Image processing

Theory

A linear transformation maps input vectors to output vectors while preserving the structure of the vector space. Key properties of a linear transformation T include additivity, which states that T(u+v)=T(u)+T(v) and homogeneity, which asserts that T(cu)=cT(u) for a scalar c. These properties ensure the preservation of vector operations during transformations.

Description

A linear transformation T:Rn→Rm can be expressed through a matrix Aof size m×nm times nm×n. For any input vector{R}^x∈Rn, the transformation yields an output vector y represented by the equation y=T(x). Here, A is the matrix associated with the transformation, highlighting how linear transformations can efficiently map vectors from one space to another. 

Theory

The negative of an image is achieved by replacing the intensity ‘i’ in the original image by ‘i-1’, i.e. the darkest pixels will become the brightest and the brightest pixels will become the darkest. Image negative is produced by subtracting each pixel from the maximum intensity value. 

Description 

In image processing, a negative image is generated by inverting the intensity of each pixel if we assume  8-bit grayscale image, the max intensity value is 255, thus each pixel is subtracted from 255 to produce the output image.
The transformation function used in image negative is :

s = T(r) = (L – 1) – r

Where L - 1 is the max intensity value, s is the output pixel value and r is the input pixel value

Theory

Contrast stretching enhances image visibility by remapping the original pixel values to a new range. The minimum intensity in the original image is assigned to the lowest possible value (typically 0), while the maximum intensity is mapped to the highest possible value.

Description

Contrast stretching, also known as normalization, is an image enhancement technique used to expand the range of pixel intensity values, improving an image contrast. By remapping the lowest and highest intensity values in the original image to new values, typically from 0 to 255 contrast stretching enhances visibility of details, making darker regions appear darker and brighter regions appear brighter. This method is commonly applied in fields like medical imaging and remote sensing, where it helps highlight features that are otherwise hard to discern in their original form. 

Theory

Grayscale slicing, also known as intensity-level slicing, is an image processing technique used to highlight specific ranges of grayscale intensities within an image. By isolating and enhancing certain pixel intensity ranges, grayscale slicing makes certain features in the image more visible while suppressing others. This technique is particularly useful in applications like medical imaging, remote sensing, and industrial inspection.

Description

In grayscale slicing, a range of intensity values in an image is chosen, and all the pixels within this range are either enhanced or assigned a specific intensity value. There are two primary approaches to grayscale slicing:

1. Binary Slicing: In this approach, all pixel values within a specified range are assigned one value (e.g., white), while all other pixel values are assigned another value (e.g., black). This creates a binary image, where only the pixels of interest are visible.

2. Slicing with Background Preservation: Here, pixel values within the specified range are enhanced, while the remaining pixel values are left unchanged. This allows the features within the desired intensity range to stand out without completely removing the background information.

Theory

A histogram in image processing is a graphical representation that shows the distribution of pixel intensity values in an image. For a grayscale image, the histogram represents the number of pixels for each intensity level, ranging from 0 (black) to 255 (white) in an 8-bit image. For color images, histograms are typically computed for each color channel (Red, Green, and Blue), showing the distribution of intensities for each color. 

Description

A histogram provides a frequency distribution of intensity values in an image. It shows how often each intensity level appears in the image. The x-axis of the histogram corresponds to the intensity levels (0 to 255 for an 8-bit image), while the y-axis corresponds to the number of pixels that have each intensity value.

Key Components of a Histogram:

1. Intensity Levels: The range of values representing pixel brightness (0 for black, 255 for white in grayscale images).

2. Frequency: The number of pixels corresponding to each intensity level.

Theory

Histogram equalization is a technique in image processing used to improve the contrast of an image by redistributing the pixel intensity values. It transforms the intensity values of an image to make the histogram more uniform, enhancing the visual quality by spreading out the pixel intensities across the full range of values. 

Before equalization: An image may appear too bright, dark, or washed out, with poor contrast. The corresponding histogram will reflect this, showing a concentration of pixel intensities in a specific range. After applying histogram equalization.

Description

Before Histogram Equalization: In images with low contrast, the pixel intensities tend to cluster in a limited range (e.g., concentrated around darker or brighter values). 

After Histogram Equalization: Histogram equalization enhances the contrast of the image by spreading the intensity values more evenly across the full intensity range (0 to 255 for an 8-bit grayscale image). This results in a more balanced histogram.