Unprojecting text with ellipses (2016)
mzucker.github.io
mzucker.github.io
"Never have unprojected text."
Not:
"I never have [...]"
The absence of an explicit subject means that another correct interpretation of the sentence is that the OP is giving you some good advice.
That's an outrageous claim that needs some sort of justification.
Im not evaluating the worth of said advice, just the grammar, to play nitpick tennis with alex_duf who graciously conceded the point.
Matrix equations are really just shorthand for several related equations. The notation can be a bit unsettling if you aren’t used to it.
In contrast, elements of a vector space have absolute properties; in particular, there is a "zero vector" whose magnitude is 0. Vectors can be added and scaled in a meaningful way.
We can obtain a vector space from an affine space of points by considering the displacements between points (i.e. arrows going from one point to another): the zero vector is the displacement from any point to itself; displacements can be added (first follow one arrow, then the other); and displacements can be scaled (changing the magnitude, whilst keeping the direction fixed).
To form a vector space of points, we can choose some arbitrary point in our affine space (which we call "the origin") and consider the vector space of displacements from our chosen origin. In other words, if we take the "arrows" from our vector space of displacements, and have them all start at the same point (our arbitrary "origin"), then each arrow corresponds to a point (its end), and each point corresponds to an arrow (ending at that point, starting at the origin). The origin point itself corresponds to the zero vector of displacements. This way, we've made points equivalent to displacements (from "the origin"), and since the latter form a vector space, so do the former.
However, despite these shenanigans it's important to keep in mind that it's still only meaningful to talk about points in a relative way. That's explicit in the affine approach; whilst the vector approach always has an implicit "relative to this particular origin".
I guess the author partly answers your question early on with discussion of the Merino-Gracia paper, which fits a quad to individual lines of text, and a comment about how that relies on first being able to detect lines of text.
Matt also doesn’t claim this method is better. He says “I’m sure its neither as accurate or as useful as the Merino-Gracia approach.“ I assume the example text “Needlessly Complex” is a bit of self-deprecating humor, acknowledging he may not be taking the easiest path there is. But the method here seems interesting and useful to me for its approach; it doesn’t have to identify word or page boundaries, or lines of text, as a prerequisite. The assumptions are simple and the optimization is simple, it’s a nice study in different ways to think about the problem.
(Having to identify page boundaries is handwaved away with "I’m going to make a huge simplifying assumption that that the image we’re processing basically contains only the text that we want to rectify")