Digital Image Processing Fundamentals

Digital Image Processing Fundamentals

• Digital image processing is covered in lecture 4 by Dr. Imran Usman.

Representing Images

• An image can be plotted as a surface.
• An image can be displayed as a visual intensity array.
• An image is shown as a 2-D numerical array.
• In the array, 0 represents black, 0.5 represents gray, and 1 represents white.

Neighbors of a Pixel

• A pixel p at coordinates (x, y) has four horizontal and vertical neighbors.
• The coordinates of these neighbors are given by (x + 1, y), (x - 1, y), (x, y + 1), and (x, y - 1).
• The set of these four pixels is called the 4-neighbors of p, denoted by N₄(p).
• Each pixel in N₄(p) is a unit distance from (x, y).
• Some neighbors of p may lie outside the digital image if (x, y) is on the border of the image.
• The four diagonal neighbors of p have coordinates (x + 1, y + 1), (x + 1, y - 1), (x - 1, y + 1), and (x - 1, y - 1).
• The four diagonal neighbors are denoted by ND(p).
• These points, together with the 4-neighbors, are called the 8-neighbors of p, denoted by N₈(p).
• Some points in ND(p) and N₈(p) may fall outside the image if (x, y) is on the border.

Adjacency

• V is the set of intensity values to define adjacency.
• In a binary image, V = {1} if referring to adjacency of pixels with value = 1.
• In grayscale images, V could be any subset of 256 values.
• Adjacencies can be one of three types: 4-adjacency, 8-adjacency, or m-adjacency (mixed adjacency).
• Two pixels p and q with values from V are 4-adjacent if q is in the set N₄(p).
• Two pixels p and q with values from V are 8-adjacent if q is in the set N₈(p).
• Two pixels p and q with values from V are m-adjacent if q is in N₄(p), or q is in ND(p) and set N₄(p) ∩ N₄(q) has no pixels whose values are from V.

Connectivity Between Pixels

• An important concept used in establishing boundaries of objects and components of regions.
• Two pixels p and q are connected if they are adjacent in some sense or if their gray levels satisfy a specified criterion of similarity.
• V is the set of gray level values used to define the criterion of similarity.
• 4-connectivity exists if gray-level p, q ∈ V, and q ∈ N4(p).
• 8-connectivity exists if gray-level p, q ∈ V, and q ∈ N8(p).
• m-connectivity (mixed connectivity) exists if gray-level p, q ∈ V, and q satisfies one of the following: q ∈ N4(p), or q ∈ ND(p) and N4(p) ∩ N4(q) has no values from V.
• Mixed connectivity can eliminate the multiple path connections that often occur in 8-connectivity.

Neighbors of a Pixel: Binary Image Example

• A binary image example with two connected components is based on 4-connectivity.
• If the connectivity were based on 8-neighbors, the two connected components would merge into one.

Path

• Coordinates of pixel p are (x, y) and of pixel q are (s, t).
• A path from p to q is a sequence of distinct pixels with coordinates (x₀, y₀), (x₁, y₁), ..., (xₙ, yₙ).
  • (x₀, y₀) = (x, y) & (xₙ, yₙ) = (s, t)
  • (xᵢ, yᵢ) is adjacent to (xᵢ-₁, yᵢ-₁) where 1 ≤ i ≤ n.
• A set of pixels in an image where all component pixels are connected.
• A boundary of a region is a set of pixels of a region R that have one or more neighbors that are not in R.
• The type of adjacency specifies 4-, 8-, or m-paths.

Connected Set

• S represents a subset of pixels in an image.
• Two pixels p and q are said to be connected in S if a path exists between them consisting entirely of pixels in S.
• For any pixel p in S, the set of pixels connected to it in S is called a connected component of S.
• If S has only one connected component, then S is called a connected set.

Region

• R is a subset of pixels in an image, defined as a region if it is a connected set.
• Two regions Ri and Rj are adjacent if their union forms a connected set.
• Regions that are not adjacent are said to be disjoint.
• When referring to regions, 4- and 8-adjacency are considered.

Foreground and Background

• An image contains K disjoint regions, Rk, where k = 1, 2, 3, ..., K, none of which touch the image border.
• Ru denotes the union of all the K regions, and Ruᶜ denotes its complement.
• All the points in Ru are called the foreground, while all points in Ruᶜ are called the background of the image.

Boundary

• The boundary (border or contour) of a region R is the set of points adjacent to points in the complement of R.
• The border of a region is the set of pixels in the region that have at least one background neighbor.
• The connectivity being used to define adjacency must be specified.

Distance Measures

• For pixels p, q, and z, with coordinates (x, y), (s, t), and (v, w) the following are true regarding D as a distance function or metric: D(p, q) ≥ 0, D(p, q) = 0 if and only if p = q, D(p, q) = D(q, p), and D(p, z) ≤ D(p, q) + D(q, z).

Euclidean Distance

• The Euclidean distance between p and q is defined as De(p, q) = √((x - s)² + (y - t)²).
• Pixels having a distance less than or equal to some value r from (x, y) are in a disk of radius r centered at (x, y)

City-Block Distance

• (D4 distance) between p and q is defined as D4(p, q) = |x - s| + |y - t|.
• Pixels having a D4 distance from (x, y) less than or equal to some value r form a diamond centered at (x, y).
• Pixels with D4 distance ≤ 2 from (x, y) form contours of constant distance.
• Pixels with D4 = 1 are the 4-neighbors of (x, y)

Chessboard Distance

• (D8 distance) between p and q is defined as D8(p, q) = max(|x - s|, |y - t|).
• Pixels with D8 distance from (x, y) less than or equal to some value r form a square centered at (x, y).
• Pixels with D8 distance ≤ 2 from (x, y) form contours of constant distance.
• Pixels with D8 = 1 are the 8-neighbors of (x, y).
• The D4 and D8 distances between p and q are independent of any paths because these distances involve only the coordinates of the points.
• The Dm distance between two points is defined as the shortest m-path between the points when considering m-adjacency.
• The distance between two pixels depends on the values of the pixels along the path and their neighbors.

Quiz Questions

• Differentiate between image enhancement and image restoration, providing suitable everyday applications.
• Outline the basic procedure for ultrasound image formation.
• Explain and highlight key application areas for reflection imaging, absorption imaging, and emission imaging encompassing the EM Spectrum.
• Describe brightness discrimination about Weber ratio and why a large value of ΔI/I means poor brightness discrimination.
• Refer to the whiteboard for question 5.