Lecture 11+x+1 Chapter 9 Morphological Image Processing

Image processing Linear –Convolution: Linear-Position invariant processing Frequency domain –Frequency domain filtering: Low pass/High pass, Notch filters, Inverse filtering, CLS filtering Statistical processing –Order statistics filtering, Weiner filtering Set theoretical approach –Morphological processing

Morphological processing Morphology – Study of form and structure of objects. Developed in 1960’s by Matheron and Serra. Deals with set of spatial coordinates rather then pixel values and therefore set theory is the ideal language for such processing. Applications include: thinning, region processing, segmentation, boundary extraction, connected components etc.

Moprhological processing It is based on set theory and can model non-linear processing. We will start by considering only digital binary images. Morphological processing can be extended to gray and color images.

Set theory Set is a collection of elements. –In our case of digital binary images, sets would consist of collection of discrete coordinates. Set of white or black pixels completely defines the image. –Working on the set of black or white pixel coordinates in the image is equivalent to working on the image.

Set theory – basic concepts Belongs to ( ), Null set ( ) Subset ( ) - If every element of one set is also an element of the other set Union ( ) - Set of elements belonging to either set Intersection ( ) - Set of elements belonging to both sets Complement ( ) - Set of elements not in A Difference (A - B) – Set of elements belonging to A, but not to B.

Set theory

The following operations do not apply to a general set: Reflection of a set: Translation of a set: Where is a point.

Fundamental operators on sets Minkowski addition: Minkowski subtraction:

Basic Morphological operations: Dilation & Erosion Consider two subsets A & B of Z 2 Dilation of A by B is defined by: –Flip/Reflect B to get. –Translate by z and check –If non-empty, then

Dilation The origin plays an important role. The operation is not commutative and hence the order in which the sets are considered is important The set B is called the structuring element. The choice of structuring element becomes critical and depends on the application/problem.

Dilation

Applications of dilation

Erosion Erosion: –Translate with z –Check, if yes then Erosion of A by B is the set of points z such that B translated by z is contained in A.

Erosion

Application of Erosion Noise removal, separates objects joined with narrow bridges. Example 1 Example 2 Example 3 What is the difference?

Dual operators: Dilation and Erosion Duality: Proof: –Where

Dilation & Erosion Is Dilation(Erosion(Image)) = Image?

Dilation and Erosion Dilation(Erosion(Image)) ImageEroded ImageDilation of Eroded Image

Two more operators Combine Dilation and Erosion to get more operators: Opening:

Opening Example Structuring Element Original Image Processed Image

Opening example Removes narrow parts, small extrusions and isolated pixels. The size of the neighborhood that is removed depends on the structuring element

Closing Definition:

Closing Closes small intrusions, small gaps and connects disconnected objects Again, definition of “small” depends on the size of the structuring element.

Closing Example Structuring Element Original Image Processed Image

Opening and Closing

Opening: Erosion followed by Dilation Closing: Dilation followed by Erosion Properties of Opening and Closing? –Duality: –Idempotent: