Basic relations between pixels (Chapter 2)

Transcription

1 Basic relations between pixels (Chapter 2) Lecture 3 Basic Relationships Between Pixels Definitions: f(x,y): digital image Pixels: q, p (p,q f) A subset of pixels of f(x,y): S A typology of relations: Neighborhood Adjacency Connectivity Region & boundary Distance 1

2 Neighbors of a Pixel A pixel p at (x,y) has 2 horizontal and 2 vertical neighbors: (x+1,y), (x-1,y), (x,y+1), (x,y-1) This set of pixels is called the 4-neighbors of p: N 4 (p) p Neighbors of a Pixel The 4 diagonal neighbors of p are: (N D (p)) (x+1,y+1), (x+1,y-1), (x-1,y+1), (x-1,y-1) p N 4 (p) + N D (p) N 8 (p): the 8-neighbors of p 2

3 Adjacent pixels Two pixels are adjacent if: They are neighbors in some sense (e.g. N 4 (p), N 8 (p), ) Their gray levels satisfy a specified criterion of similarity V (e.g. equality, ) V is the set of gray-level values used to define adjacency (e.g. V={1} for adjacency of pixels of value 1) Adjacency (cont.) p is adjacent to q if: { } q N 4 ( p),n 8 ( p),... f ( p) V f ( q) V We consider three types of adjacency: 4-adjacency: two pixels p and q with values from V are 4- adjacent if q is in the set N 4 (p) 8-adjacency: two pixels p and q with values from V are 8- adjacent if q is in the set N 8 (p) 3

4 Adjacency (cont.) The third type of adjacency: m-adjacency: p and q with values from V are m-adjacent if: q is in N 4 (p) or q is in ND (p) and the set N 4 (p) N 4 (q) has no pixels with values from V Mixed adjacency is a modification of 8-adjacency and is used to eliminate the multiple path connections that often arise when 8- adjacency is used. Adjacency An example N 4 (p) N 4 (q) 4

5 Adjacency - An example N 4 (p) N 4 (q) Subset Adjacency Two image subsets S1 and S2 are adjacent if some pixel in S1 is adjacent to some pixel in S2. 5

6 Paths A path (curve) from pixel p with coordinates (x,y) to pixel q with coordinates (s,t) is a sequence of distinct pixels: Q : (x 0,y 0 ), (x 1,y 1 ),, (x n,y n ) where (x 0,y 0 )=(x,y), (x n,y n )=(s,t), and (x i,y i ) is adjacent to (x i-1,y i-1 ), for 1 i n n is the length of the path ( Q ). If (x 0, y 0 ) = (x n, y n ), then Q is a closed path 4-, 8-, and m-paths can be defined depending on the type of adjacency specified. Connectivity Connectivity between pixels is important for establishing boundaries of objects and components of regions in an image 6

7 Connectivity Two pixels (p S, q S) are 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 in S that are connected to p is a connected component of S. If S has only one connected component then S is called a connected set. Region & Boundary Let R be a subset of pixels: R is a region if R is a connected set. Its boundary (border, contour) is the set of pixels in R that have at least one neighbor not in R An edge can be the region boundary (in binary images) 7

8 Distance Measures For pixels p,q,z with coordinates (x,y), (s,t), (u,v), D is a distance function or metric if: D(p,q) 0 (D(p,q)=0 iff p=q) D(p,q) = D(q,p) and D(p,z) D(p,q) + D(q,z) Distance Measures Euclidean distance: D e (p,q) = [(x-s) 2 + (y-t) 2 ] 1/2 Points (pixels) having a distance less than or equal to r from (x,y) are contained in a disk of radius r centered at (x,y). 8

9 Distance Measures D 4 distance (city-block distance): D 4 (p,q) = x-s + y-t forms a diamond centered at (x,y) e.g. pixels with D 4 2 from p D 4 = 1 are the 4-neighbors of p Distance Measures D 8 distance (chessboard distance): D 8 (p,q) = max( x-s, y-t ) Forms a square centered at p e.g. pixels with D 8 2 from p D 8 = 1 are the 8-neighbors of p 9

10 Distance Measures D 4 and D 8 distances between p and q are independent of any paths that exist between the points because these distances involve only the coordinates of the points (regardless of whether a connected path exists between them). However, using m-adjacency the value of the distance (length of path) between two pixels depends on the values of the pixels along the path and those of their neighbors. Distance Measures e.g. assume p, p 2, p 4 = 1 or 1 p 1, p 3 = can have either 0 If only connectivity of pixels valued 1 is allowed, and p 1 and p 3 are 0, the m- distance between p and p 4 is 2. If either p 1 or p 3 is 1, the distance is 3. If both p 1 and p 3 are 1, the distance is 4 (p p1 p2 p3 p4) 10

