Made to measure morphological filters Fernand Meyer Center

Made to measure morphological filters Fernand Meyer Center of mathematical morphology Mines-Paris. Tech France

Non edge preserving filters The adjunction erosion/dilation Openings/closings The simplest filters Alternate sequential filters

EROSIONS AND DILATIONS OF INCREASING SIZES

Erosions of increasing size Regional minima get larger New maxima may appear. Most maxima vanish, others get larger.

Dilations of increasing size New minima may appear. Most mainima vanish, others get larger. Regional maxima get larger, some coalesce, others disappear.

Erosion = minimal value in a window darker image Dilation = maximal value in a window brighter image INCREASING WINDOW + STRONGER EFFECT Increasing erosions Increasing dilations

COMBINING AN EROSION AND A DILATION IN SEQUENCE, IN ORDER TO GET OPENINGS OR CLOSINGS.

An erosion Darker Followed by A dilation Brighter = opening Darker

openings of increasing size Each reg. min. of a large closing contains a reg. Min. of each smaller opening Regional maxima get larger, some coalesce, others disappear.

A dilation Brighter Followed by An erosion Darker = closing Brighter

openings of increasing size Regional minima get larger, some coalesce, others disappear. Each reg. max. of a large closing contains a reg. Max. of each smaller closing

OPEN-CLOSE, CLOSE-OPEN, OPEN-CLOSE-OPEN, CLOSE-OPEN-CLOSE

An opening Darker Followed by A closing Brighter = morphological filter Similar grey tone

A closing Brighter Followed by An opening Darker = morphological filter Similar grey tone

Opening followed by closing of increasing sizes Regional minima Regional maxima

Closing followed by opening of increasing sizes Regional minima Regional maxima

Opening followed by closing of increasing sizes Closing followed by opening of increasing sizes The two are not comparable

Open close open of increasing sizes Close open close of increasing sizes Close-open-close open-close-open

Closing followed by opening followed by closing of increasing sizes Regional minima Regional maxima

A small closing followed by a small opening = small effects A large closing followed by a large opening = too crude Alternate sequential filter = sequence of increasing pairs of an opening followed by a closing

Alternate sequential filter starting with an opening of increasing sizes Regional minima Regional maxima

Alternate sequential filter starting with a closing of increasing sizes Regional minima Regional maxima

Partial conclusion Non edge preserving filters simplify the images but the displacement of the contours is annoying in many cases and in particular if the filter precedes image segmentation We will now introduce edge preserving filters : Razings Floodings Levelings