Role of Mathematics in Image Processing
Most commonly used Image Processing tasks include; Image Restoration, Image Segmentation, Image Enhancement, De-Blurring, De-noising, etc Mathematics is quite inherent and deeply connected to various core Image Processing tasks: just like, De-noising, De-blurring, Enhancement, Segmentation and Edge Detection etc. The study of such Image Processing tasks provides a unique opportunity of incorporating Mathematical tools and techniques to address several of the Image Processing applications in various scientific fields of study.
1) Histogram Equalization
The operation of histogram equalization is used to reassign grey values in the image, such that the histogram of the resultant image is (theoretically) flat. I say theoretically because practically, image intensities are discrete, and it may not be possible to reassign values equally across greyscale. Practically speaking, this also means that the contrast is enhanced in the image, which is most of the times, a good thing, depending on your application. By contrast enhancement, we mean that intermediate greyscale intensities that were not well represented in the image earlier, are now distributed better, and the intensities that occurred with very high frequency are de-emphasized. Mostly the histogram isn't perfectly flat and perfectly distributed after equalization, and the mean will be close to the middle grey level. Closer to the middle grey level after histogram equalization than it was earlier.
2) Spatial Linear Filters
There are two classes of spatial linear filters: (i) smoothing linear spatial filters (ii) sharpening linear spatial filters
Smoothing linear spatial filters Smoothing is known to cause local averaging (or blurring), which is similar to spatial summation or spatial integration. During the smoothing process, small details and noise get lost, but sharp edges become blurry. Sharpening and smoothing operations result in opposite effects. Using a smoothing, a blurry image f is made sharper, where the edge of the image f would get enhanced
Sharpening linear spatial filters Sharpening linear filters for image enhancement uses the Laplacian technique. It is assumed that image function f has second-order partial derivatives.
3)Discrete Cosine Transform (DCT)
This technique is used for performing image compression step 1 find out DCT matrix. For first row formula is given by 1/sqrt(n) For next rows of matrix sqrt(2/n)cos[(2v+1)PIu/2n] u and v are indices of rows and matrix. For more information on DCT compression technique refer the video link given in reference Step 2 multiply input image with DCTmatrix step3 resultant will be a matrix of the compressed image. Plot the matrix using matplotlib imshow function. for practical implementation please refer code
Original Image
Image after multiplication with DCT (compressed image)
About DCT -
With the advent of the web and industry-wide digital transformation, the JPEG (Joint Photographic Experts Group) image format has emerged as a most commonly used format for sharing images with lossy compression over the web. Discrete Cosine Transform (DCT) is one of the powerful techniques in the domain of signal processing.
DCT performs execution using a mathematical operation technique known as Fast Fourier Transform (FFT). FFT takes one signal as an input and then transforms the representation from one type to another. In this transformation process, a signal from the spatial domain gets transformed to the frequency domain. During this process, information redundancy is kept minimum because of kernel functions (cosines) comprise an orthogonal basis. This technique was first developed by J.W. Cooley and J.W. Tukey in 1965. FFT found wide acceptance since it helped in reducing the computation time and increased the ease of performing the digital Fourier analysis much more feasible.
4) Laplacian Distribution
The Laplacian distribution is one of the most commonly used approaches used for performing image analysis. Since the Laplacian approach uses invariant Gaussian kernels, there are known issues with regard to the inability to represent the edges of images suitably. Hence, this approach is not recommended for edge centric operations like preservation of image edges, image, smoothing and tone mapping.
Laplacian operator is also known as a derivative operator which is used to find edges in an image. The major difference between Laplacian and other operators like Prewitt, Sobel, Robinson and Kirsch is that these all are first-order derivative masks but Laplacian is a second-order derivative mask. In this mask we have two further classifications one is Positive Laplacian Operator and other is Negative Laplacian Operator. Another difference between Laplacian and other operators is that unlike other operators Laplacian didn’t take out edges in any particular direction but it takes out edges in the following classification. • Inward Edges • Outward Edges
Original Image Histogram
Laplace Image Histogram
Reference:
Research Paper Name :Role of Mathematics in Image Processing
Other Links https://theailearner.com/2019/04/21/spatial-filtering/
https://docs.opencv.org/master/d4/d13/tutorial_py_filtering.html
https://medium.com/@almutawakel.ali/opencv-filters-arithmetic-operations-2f4ff236d6aa
https://www.youtube.com/watch?v=tW3Hc0Wrgl0
Github Code Link
https://github.com/Krishna5996/Role-of-Mathematics-in-Image-Processing