11 Image Enhancements in the spatial domain: Intensity Transformations (chapter 3) (Lecture 3) Image statistics Consider an image f as a series of random samples, where each pixel, f(x,y), is a sample at (x,y). The samples can be described by: The mean value (a measure of the overall brightness): f = 1 MN f ( x, y) The standard deviation (a measure of the overall contrast): M N σ = 1 MN M N ( f ( x, y) f ) 2 Other summary statistics may also be used (min, max, median, ) 11

12 Image Enhancement Processing an image so that the result is more suitable than the original image for a specific application. Problem oriented May be subjective There is no general theory of image enhancement Two primary categories (may be combined): Spatial domain methods frequency domain methods Spatial Domain Methods Procedures that operate directly on the aggregate of pixels composing an image A neighborhood about (x,y) is defined by using a square (or rectangular) sub-image area centered at (x,y). 12

13 Spatial Domain Methods Mask processing or filtering: when the values of f in a predefined neighborhood of (x,y) determine the value of g at (x,y). Through the use of masks (or kernels, templates, or windows, or filters). Spatial Domain Methods When the neighborhood is 1 x 1 then g depends only on the value of f at (x,y) and T becomes a gray-level transformation (or mapping) function (point operation): s=t(r), T:Z + Z + r and s are the gray levels of f(x,y) and g(x,y) at (x,y), where: r denotes the pixel intensity before processing s denotes the pixel intensity after processing. Point processing techniques (e.g. contrast stretching, thresholding) 13

14 Contrast Stretching & Thresholding Contrast Stretching Thresholding Some Simple Intensity Transformations Image negatives Piecewise-Linear Transformation Functions: Contrast stretching Gray-level slicing Bit-plane slicing Implemented via Look-Up Tables (LUT) where values of T are stored in a 1-D array (for 8-bit, the LUT will have 256 values) 14

15 Some basic transformations Image Negatives Function reverses the order from black to white so that the intensity of the output image decreases as the intensity of the input increases. Used mainly in medical images and negative scanning. 15

16 Negative of an image The negative is obtained by using the following transformation function s=t(r): [0,L-1] the range of gray levels S= L-1-r Negative - example Aerial image The negative 16

17 Log Transformations s = c log(1+r) where r 0 and c is a constant Compresses the dynamic range of images with large variations in pixel values Log transform - example Dynamic range: 0 1.5x10 6 Dynamic range:

18 Power-Law Transformations C, γ : positive constants γ=c=1: Identity transformation Gamma correction Used to correct the output of devices that have a power-law response For example: CRT monitors Most computer displays have a intensity-to-voltage response curve which is roughly a 1.8 to 2.5 power function. Consequently, such displays will generate a darker image than the original. To correct this, we could use the inverse power function, for example: S=r 1/2.5 =r

19 Gamma Correction - example Gamma Correction - example Original image Brightness Gamma correction 19

20 Piecewise-Linear Transformation Functions - Contrast Stretching To increase the dynamic range of the gray levels in the image being processed. αr s = β r r 1 γ r r 2 0 r r ( ) + s 1 r 1 < r r 2 ( ) + s 2 r 2 < r L 1 20

21 Contrast Stretching The locations of (r 1,s 1 ) and (r 2,s 2 ) control the shape of the transformation function. If r 1 = s 1 and r 2 = s 2 the transformation is a linear function and produces no changes. If r 1 =r 2, s 1 =0 and s 2 =L-1, the transformation becomes a thresholding function that creates a binary image. Contrast Stretching More on function shapes: Intermediate values of (r 1,s 1 ) and (r 2,s 2 ) produce various degrees of spread in the gray levels of the output image, thus affecting its contrast. Generally, r 1 r 2 and s 1 s 2 is assumed. 21

22 Gray-Level Slicing To highlight a specific range of gray levels in an image (e.g. to enhance certain features). One way is to display a high value for all gray levels in the range of interest and a low value for all other gray levels (binary image). 22

23 Gray-Level Slicing The second approach is to brighten the desired range of gray levels but preserve the background and gray-level tonalities in the image: 23

24 Bit-Plane Slicing Analyzing the contribution made to the total image appearance by specific bits. i.e. Assuming that each pixel is represented by 8 bits, the image is composed of 8 1-bit planes. Plane 0 contains the least significant bit and plane 7 contains the most significant bit. Bit-plane slicing is useful for image compression Bit-Plane Slicing Usually, only the higher order bits (top four) contain visually significant data. The other bit planes contribute the more subtle details. Plane 7 corresponds exactly with an image thresholded at gray level

25 Bit-plane slicing An example f(x,y)

26 Bit-plane slicing: example Bit-plane slicing: example (cont.) 26

27 Bit 1 Bit 2 Bit 4 Bit 5 Bit 7 Bit 8 Bit 3 Bit 6 27

28 Bit-plane slicing: example 28