01 · THE REPRESENTATION

One correspondence. Four numbers.

A flow field supplies a displacement at an image location. A sparse keypoint pair supplies the same information: subtract its starting coordinate from its matching coordinate. BFV puts both into a common space:

Position tells us where; displacement tells us how far and in which direction. Magnitude is √(Δx̂² + Δŷ²), and direction is atan2(Δŷ, Δx̂). Encoding the two displacement components retains both without choosing angle or magnitude bins. A zero displacement has zero magnitude and no defined direction.

This is a compact description of annotated image-space motion. It requires no RGB pixels, learned descriptor or object category. For semantic keypoints, “motion” means the displacement between corresponding parts across images, even when they belong to different objects.

02 · FROM PIXELS TO SHARED UNITS

Normalize the coordinate system, not the arrow’s length.

The paper crops images to a square, then expresses locations and displacements in units of that image. Coordinates must refer to the cropped frame, with correspondence annotations transformed consistently.

The image center maps to (0, 0); its edges map to −1 and +1. A displacement of +1 means half an image width or height, not one pixel. The displacement components can exceed 1: they are differences between normalized coordinates. We do not rescale every arrow to unit length, which would erase magnitude.

Resolution invariance means the same relative correspondence has the same encoding after a consistent resize. A different crop or a different set of sampled frames can still change the distribution. Frame order matters too: swapping a correspondence moves its anchor to the other endpoint and reverses its displacement.

03 · WHY KEEP POSITION?

The same arrows can describe different motion fields.

A direction histogram could count equally many leftward and rightward vectors in both an expanding and a contracting field. Their magnitude distributions can match too. What differs is where those vectors occur. BFV retains that relationship by keeping position and displacement together in each row.

BFV has no predefined spatial cells. Nearby correspondences can be compared continuously in four dimensions. It preserves the relation between a vector and its location, but does not retain which neighboring vectors came from the same image.

04 · FROM ONE PAIR TO A DATASET

The bag is a dataset-level empirical distribution.

Pool the retained annotation vectors across the dataset. One point in BFV space represents one correspondence—not an entire image pair. The bag discards row order and image identity, but preserves the observed distribution of positions and displacements. It is not a mean arrow: opposing motions do not cancel into a stationary dataset.

Dense flow contributes vectors at labeled pixels; sparse matching contributes vectors only at its annotated keypoints. Unannotated locations do not become zero-motion examples. Annotation validity, extraction filters and sampling determine which vectors enter the bag. Pooling by vector is also different from assigning equal weight to every image, so supervision density and the sampling protocol matter.

The representation lets a synthetic source and a real benchmark live in the same motion space, even if one contains rendered objects and the other contains sparse animal keypoints. That does not make the tasks identical; it makes their annotated displacement statistics comparable.

05 · WHY THIS HELPS TEST THE THESIS

Compare what datasets supply before training a matcher.

BFV isolates a compact motion hypothesis: perhaps transfer depends on the displacements a source supplies, not only on the appearance of its images. The paper separately represents appearance with DINOv3 patch features and compares both representations with distribution distances. BFV itself is a representation, not a transfer score.

Next, coverage asks whether each target vector has a nearby source vector. The reverse direction asks how much source motion is unlike the target. These are dataset comparisons over the bags; no correspondence between individual source and target images is required.