Morphological Image Processing การทำงานกับรูปภาพด้วยวิธีมอร์โฟโลจิคัล Chapter 9 Morphological Image Processing การทำงานกับรูปภาพด้วยวิธีมอร์โฟโลจิคัล

Meaning of “Morphology” Commonly a branch of biology that deals with the form and structure of animals and plants. “mathematical morphology” as a tool for extracting image components that representation and description of region shape, such as boundaries, skeletons, and the convex hull. Digital Image Processing by K.Ratchadaporn

Mathematical morphology The language of mathematical morphology is set theory. Sets in mathematical morphology represent objects in an image. Example: the set of all back pixels in a binary image is a complete morphological description of the image. Each element of set is a tuple(2D vector) whose coordinates are the (x,y) coordinates of a black pixel in the image. Digital Image Processing by K.Ratchadaporn

Digital Image Processing by K.Ratchadaporn Example Binary Image Set A is set of black pixels 7 A = {(3,1),(4,1),(2,2),(5,2), (2,3),(5,3),(1,4),(2,4),(3,4), (4,4),(5,4),(6,4),(1,5),(6,5), (1,6),(6,6)} 7 Digital Image Processing by K.Ratchadaporn

Basic Concepts of Set Theory Definition of Elements What Subset is Union Operation Intersection Operation Mutually exclusive Property Complement Operation Difference Operation Digital Image Processing by K.Ratchadaporn

Definition of Elements Let A be a set in Z2. If a = (a1,a2) is an element of A, then we write Similarly, if a isn’t an element of A we write An arbitrary set in Zn has elements n-tuples as (z1,z2,. . .,zn) The set with no elements is called the null or empty set Digital Image Processing by K.Ratchadaporn

Digital Image Processing by K.Ratchadaporn What subset is If every element of a set A is also an element of another set B then A is said to be a subset of B, denoted as Example: X={(1,1),(1,2),(1,3),(2,1),(2,2), (2,3),(3,1),(3,2),(3,3)} and Y={(1,2),(2,1),(2,2),(2,3),(3,2)} So, Digital Image Processing by K.Ratchadaporn

Digital Image Processing by K.Ratchadaporn Union Operation The union of two sets A and B denoted by Set C is the set of all elements belonging to either A, B, or both Digital Image Processing by K.Ratchadaporn

Intersection Operation The intersection of two sets A and B denoted by Set D is the set of all elements belonging to both A and B Digital Image Processing by K.Ratchadaporn

Mutually Exclusive Property Two sets A and B is disjoint or mutually exclusive if they have no common elements A B Digital Image Processing by K.Ratchadaporn

Complement & Difference The complement of a set A is the set of elements not contained in A: Difference of two sets A and B, denoted A-B, is defined as This is the set of elements that belong to A, but not to B. Digital Image Processing by K.Ratchadaporn

Digital Image Processing by K.Ratchadaporn Summary Digital Image Processing by K.Ratchadaporn

Digital Image Processing by K.Ratchadaporn Addition definition Two additional definition that are used extensively in morphology The reflection of set B is defined as The translation of set A by point z=(z1,z2) is defined as Digital Image Processing by K.Ratchadaporn

Digital Image Processing by K.Ratchadaporn Reflection Digital Image Processing by K.Ratchadaporn

Digital Image Processing by K.Ratchadaporn Translation z1 z2 Digital Image Processing by K.Ratchadaporn

Logic Operations & Binary Images The principal logic operations used in image processing are AND, OR, and NOT(Complement) Logic operations are preformed on a pixel by pixel basis between corresponding pixels of two or more images(except NOT) Digital Image Processing by K.Ratchadaporn

Logic Operations & Binary Images AND OR Digital Image Processing by K.Ratchadaporn

Logic Operations & Binary Images NAND XOR Digital Image Processing by K.Ratchadaporn

Fundamental morphological processing Two Operation are fundamental to morphological processing: Dilation Erosion Many of the morphological algorithms are based on these two primitive operations Digital Image Processing by K.Ratchadaporn

Digital Image Processing by K.Ratchadaporn Dilation Let A and B as set in Z2, The dilation of A by B is defined as Then it is the set of all displacements, z Such that B and A overlap by at least one element Note : Set B is commonly referred to as the “structuring element” Digital Image Processing by K.Ratchadaporn

Digital Image Processing by K.Ratchadaporn Example : Dilation d d x y Digital Image Processing by K.Ratchadaporn

Digital Image Processing by K.Ratchadaporn Example : Dilation Digital Image Processing by K.Ratchadaporn

Application : Dilation 1 Digital Image Processing by K.Ratchadaporn

Digital Image Processing by K.Ratchadaporn Erosion Let A and B as set in Z2, The erosion of A by B is defined as Then it is the set of all points z Such that B translated by z, is contained in A Digital Image Processing by K.Ratchadaporn

Digital Image Processing by K.Ratchadaporn Example : Erosion d d x y Digital Image Processing by K.Ratchadaporn

Digital Image Processing by K.Ratchadaporn Example : Erosion Digital Image Processing by K.Ratchadaporn

Digital Image Processing by K.Ratchadaporn Application : Erosion (a) Image of squares of size 1,3,5,7,9 and 15 pixels on the side (b) Erosion of (a) with a square structuring element of 1’s, 13 pixels on the side (c) Dilation of (b) with a same structuring element Digital Image Processing by K.Ratchadaporn

Digital Image Processing by K.Ratchadaporn Erosion Complement Dilation and Erosion are duals of each other with respect to set complementation and reflection Digital Image Processing by K.Ratchadaporn

Digital Image Processing by K.Ratchadaporn Proving Starting with the definition of erosion If set (B)z is contained in set A, then thus Digital Image Processing by K.Ratchadaporn

About Dilation & Erosion Dilation expands an image. Erosion shrinks an image. Digital Image Processing by K.Ratchadaporn

Digital Image Processing by K.Ratchadaporn Opening and Closing Opening generally smoothes the contour of an object, breaks narrow isthmuses, and eliminates thin protrusions. Closing also tends to smooth sections of contours but, as opposed to opening, it generally fuses narrow breaks and long thin gulfs, eliminates small holes, and fills gaps in the contour Digital Image Processing by K.Ratchadaporn

Digital Image Processing by K.Ratchadaporn Opening The opening of set A by structuring element B is defined as Thus, the opening A by B is the erosion of A by B, followed by a dilation of the result by B. Digital Image Processing by K.Ratchadaporn

Digital Image Processing by K.Ratchadaporn Opening A B Digital Image Processing by K.Ratchadaporn

Digital Image Processing by K.Ratchadaporn Closing The closing of set A by structuring element B is defined as Thus, the opening A by B is the dilation of A by B, followed by a erosion of the result by B. Digital Image Processing by K.Ratchadaporn

Digital Image Processing by K.Ratchadaporn Closing A B Digital Image Processing by K.Ratchadaporn

Digital Image Processing by K.Ratchadaporn Opening and Closing Digital Image Processing by K.Ratchadaporn

Digital Image Processing by K.Ratchadaporn Apply for Problem Digital Image Processing by K.Ratchadaporn

HIS-or-MISS Translation Digital Image Processing by K.Ratchadaporn

HIS-or-MISS Translation If B denotes the set composed of X and its background, The match (or set of matches) of B in A, denoted is If B1=X and B2=(W-X) By using the definition of set differences given Digital Image Processing by K.Ratchadaporn