Detailed Notes on Histogram Processing

Histogram Processing

Overview

• Histogram Processing: A technique used in digital image processing for manipulating the histogram of an image to enhance its visual quality.

• Topics Covered: Histogram equalization, histogram matching, local histogram processing, and using histogram statistics for image enhancement.

Histograms

• Definition: A graphical representation of the distribution of pixel intensities in a digital image.

• Function: Plots the number of pixels for each intensity value.

• Application: Used in digital cameras to avoid over-exposed pictures.

• Usage: Photographers use histograms to analyze the distribution of captured intensities.

Histogram Axes

• Horizontal Axis: Represents intensity variations (gray levels).

• Vertical Axis: Represents the number of pixels for each intensity value.

Intensity Representation

• Left Side: Black and dark areas.

• Middle: Medium grey color.

Histogram Example

• Image: A sample image with 256 distinct gray levels (8 bits).

• Histogram: Shows the frequency of each gray level.

Statistical Measures

• Count: 1,920,000 pixels.

• Mean: 118.848.

• Standard Deviation: 59.179.

• Minimum: 0.

• Maximum: 251.

• Mode: 184 (occurs 30,513 times).

Histogram Properties

• Statistical Information: Histograms provide statistical information about the image.

• Location Insensitivity: Histograms do not indicate the location of pixels with specific intensity values.

• Example: In an image with $L=16$ gray levels, 10 pixels have an intensity value of 2.

Identical Histograms

• Multiple Images: Different images can have the same histogram.

• Example: Images with half of the pixels gray and half white will have the same histogram, regardless of pixel arrangement.

• Implication: Same histogram implies same statistics but potentially different intensity distributions.

Histogram Definition

• Range: For a grayscale image with intensity values in the range $I(x, y) = [0, L-1]$, the histogram contains $L$ entries.

• Example: For an 8-bit grayscale image, $L = 2^8 = 256$.

• Entry Definition: Each histogram entry is defined as:
  h(i) = \text{number of pixels with intensity } i \text{ for all } 0 <= i < L
  Example: $h(255) = \text{number of pixels with intensity } = 255$

Image Contrast

• Definition: Indicates how easily objects in an image can be distinguished.

• High Contrast: Image with many distinct intensity values.

• Low Contrast: Image that uses few intensity values.

Histogram and Contrast

• Low Contrast Image:Appears dull with most pixels concentrated in a narrow range of gray levels.

• Normal Contrast Image:Utilizes a broader range of intensity values.

• High Contrast Image: Spreads intensity values across the entire spectrum.

Histogram Analysis

• More Black Pixels: Histogram is concentrated on the left side.

• More White Pixels: Histogram is concentrated on the right side.

• Pixels of All Intensity Levels: Histogram is evenly distributed.

Histogram Processing

• Discrete Function: The histogram of a digital image $f$ (with intensities $[0, L-1]$) is a discrete function.

  $h(r<em>k) = n</em>k$

  where $r<em>k$ is the kth intensity value and $n</em>k$ is the number of pixels in $f$ with intensity $r_k$.

• Normalization: Normalizing the histogram is a common practice.

• Normalization Formula: Divide the components by the total number of pixels in the image.

  $p(r<em>k) = \frac{n</em>k}{MN} \text{ for } k = 0, 1, 2, …, L-1$

  where $M$ and $N$ are the dimensions of the image.

• Probability Estimation: $p(r_k)$ estimates the probability of occurrence of intensity level $r$ in an image.

  $\sum p(r_k) = 1$

Visual Examples of Histograms

• Dark Image: Histogram is concentrated on the lower end of the gray scale.

• Light Image: Histogram is concentrated on the higher end of the gray scale.

• Low-Contrast Image: Histogram is narrow and concentrated.

• High-Contrast Image: Histogram covers a broad range of gray levels.

Histogram Equalization

• Definition: A process for increasing the contrast in an image by spreading the histogram out to be approximately uniformly distributed.

• Effect: Gray levels of an image subjected to histogram equalization are spread out and tend to reach white.

• Benefit: Increase of dynamic range produces an increase in contrast.

• Drawback: For images with low contrast, histogram equalization can increase visual graininess.

Histogram Equalization Steps

1. Count Pixel Intensities ($n_k$):

   • Find the count of each pixel intensity value.

2. Normalize Values (PDF):

   • Calculate the Probability Density Function (PDF) by dividing each count by the total number of pixels.

