Matchsticks have an axis before they have an arrow
Scatter short matchsticks on a table and comb them into parallel rows. Ignore which end has the match head. A stick facing left now has the same orientation as one facing right. This is nematic order: nearby long objects share an axis without agreeing on a head-to-tail direction.
An ordinary average of arrows would fail here. Half the arrows could point left and half right, giving zero even in a perfectly aligned patch. A nematic order tensor solves this problem by recording the axis and how strongly the rods share it. In two dimensions, it can be stored as two numbers at every point.
The local preferred axis is called the director. In a real image, estimate it over a small patch containing several rods. Tiny patches are noisy; very large patches smear out sharp turns. Choosing a measurement scale is part of defining the field.
Carry a pencil around the centre
Look at the comet-shaped texture of a positive half defect. Place a pencil along the local axis, then carry it slowly around the centre, keeping it aligned with the rods. Keep track of the rotation continuously; never reset the pencil merely because an angle crosses the edge of a plotting scale.
After one circuit, the pencil has turned half a revolution. It points the other way, but it lies along the same axis. Around a negative half defect, the continuous rotation is a half revolution in the opposite sense. The surrounding pattern has three arms instead of a comet’s head and tail.
Charge counts this net rotation in units of a full revolution. The sign depends on traversing the loop counterclockwise in a fixed right-handed plane. A screen often measures height downward, while a physical angle is measured upward. Mixing those two coordinate conventions reverses a computed sign; reflecting both coordinates and rods consistently preserves the charge.
Worked example
Count an example without trusting the picture
Walk counterclockwise around four corners of a small square. Suppose the director rotates smoothly by 45 degrees along each edge. Four changes add to 180 degrees: the enclosed charge is positive one half. The final axis matches the initial one.
At each edge, choose the equivalent director rotation whose magnitude is less than 90 degrees. If the field turns farther between samples, the sign becomes ambiguous. Add more samples or use a larger, well-resolved loop around the core. Inside the core, local alignment weakens and a director may be poorly defined.
A detector finds winding, then a tracker connects it
A sampled-field detector adds these small angle changes around every grid cell. Cells with half-turn winding mark candidate defects. Its input is the recorded orientation field, not the locations used to prepare a synthetic texture. An analytic test texture is useful precisely because its known charge lets us check the detector.
Tracking is another operation. A positive defect in one frame is connected to a nearby positive defect in the next, within a stated maximum displacement. If two defects pass too closely, merge or disappear, a trajectory can become ambiguous. Mark a track as uncertain whenever the association is ambiguous.
On a flat periodic plane with no holes, all charges sum to zero. Defects can appear or disappear in opposite-charge pairs. A nonzero measured total is a warning to inspect the resolution, seam treatment and order threshold. A bounded region has a different relation: the orientation around its boundary supplies the total winding.