More ways to Rome.

You have two things: triangle coordinates and texture coordinates. In the regular image these coincide. In the exploded image these don't coincide. The texture coordinates are the same, but the triangle coordinates have changed. So my idea would be to use a beginShape-endShape for each triangle. Use a texture (the original) image and the original texture coordinates. You could change the coordinates in the Shape directly (for example to deform the image). But in this case you probably want to keep the image within the triangle as it was (only move those triangles). In that case it would be easier to just use the original coordinates for the Shape as well, but additionally translate and rotate each Shape however you want in order to create the explosion effect.

PseudoCode Example

1. textureMode(IMAGE);
   
2. for (DelauneyTriangle d : allDelauneyTriangles) {
3.       translate(thisTriangleTranslation);
4.       rotate(thisTriangleRotation);
   
5.       beginShape();
6.       texture(original_image);
7.       vertex(d1.x, d1.y, d1.x, d1.y);
   
8.       vertex(d2.x, d2.y, d2.x, d2.y);
   
9.       vertex(d3.x, d3.y, d3.x, d3.y);
   
10.       endShape();
11. }