     $P<em>k = \frac{n</em>k}{\text{sum}}$

3. Find Running Sum (CDF):

   • Compute the running sum of the PDF values, resulting in the Cumulative Distribution Function (CDF), denoted as $S_k$.

4. Scale and Round Off:

   • Multiply the CDF values by the maximum gray level value and round off to the nearest integer.

5. Map Gray Levels:

   • Map the original gray level values to the results from step (4) using a one-to-one correspondence.

Histogram Modification

• Examination of the histogram is one of the most useful tools for image enhancement.

• Histogram modification makes it easy to modify images to improve them.

• Mapping functions can stretch, shrink (compress), or slide histograms.

• Histogram stretching and Histogram shrinking are forms of gray scale modification referred to as histogram scaling.

Histogram Stretch

• Mapping Function Equation:

  $I<em>{\text{out}}(r, c) = \frac{I</em>{\text{in}}(r, c) - I<em>{\text{MIN}}}{I</em>{\text{MAX}} - I_{\text{MIN}}} \cdot (\text{MAX} - \text{MIN}) + \text{MIN}$

  • $I(r,c)_{\text{MAX}}$ is the largest gray level value in the image $I(r,c)$
  • $I(r,c)_{\text{MIN}}$ is the smallest gray level value in $I(r,c)$
  • $\text{MAX}$ and $\text{MIN}$ correspond to the maximum and minimum gray level values possible (for an 8-bit image these are 0 and 255).

• Effect: This equation stretches the histogram across the entire gray level range, increasing the contrast of a low contrast image.

• Clipping: To allow a small percentage of pixel values to be clipped at the low and high end of the range is useful, this involves truncating values at 0 and 255 for an 8-bit image.

Histogram Shrink

• Mapping Function Equation:

  $I<em>{\text{out}}(r, c) = \frac{I</em>{\text{in}}(r, c) - I<em>{\text{MIN}}}{I</em>{\text{MAX}} - I_{\text{MIN}}} \cdot (\text{ShrinkMAX} - \text{ShrinkMIN}) + \text{ShrinkMIN}$

  • $I(r,c)_{\text{MAX}}$ is the largest gray level value in the image $I(r,c)$
  • $I(r,c)_{\text{MIN}}$ is the smallest gray level value in $I(r,c)$
  • $\text{ShrinkMAX}$ and $\text{ShrinkMIN}$ correspond to the maximum and minimum desired in the compressed histogram.

• Effect: Decreases image contrast by compressing the gray levels.

• Usefulness: It may not be useful as an image enhancement tool by itself, but it is used in an image sharpening algorithm (unsharp masking) as a part of an enhancement technique.

Histogram Slide

• Purpose: Used to make an image either darker or lighter while retaining the relationship between gray level values.

• Method: Accomplished by adding or subtracting a fixed number from all gray level values.

• Equation:

  $I<em>{\text{out}}(r, c) = I</em>{\text{in}}(r, c) + \text{OFFSET}$

  • The OFFSET value is the amount to slide the histogram.

  • Any values slid past the minimum and maximum values will be clipped to the respective minimum or maximum.

• Positive OFFSET: Increases overall brightness.

• Negative OFFSET: Creates a darker image.

Histogram Specification/Histogram Matching

• Definition: A method to transform the intensity distribution of an image into a desired or a specified histogram.

• Main Aim: To improve the visual quality of an image by changing its contrast, brightness, or color.

• Special Case: Well-known histogram equalization method is a special case in which the specified histogram is uniformly distributed.

Applications of Histograms

• Histograms are used for simple calculations in software and to analyze an image.

• Properties of an image can be predicted by the detailed study of the histogram.

• Brightness and contrast of the image can be adjusted.

• Used for image equalization to produce a high contrast image.

• Histograms are used in thresholding as it improves the appearance of the image.

• If we have input and output histogram of an image, we can determine which type of transformation is applied in the algorithm.

• Histogram-based techniques are a popular tool for real-time image processing.

Uses of Histogram Processing

• Image enhancements
• Image statistics
• Image compression
• Image segmentation
• Simple to calculate and economic hardware implementations
• Popular tool in real-time image processing
• Provides a global description of the appearance of the image and gives useful information for contrast enhancement.

Types of basic image and corresponding histograms

• Dark
• Light
• Low contrast
• High contrast

Histograms are commonly viewed in plots as

• $h(r) = n \text{ versus } r_k$
• $p(r) = \frac{n<em>k}{MN} \text{ versus } r</em>k$