# Basic Relationships Between Pixels

**Advanced Image Processing**

**Basic Relationships Between Pixels**

- Neighborhood
- Adjacency
- Paths
- Connectivity
- Regions
- Boundaries

**Neighbors of a pixel – N****4****(p)**

- Any pixel p
*(x, y)*has two vertical and two horizontal neighbors, given by

*(x+1, y),*

*(x-1, y),*

*(x, y+1),*

*(x, y-1)*

- This set of pixels are called the 4-neighbors of P, and is denoted by N4(P).

x , y+1 | ||

x-1 , y | x,y |
x+1 , y |

x , y-1 |

**Neighbors of a pixel – N****D****(p)**

- Any pixel p
*(x, y)*has four diagonal neighbors, given by

*(x+1, y+1), (x+1, y-1), (x-1, y+1), (x-1 ,y-1)*

- This set is denoted by ND(p).

x-1 , y+1 | x+1, y+1 | |

x,y |
||

x-1, y-1 | x+1,y-1 |

**Neighbors of a pixel – N****8****(p)**

- ND(p) and N4(p) are together known as 8-Neighbors and are denoted by N8(p)
- ND(p) U N4(p) = N8(p)
*What about when p(**x,y**) is a border pixel of the image ?*

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 |

**Adjacency **

- Let V be the set of intensity values used to define adjacency
- For binary images à V = {1}
- A particular grayscale image à V = {1,3,5,…,251,253,255}
**4-adjacency:**Two pixelsand*p*with values from*q***V**are 4-adjacent if**q**is in the set**N****4****(***p***)**.**8-adjacency:**Two pixelsand*p*with values from*q***V**are 8-adjacent if**q**is in the set**N****8****(***p***)**.**m-adjacency:**Two pixels*p*and*q*with values from V are m-adjacent if,

**q is in N****4****(p)**

OR

**q is in N****D****(p)** AND **N****4****(p)**∩**N****4****(q) **has no pixels whose values are from V

**Path**

- set of pixels lying in some adjacency definition
- 4-adjacency à 4-path
- 8-adjacency à 8-path
- m-adjacency à m-path
- path length ?
- Number of pixels involved

**Connectivity**

- Let Sà subset of pixels in an image
- Two pixels
**p**and**q**are said to be connected in**S**if there exist 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 connected component of**S**. - If S has only one connected component, then it is called connected set.

**Region**

- A connected set is also called a Region.
- Two regions (let Ri and Rj) are said to be adjacent if their union forms a connected set. Adjacent Regions or joint regions
- Regions that are not adjacent are said to be disjoint regions.
- 4- and 8-adjacency is considered when referring to regions (author)
- Discussing a particular region, type of adjacency must be specified.
- Fig2.25d the two regions are adjacent only if 8-adjacency is considered

**Foreground and Background**

- Suppose an image contain K disjoint regions Rk , k=1,2,3,…K, none of which touches the image border
- Let Ru denote the union of all the K regions.
- Let (Ru)c denote its compliment.
- We call all the points in Ru the foreground and all the points in (Ru)c the background

**Boundary**

**The boundary (border or contour) of a region R is the set of points that are adjacent to the points in the complement of R.****Set of pixels in the region that have at least one background neighbor.****The boundary of the region R is the set of pixels in the region that have one or more neighbors that are not in R.**- Inner Border: Border of Foreground
- Outer Border: Border of Background
- If R happens to be entire Image?
- There is a difference between boundary and edge in Digital Image Paradigm. The author refers this discussion to chapter 10.

**Distance Measures**

- Euclidean Distance:
*D**e**(p, q) = [(x-s)**2**+ (y-t)**2**]**1/2* - City Block Distance:
*D**4**(p, q) = |x-s| + |y-t|* - Chess Board Distance:
*D**8**(p, q) = max(|x-s|, |y-t|)*

**Sample Problem from exercise**

**Histogram Representation**

- Histograms plots how many times (frequency) each intensity value in image occurs
- Image below (left) has 256 distinct gray levels (8 bits)
- Histogram (right) shows frequency (how many times) each gray level occurs

- E.g.
*K*= 16, 10 pixels have intensity value = 2 - Only statistical information
- No indication of
**location**of pixels

**Rough guess about the histogram of these images ?**

**Histogram Representation**

- Different images can have
**same**histogram - 3 images below have same histogram
- Half of pixels are gray, half are white
- Same histogram = Same statistics
- Distribution of intensities could be different

**Histogram Representation**

- Many cameras display real time histograms of scene
- Helps taking pictures according to your requirement
- Also easier to detect types of processing applied to image

**?**

- Can we reconstruct image from histogram ?

**Histogram**

- Histograms help detect image acquisition issues
- Histogram representation of an image can be useful in following characteristics of an image.
- Exposure:
**amount of light per unit area**reaching the image sensor - Brightness:
**average intensity**of all pixels in image - Contrast:
**difference of foreground**and**background**(objects distinction) - Dynamic Range: Number of
**distinct pixels**in image - Artifacts: Image
**alteration**after it is being captured

**Histogram Representation**

Four basic image types: dark, light, low contrast, high contrast and their corresponding histograms.

**Histogram Representation**

**Histogram Representation**

**Histogram Representation**

**Histogram Representation**

**Contrast**

**Good Contrast?**- Widely spread intensity values
- Large difference between min and max intensity values

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