![Distance measures
If we have 3 pixels: p,q,z:
p with (x,y)
q with (s,t)
z with (v,w)
Then:
D(p,q) = 0 iff p = q
D(p,q) = D(q,p)
D(p,z) ≤ D(p,q) + D(q,z)
Euclidean distance between p and q: De(p,q) = [(x-s)2 + (y-t)2]1/2
D4 distance: D4(p,q) = |x-s| + |y-t|
D8 distance: D8(p,q) = max (|x-s| , |y-t|)
D4 and D8 distances between p and q are independent of any paths
that might exist between the points.
For m-adjacency, Dm distance between two points is defined as the
shortest m-path between the points.
Ch2, lesson3: Some basic Relationships between pixels](https://image.slidesharecdn.com/imageprocessing-ch2part2-240202142530-a1f8bbdf/85/Image_Processing-ch2surface-r_part_2-ppt-11-320.jpg)
Image_Processing-ch2surface r_part_2.ppt
- 1. Image Processing Ch2: Digital image Fundamentals Part 2 Prepared by: Tahani Khatib
- 2. Zooming (over sampling) images Ch2, lesson2: Zooming and shrinking Zoomed by using nearest neighbor Zoomed by using Bilinear
- 3. Zooming (over sampling) images Ch2, lesson2: Zooming and shrinking Zooming requires 2 steps: The creation of new pixel locations. The assignment of gray levels to these new locations. Two techniques for zooming: 1. Nearest neighbor interpolation 2. Bilinear interpolation
- 4. Nearest neighbor interpolation Ch2, lesson2: Zooming and shrinking Example: Suppose A 2x2 pixels image will be enlarged 2 times by the nearest neighbor method: 1. Lay an imaginary 4*4 grid over the original image.. 2. For any point in the overlay, look for the closest pixel in the original image, and assign its gray level to the new pixel in the grid. (copy) 3. When all the new pixels are assigned values, expand the overlay grid to the original specified size to obtain the zoomed image. • Pixel replication (re sampling) is a special case that is applicable when the size of the image needs to be increased an integer number of times (like 2 times not 1.5 for example). + ve : Nearest neighbor is fast -ve: it produces a checkerboard effect like this!
- 5. shrinking Similar to image zooming. Shrinking an image an integer number of times Pixel replication is replaced by row&column deletion. Shrinking an image by a non-integer factor Expand the grid to fit over the original image. Do gray-level interpolation (nearest neighbor or bilinear). Shrink the grid back to its original specified size. Ch2, lesson2: Zooming and shrinking
- 6. Neighbors of a pixel Ch2, lesson3: Some basic Relationships between pixels
- 7. Adjacency Ch2, lesson3: Some basic Relationships between pixels V: set of gray level values (L), (V is a subset of L.) 3 types of adjacency 4- adjacency: 2 pixels p and q with values from V are 4- adjacent if q is in the set N4(p) 8- adjacency: 2 pixels p and q with values from V are 8- adjacent if q is in the set N8(p) m- adjacency: 2 pixels p and q with values from V are madjacent if 1. q is in N4(p), or 2. q is in ND(p) and the set N4(p) ∩ N4(q) has no pixels whose values are from V
- 8. connectivity A digital path from pixel p with coordinates (x,y) to pixel q with coordinates (s,t) is a sequence of distinct pixels with coordinates (x0,y0), (x1,y1), …, (xn,yn), where (x0,y0)= (x,y) and (xn,yn)=(s,t), and pixels (xi,yi) and (xi-1,yi-1) are adjacent for 1 ≤ i ≤ n. S: a subset of pixels in an image. Two pixels p and q are said to be connected in S if there exists a path between them consisting entirely of pixels in S. For any pixel p in S, the set of pixels that are connected to it in S is called a connected component of S. If S has only one connected component, it is called a connected set. Ch2, lesson3: Some basic Relationships between pixels
- 9. Regions and boundaries R: a subset of pixels in an image. R is a region of the image if R is a connected set. The boundary of a region R is the set of pixels in the region that have one or more neighbors that are not in R. Ch2, lesson3: Some basic Relationships between pixels
- 10. Foreground and background Suppose that the image contains K disjoint regions Rk none of which touches the image border . Ru : the union of all regions . (Ru)c : is the complement . so Ru is called foreground , and (Ru)c : is the background .
- 11. Distance measures If we have 3 pixels: p,q,z: p with (x,y) q with (s,t) z with (v,w) Then: D(p,q) = 0 iff p = q D(p,q) = D(q,p) D(p,z) ≤ D(p,q) + D(q,z) Euclidean distance between p and q: De(p,q) = [(x-s)2 + (y-t)2]1/2 D4 distance: D4(p,q) = |x-s| + |y-t| D8 distance: D8(p,q) = max (|x-s| , |y-t|) D4 and D8 distances between p and q are independent of any paths that might exist between the points. For m-adjacency, Dm distance between two points is defined as the shortest m-path between the points. Ch2, lesson3: Some basic Relationships between pixels
- 13. Example q p 1 2 3 1 2 3 Compute the distance between the two pixels using the three distances : q:(1,1) P: (2,2) Euclidian distance : ((1-2)2+(1-2)2)1/2 = sqrt(2). D4(City Block distance): |1-2| +|1-2| =2 D8(chessboard distance ) : max(|1-2|,|1-2|)= 1 (because it is one of the 8-neighbors )
- 14. Distance measures Example : Use the city block distance to prove 4- neighbors ? Pixel A : | 2-2| + |1-2| = 1 Pixel B: | 3-2|+|2-2|= 1 Pixel C: |2-2|+|2-3| =1 Pixel D: |1-2| + |2-2| = 1 Now as a homework try the chessboard distance to proof the 8- neighbors!!!! d c p a b 1 2 3 1 2 3