Mathematical Morphology and Image Interpolation



Image interpolation is a general term for a set of techniques used in image synthesis to generate intermediary pictures between two successive images. Various tools may be used to achieve this: linear tools (average of more sophisticated weighting functions), interactive means (morphing techniques), etc.
In this page, we present some interpolations performed by using mathematical morphology transformations. In these examples, the interpolations are based on a transformation named SKIZ (Skeleton by Influence Zones). More complex transformations, such as geodesic operators or Hausdorff distance computations, can also be used.
Besides the funny aspect of these interpolations, they are also used to solve more "serious" image analysis problems in particular in image compression, image coding, object tracking, as it is illustrated in some animations available in this page.


Set interpolations

Here are some examples of set interpolations. The first and last images of each example are displayed. Click on the corresponding link to show the intermediary sets (animated GIF).

Animation

Animation

Animation

Animation

Animation


Greytone image interpolations

This interpolation technique can also be used on greytone images. It is still a set interpolation. The sets which are interpolated are the sets under the graph of the starting and ending functions (3D sets). This is illustrated in the following example where the initial and final images are the same as in the last set interpolation, but where they are considered as greytone images (click on the left picture to see the animation).
This approach can be compared with a classical arithmetic interpolation (click on the right picture). The morphological interpolation better preserves the topological (connectivity) properties of the images (although a little bit altered compared to the set interpolation).




Partition interpolations

Morphological interpolation can also be used to interpolate partitions. Each corresponding cell in the initial and final partitions is interpolated, its color or grey value remains unchanged.

Animation

This kind of interpolation is particularly useful in image compression. Many techniques exist to generate a partition from any greytone or color image. We can, for instance, define a mosaic image, by means of the watershed transformation. A very simple solution to increase dramatically the compression rate of a sequence is to transmit only one image over n in the sequence and to interpolate the missing images.
The following animations show an original sequence (on the left) and the final result (on the right) when only one image out of eight have been transmitted, the others being interpolated.



The result is not perfect (mouth and left eye in particular) but it can be greatly enhanced if we transmit and add to the final sequences the small white and black features detected with a top-hat transformation.



These interpolation techniques are patented (Patent n94-14162), inventors S. Beucher, F. Meyer, J. Serra.
Images and animations copyrighted 1996, Serge Beucher, All rights reserved.


Serge BEUCHER Home Page

CMM Home Page