The convex hull answers one question: at a given bitrate, which resolution gives the best picture? Because a low resolution wins when bits are scarce (a clean small frame upscaled beats a starved large one) and a high resolution wins when bits are plentiful, the resolution curves of quality-versus-bitrate cross, and the best choice is the upper envelope stitched from whichever resolution is currently winning. That envelope is the convex hull, and every rung of a per-title ladder sits on it. To build it you encode a grid, score each encode, drop the dominated points, and read the resolution off the frontier at each bitrate. The load-bearing catch is the measurement: full-reference metrics need both frames the same size, so a lower-resolution encode must be upscaled to the source resolution before scoring, and the upscaler must be held constant. It is the geometry behind per-title encoding and the curve BD-rate is computed on.
The handbook · 7 volumes
Fora Soft Video Engineering Handbook
The convex hull keeps only the resolution and bitrate points that win on quality, which is the math that makes per-title encoding save bandwidth. We wrote seven volumes covering the whole video stack, and for the convex hull start with Volume 1: rate-quality curves, resolution and scaling, per-title encoding and quality metrics, with Volume 3 on bitrate ladders and ABR. Written by Nikolay Sapunov, CEO at Fora Soft.
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Send us a sample of titles: we build the convex hull per title, derive a ladder from it and estimate the bitrate and storage you save. 250+ video products shipped since 2005.
