Digital Image Fundamentals

Digital Image Fundamentals Faculty of Science Silpakorn University

Overviews • Elements of Visual Perception • Term of image • Sampling and Quantization

Term of image • Pixel • Resolution • Neighbor of pixels • Space and Frequency • Vector

PIXEL • Pixel is a smallest component of digital image • Pixel is a color point of digital image • An image should be comprised of many Pixels.

PIXEL

RESOLUTION • How quality of image • With the same size of picture • If high resolution, high memory is required to store data • If low resolution, less memory is required to store data • Its unit is call “point per inch”

RESOLUTION 72 36 18 9

Digital Image Representation y f(x,y) x

Neighbors of a pixel • Neighbors of pixel are the pixels that are adjacent pixels of an identified pixel. • Each pixel is a unit distance from the particular pixel. • Some of the neighbors of pixel lie outside the digital image if its position is on the border of the image.

Pixel at coordinate (column x, row y) can be represented by f(x,y) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Pixel at the 7th column and 4th row is yellow color Zoom 1600%

4-neighbors of pixel • 4-neighbors of pixel is denoted by N4(p) • It is set of horizontal and vertical neighbors f(x,y) is a yellow circle f(x,y-1) is top one f(x-1,y) is left one f(x+1,y) is right one f(x,y+1) is bottom one (x) (x+1) (x-1) (y-1) (y) (y+1)

Diagonal neighbors of pixel • Diagonal neighbors of pixel is denoted by ND(p) • It is set of diagonal neighbors f(x,y) is a yellow circle f(x-1,y-1) is top-left one f(x+1,y-1) is top-right one f(x-1,y+1) is bottom-left one f(x+1,y+1) is bottom-right one (x) (x+1) (x-1) (y-1) (y) (y+1)

8-neighbors of pixel • 8-neighbors of pixel is denoted by N8(p) • 4-neighbors and Diagonal neighbors of pixel f(x,y) is a yellow circle (x-1,y-1), (x,y-1),(x+1,y-1), (x-1,y), (x,y), (x+1,y), (x-1,y+1),(x,y+1), (x+1,y+1) (x) (x+1) (x-1) (y-1) (y) (y+1)

Connectivity • Establishing boundaries of objects and components of regions in an image. • Group the same region by assumption that the pixels being the same color or equal intensity will are the same region

Connectivity • Let C is the set of colors used to define • There are three type of connectivity: • 4-Connectivity : 2 pixels (p and q) with value in C are 4-connectivity if q is in the set N4(p) • 8-Connectivity : 2 pixels (p and q) with value in C are 8-connectivity if q is in the set N8(p) • M-Connectivity : 2 pixels (p and q) with value in C are 8-connectivity if • Q is in N4(p), or • Q is in ND(p) and the set N4(p) ∩ N4(q) is empty

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 Binary Image Represent

0 1 1 0 1 1 0 1 0 0 1 1 Example 4-Connectivity • Set of color consists of color 1 ; C ={1}

0 1 1 0 1 1 0 1 0 0 1 1 Example 8-Connectivity • Set of color consists of color 1 ; C ={1}

0 1 1 0 1 1 0 1 0 0 1 1 Example M-Connectivity • Set of color consists of color 1 ; C ={1}

Distance Measure • For pixels p, q, and z, with coordinates (x,y), (s,t) and (u,v) respectively, D is a distance function or metric if • D(p,q) ≥ 0 and • D(p,q) = 0 iff p = q and • D(p,q) = D(q,p) and • D(p,z) ≤ D(p,q) + D(q,z)

2 2 1 2 2 1 0 1 2 2 1 2 2 The D4 Distance • Also called city-block distance • Calculate between p and q is defined as D4(p,q) = | px- qx | + | py- qy |

2 2 2 2 2 1 1 1 2 2 2 1 0 1 2 1 1 1 2 2 2 2 2 2 2 The D8 Distance • Also called city-block distance • Calculate between p and q is defined as D8(p,q) = max(| px- qx |,|py- qy |)

Labeling of connected Components • Scan an image pixel by pixel from left to right and top to bottom • There are 2 types of connectivity interested • 4-connectivity • 8-connectivity • Equivalent labeling

Step: 4-connected components • P is pixel scanned process • If pixel p is color value 0 move on the next scaning • If pixel p is color value 1 examine pixel top and left • If top and left were 0, assign a new label to p • If only one of them’re 1, assign its label to p • If both of them’re 1 and have • the same number, assign their label to p • Different number, assign label of top to p and make note that two label is equivalent • Sort all pairs of equivalent labels and assign each equivalent to be same type

1 0 1 1 0 1 1 0 1 1 0 0 1 1 3 Example 4-connected components 1 Equivalent Table 2 3 4

Step: 8-connected components • Steps are same as 4-connected components • But the pixel that are consider is 4 previous pixels ( top-left, top, top-right, and left )

1 0 1 1 0 1 1 0 1 1 1 1 0 0 1 1 1 1 Example 8-connected components 1

Multi-Color Labeling • Well define the group of color clearly and independency. • Then label each group independency.

Example Multi-Color labeling Group of color 1st Group consists of 1 2nd Group consists of 2 3rd Group consists of 3 แทน 0 แทน 1 แทน 2 แทน 3

1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Color Labeling 1st Group

1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 5 5 5 5 5 5 Color Labeling 2st Group 2 4

1 1 1 1 1 1 2 3 3 3 1 1 1 4 2 3 3 3 3 1 1 1 4 2 3 1 1 1 3 1 1 1 4 5 5 5 5 5 5 5 5 5 6 Color Labeling 3st Group

Exercise • If we want to label red and yellow is the same group. • How many label we can get?

Digital Image Fundamentals