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Is there any way to cleanup/remove empty vertex weights? Namely, on each vertex in a mesh, remove any vertex weights that have a value of zero or something close to it.

Note that I am referring to something more distinct than vertex groups. When you have a vertex selected, a "Vertex Weights" tab will appear in the Properties Panel, and it will list all of the vertex weights affecting a particular vertex.

enter image description here

Stuff like that in the image. I'd like something to remove all of the vertex weights that are at 0 from each vertex. Or any that have a weight of less than 0.001, since the display there only seems to go out to 3 digits past the decimal point.

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You want to be in Weight Paint Mode. From there, goto Weights >> Clean. The settings for which threshold you want (if its other than zero) can be found on the tool shelf, on the Clean Vertex group Properties.

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    $\begingroup$ In Blender 2.8X, the "Clean" option in Weight Paint Mode has a slight small issue if set to 0.00 value. While the note tip appearing state that it will remove any weight group (on selected verticles) with a value lower or equals to the number, there's an internal error margin (float precision) for the 0.00 value which return a value below zero (like -0.0000134), hence doesn't remove the 0 weight groups. Setting it to a super low value like 0.001 is a work around. $\endgroup$
    – user3345048
    Commented Aug 21, 2021 at 16:44

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