x-c_x, y-c_y, f)$ in camera coordinates, where $f$ is the focal length and $(c_x,c_y)$ is the principal point (image center).

• Cylindrical projection geometry Points are projected onto a cylinder of radius $f$ around the camera’s optical axis:

• Horizontal: use angle \theta = \arctan\left(\frac{x-c_x}{f}\right), then $x' = f \,\theta$.

• Vertical: project the ray onto the cylinder’s tangent plane vertically, giving

$$y' = f \cdot \frac{y - c_y}{\sqrt{(x-c_x)^2 + f^2}}.$$

This preserves horizontal angles and avoids unbounded growth of coordinates as you move away from the center.

• Why this helps stitching For a purely rotating camera (no translation), images are related approximately by horizontal shifts in cylindrical coordinates, rather than full projective transforms. That simplifies alignment (e.g., using 1D translation estimation or simple homographies in cylindrical space) and allows seamless 360° panoramas.

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2. Algorithm / Approach

The standard pattern for cylindrical projection in stitching is:

1. Forward model: Use the given equations to describe how an input image point $(x,y)$ maps to cylindrical coordinates $(x',y')$.
2. Inverse mapping for implementation: For each pixel in the cylindrical panorama canvas, compute where it came from in the original image (inverse projection), then sample the source image (interpolation).
3. Warp all images: Apply the cylindrical warp to each input image into its own cylindrical canvas.
4. Estimate relative shifts: Align warped images (e.g., via feature matching + RANSAC, or simpler translation estimation) in the cylindrical domain.
5. Blend and composite: Place each warped image into a global panorama canvas and apply blending along overlaps.

The core of this problem is implementing the cylindrical projection warp for a single image, not the full stitching pipeline.

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3. Step-by-Step Strategy

Assume you are given an input image I, focal length f, and know or can compute center (cx, cy).

Step 1: Choose coordinate conventions

• Use pixel coordinates such that:
• $x \in [0, W-1]$, $y \in [0, H-1]$.
• Center: $c_x = (W-1)/2$, $c_y = (H-1)/2$ (or given).
• Decide the cylindrical output canvas size:
• Horizontal span roughly:

$$x'_{\min}, x'_{\max} \approx f \cdot \arctan\left(\frac{-c_x}{f}\right),\ f \cdot \arctan\left(\frac{W-1-c_x}{f}\right)$$

• Vertical span similarly from min/max $y'$.

In practice, many implementations pick output width and height heuristically (e.g., same height, wider width) and then map backward.

Step 2: Derive the inverse mapping

The given forward mapping is:

$$x' = f \cdot \arctan\left(\frac{x-c_x}{f}\right),\quad y' = f \cdot \frac{y-c_y}{\sqrt{(x-c_x)^2 + f^2}}.$$

For implementation, you want for each output $(x',y')$ to find input $(x,y)$.

From the 3D geometry:

1. Compute angle:

$$\theta = \frac{x'}{f}.$$

2. Horizontal direction:

$$X = \sin\theta,\quad Z = \cos\theta.$$

3. Vertical:

$$Y = \frac{y'}{f} \cdot \sqrt{X^2 + Z^2} = \frac{y'}{f}.$$

(because $X^2 + Z^2 = 1$ on the unit cylinder.)

4. Project back to image plane:

$$x = f \cdot \frac{X}{Z} + c_x = f \tan\theta + c_x,$$ $$y = f \cdot \frac{Y}{Z} + c_y = f \cdot \frac{y'/f}{\cos\theta} + c_y = \frac{y'}{\cos\theta} + c_y.$$

So your inverse mapping for sampling is:

$$\theta = \frac{x'}{f},\quad x = f \tan\theta + c_x,\quad y = \frac{y'}{\cos\theta} + c_y.$$

Step 3: Implement the warp (inverse mapping loop)

For each pixel (u, v) in the cylindrical output image:

1. Convert (u, v) to continuous cylindrical coordinates (x', y'):

x_prime = u - cx_out # centering if needed
y_prime = v - cy_out

or adjust based on how you defined the output origin.

2. Compute:

theta = x_prime / f
X = math.tan(theta) # since X/Z = tan(theta)
cos_theta = math.cos(theta)
if abs(cos_theta) < eps: continue # avoid division by zero

x_src = f * X + cx # x = f * tan(theta) + cx
y_src = y_prime / cos_theta + cy

3. If (x_src, y_src) is inside the source image bounds, sample from I:

• Use bilinear interpolation for better quality:

color = bilinear_sample(I, x_src, y_src)
out[v, u] = color

Step 4: Apply per-image, then stitch

• Call this warp for each input image to get its cylindrical version.
• Downstream, you:
• Detect/match features in cylindrical images.
• Estimate horizontal translations (and possibly slight vertical corrections).
• Place each cylindrical image into a large panorama canvas and blend.

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4. Common Pitfalls

• Forward vs inverse mapping confusion Implementing forward mapping (loop over source pixels, compute (x',y')) leaves holes and requires splatting; inverse mapping (loop over destination, sample source) is cleaner and standard in image warping.

• Wrong coordinate center Forgetting to subtract/add (c_x, c_y) correctly leads to strong distortions and misalignment. Always treat rays relative to the optical center.

• Focal length scale issues If f is in pixels, but you treat it as normalized or mis-estimate it, the panorama will look excessively stretched or compressed. Often f comes from EXIF or camera calibration; if not, you may have to tune it.

• Output canvas sizing If your cylindrical canvas is too small, parts of the image get clipped; too large and you waste memory. Derive reasonable bounds from the mapping or start with a safe upper bound and crop later.

• Interpolation and aliasing Using nearest-neighbor leads to jagged edges and aliasing. Use bilinear interpolation; for high-quality results, consider anti-aliasing when downsampling.

• Angle discontinuities When going close to 360°, consider how you handle the wrap-around at $\theta = -\pi$ and $\theta = \pi$. In practice, you may normalize angles or use modulo arithmetic when stitching many views.

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5. Time & Space Complexity

Assuming:

• Input image: width $W$, height $H$.

• Output cylindrical image: width $W'$, height $H'$ of the same order as $W, H$.

• **Time complexity (per image):