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Using the Triangle To Quad option on this mesh produces a bunch of disconnected triangles, like in the photo.

enter image description here

What is the easiest way to remove these so that they are replaced by quads?

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    $\begingroup$ What do you mean by "disconnected"? $\endgroup$ Commented Oct 7, 2015 at 18:28
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    $\begingroup$ In this very situation, it may be easier to delete all the faces first, then fill with Ctrl+F > Grid Fill. In a more complicated case, it can be done by hand, filling Ngon and cutting it with J. $\endgroup$ Commented Oct 7, 2015 at 18:55

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From that exact view (all those exact faces selected):

  • Hit X, then faces to delete the mess you have.
  • In edge mode,Alt+Right Click an edge, to select the loop around the hole.
  • Hit F to fill with an ngon.
  • Then hit Ctrl+T to Triangulate.
  • Then hit Alt+J to do a Triangle to Quad

EDIT:

Mr Zak's suggestion to use Grid Fill greatly improves this answer by reducing the last three steps to just one:

  • Hit X, then faces to delete the mess you have.
  • In edge mode,Alt+Right Click an edge, to select the loop around the hole.
  • Hit Ctrl+F then Grid Fill
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