Warping Stratospheric Views Into Perfect Panoramas

Your stratospheric balloon just transmitted 47 overlapping images from 30 kilometres up. Each frame captures a slightly different perspective as the payload slowly rotates, creating a jigsaw puzzle of the Earth’s curve that needs algorithmic assembly.

The secret lies in homographic transformations—3×3 matrices that map any quadrilateral to any other quadrilateral through perspective projection. When your balloon’s camera shifts from one angle to another, a homography captures exactly how each pixel should move to align with the reference frame.

class HomographyStitcher {
    fun transform(x: Double, y: Double, h: Array<DoubleArray>): Pair<Double, Double> {
        val w = h[2][0] * x + h[2][1] * y + h[2][2]
        val newX = (h[0][0] * x + h[0][1] * y + h[0][2]) / w
        val newY = (h[1][0] * x + h[1][1] * y + h[1][2]) / w
        return Pair(newX, newY)
    }
}

Kotlin’s matrix operations feel natural here—each pixel coordinate gets multiplied through the homography matrix, then normalized by the third coordinate to handle perspective distortion.

: homography-transform ( x y h-addr -- x' y' )
    >r                          \ save matrix address
    r@ 6 + @ rot *              \ h[2][0] * x  
    r@ 7 + @ rot * +            \ + h[2][1] * y
    r@ 8 + @ +                  \ + h[2][2] = w
    dup >r                      \ save w
    r@ @ rot *                  \ h[0][0] * x
    r@ 1 + @ rot * +            \ + h[0][1] * y
    r@ 2 + @ + r> /             \ + h[0][2], divide by w
    swap r@ 3 + @ rot *         \ h[1][0] * x
    r@ 4 + @ rot * +            \ + h[1][1] * y  
    r> 5 + @ + r> /             \ + h[1][2], divide by w
 ;

Forth’s stack manipulation makes the math explicit—every intermediate calculation sits on the stack, ready for the next operation. It’s verbose but brutally efficient, perfect for embedded telemetry processors with tight memory constraints.

The magic happens when you identify corresponding feature points between overlapping images. Four matched points define the homography completely, transforming one image’s coordinate system to match another’s. Your balloon’s gentle rotation becomes a precise geometric dance, each frame finding its place in the final panoramic mosaic.