Jigsaw Sudoku

A Jigsaw Sudoku does not have 3x3 boxes, but regions with an irregular shape.

Jigsaw is a very popular Sudoku Variant.

Example

In a forum, this puzzle can be presented like this:

.-----------.---.-------------------.
| .   .   . | . | .   2   .   .   . |
|       .---'   '-------.           |
| .   . | .   3   .   . | .   9   . |
|       :-----------.   '---.---.   |
| .   1 | .   .   . | .   . | . | . |
|   .---'   .-------'---.   |   '---:
| 7 | .   . | .   .   . | 4 | .   . |
|   |   .---:           |   |   .---:
| . | . | . | .   .   . | . | 2 | 5 |
:---'   |   |           :---'   |   |
| .   6 | . | .   .   . | .   . | . |
:---.   |   '---.-------'   .---'   |
| . | 7 | .   . | .   .   . | .   6 |
|   '---'---.   '-----------:       |
| .   9   5 | .   .   .   . | .   . |
|           '-------.   .---'       |
| .   .   .   2   . | . | .   4   . |
'-------------------'---'-----------'

The irregular regions allow the player to use the Law of Leftovers, a solving technique which can only be used on irregular regions.