The Fast Fourier Transform (FFT) is a divide and conquer algorithm to recursively compute the DFT much quicker (with O (N.log2N) time complexity) than the much slower O (N2) naive computation for an n x n image. In Python, both the numpy and scipy libraries provide functions to compute 2D DFT/IDFT using the FFT algorithm. Let's see a few examples.
The Fast Fourier Transform algorithm to compute the DFT
by Sandipan Dey
Hands-On Image Processing with Python
- Title Page
- Copyright and Credits
- Dedication
- About Packt
- Contributors
- Preface
- Getting Started with Image Processing
- What is image processing and some applications
- The image processing pipeline
- Setting up different image processing libraries in Python
- Image I/O and display with Python
- Dealing with different image types and file formats and performing basic image manipulations
- Dealing with different image types and file formats
- Basic image manipulations
- Image manipulations with numpy array slicing 
- Image manipulations with PIL
- Cropping an image
- Resizing an image
- Negating an image
- Converting an image into grayscale
- Some gray-level transformations
- Some geometric transformations
- Changing pixel values of an image
- Drawing on an image
- Drawing text on an image
- Creating a thumbnail
- Computing the basic statistics of an image
- Plotting the histograms of pixel values for the RGB channels of an image
- Separating the RGB channels of an image 
- Combining multiple channels of an image
- α-blending two images
- Superimposing two images
- Adding two images
- Computing the difference between two images
- Subtracting two images and superimposing two image negatives
- Image manipulations with scikit-image
- Image manipulation with Matplotlib
- Image manipulation with the scipy.misc and scipy.ndimage modules
- Summary
- Questions
- Further reading
- Sampling, Fourier Transform, and Convolution
- Convolution and Frequency Domain Filtering
- Image Enhancement
- Image Enhancement Using Derivatives
- Image derivatives – Gradient and Laplacian
- Sharpening and unsharp masking
- Edge detection using derivatives and filters (Sobel, Canny, and so on)
- With gradient magnitude computed using the partial derivatives
- Sobel edge detector with scikit-image
- Different edge detectors with scikit-image – Prewitt, Roberts, Sobel, Scharr, and Laplace
- The Canny edge detector with scikit-image
- The LoG and DoG filters
- Finding and enhancing edges with PIL
- Image pyramids (Gaussian and Laplacian) – blending images
- Summary
- Questions
- Further reading
- Morphological Image Processing
- Extracting Image Features and Descriptors
- Feature detectors versus descriptors
- Harris Corner Detector
- Blob detectors with LoG, DoG, and DoH
- Histogram of Oriented Gradients
- Scale-invariant feature transform
- Haar-like features
- Summary
- Questions
- Further reading
- Image Segmentation
- What is image segmentation?
- Hough transform – detecting lines and circles
- Thresholding and Otsu's segmentation
- Edges-based/region-based segmentation
- Felzenszwalb, SLIC, QuickShift, and Compact Watershed algorithms 
- Active contours, morphological snakes, and GrabCut algorithms
- Summary
- Questions
- Further reading
- Classical Machine Learning Methods in Image Processing
- Supervised versus unsupervised learning
- Unsupervised machine learning – clustering, PCA, and eigenfaces
- Supervised machine learning – image classification
- Supervised machine learning – object detection
- Summary
- Questions
- Further reading
- Deep Learning in Image Processing - Image Classification
- Deep Learning in Image Processing - Object Detection, and more
- Additional Problems in Image Processing
- Other Books You May Enjoy
The Fast Fourier Transform algorithm to compute the DFT
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