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The document discusses the Fourier transform and its applications in image processing. It begins with an introduction to the Fourier transform and its inventor. It then explains that the discrete Fourier transform (DFT) decomposes an image into sine and cosine components, representing the image in the frequency domain. The document provides details on how the DFT works, including using a fast Fourier transform to improve efficiency. It also describes how the Fourier transform output contains magnitude and phase information and discusses various applications of the Fourier transform in fields like signal and image processing.
Nabaa
- 1. FOURIER TRANSFORM IN IMAGE PROCESSES Arranged by: Nabaa Badeea Ryiam Abd -Aljabar Alaa Zaki Sundus Ali Shahad Saad Aula Maad
- 2. INTRODUCTION Fourier Transform : It is convert a function from one domain to another with no loss of information (converts a function from time domain to the frequency domain). Jean Baptiste Joseph Fourier (1768-1830), a French mathematician and physicist.
- 3. THE DISCRETE FOURIER TRANSFORM IN I M AG E P RO C E S S I N G . . The Fourier Transform is an important image processing tool which is used to decompose an image into its sine and cosine components. The output of the transformation represents the image in the Fourier or frequency domain, while the input image is the spatial domain equivalent.
- 4. CONT… In the Fourier domain image, each point represents a particular frequency contained in the spatial domain image. In image processing, The Fourier Transform is used in a wide range of applications, such as image analysis, image filtering, image reconstruction and image compression.
- 5. HOW DOES IT WORK? From the variants of the Fourier Transform Discrete Fourier Transform (DFT) is the variant used in digital image processing. The DFT is the sampled Fourier Transform. That is why DFT does have all the frequencies which form the image, but only a set of samples which is large enough to fully describe the spatial domain image. The number of frequencies corresponds to the number of pixels in the spatial domain image, i.e. the image in the spatial and Fourier domain are of the same size.
- 6. CONT… For a square image of size N×N, the two-dimensional DFT is given by: f(i,j) is the image in the spatial domain and the exponential term is the basis function corresponding to each point F(k,l) in the Fourier space. The equation can be interpreted as: the value of each point F(k,l) is obtained by multiplying the spatial image with the corresponding base function and summing the result.
- 7. CONT… The basis functions are sine and cosine waves with increasing frequencies, i.e. F(0,0) represents the DC-component of the image which corresponds to the average brightness and F(N-1,N-1) represents the highest frequency. In a similar way, the Fourier image can be re-transformed to the spatial domain. The inverse Fourier transform is given by:
- 8. CONT… To obtain the result for the previous equations, a double sum has to be calculated for each image point. However, because the Fourier Transform is separable, it can be written has: Where:
- 9. CONT… Using those last two formulas, the spatial domain image is first transformed into an intermediate image using N one-dimensional Fourier Transforms. This intermediate image is then transformed into the final image, again using N one-dimensional Fourier Transforms. Expressing the two-dimensional Fourier Transform in terms of a series of 2N one-dimensional transforms decreases the number of required computations
- 10. CONT… The ordinary one-dimensional DFT still has complexity which can be reduced with the use of Fast Fourier Transform (FFT) to compute the one dimensional DFTs. It is a significant improvement, in particular for large images. There are various forms of the FFT and most of them restrict the size of the input image that may be transformed, often to where n is an integer.
- 11. H OW D O E S I T WO R K ? . . . M AG N I T U D E A N D PHASE The Fourier Transform produces a complex number valued output image which can be displayed with two images, either with the real and imaginary part or with magnitude and phase. In image processing, often only the magnitude of the Fourier Transform is displayed, as it contains most of the information of the geometric structure of the spatial domain image.
- 12. CONT… The Fourier image can also be re-transformed into the correct spatial domain after some processing in the frequency domain...(both magnitude and phase of the image must be preserved for this). The Fourier domain image has a much greater range than the image in the spatial domain. Hence, to be sufficiently accurate, its values are usually calculated and stored in float values.
- 13. FA S T F O U R I E R T R A N S F O R M How the FFT works? 1) Decomposed an N point time domain signal into N time domain signals each composed of a single point. 2) Calculate the N frequency spectra corresponding to these N time domain signals. 3) The N spectra are synthesized into a single frequency spectrum
- 14. FOURIER TRANSFORM APPLICATIONS Used in many scientific areas 1) Physics 2) Number theory 3) Signal processing (inc. image processing) 4) Probability theory 5) Statistics 6) Cryptography 7) Acoustics 8) Optics ,Etc, etc…
- 15. FOURIER TRANSFORM IN SIGNAL PROCESSING Includes: (for example) audio and image signal processing Audio and image signal are very alike (1D vs. 2D): 1) Both can be continuous (analog) or discrete (digital) 2) Same kind of filters can be used Fourier transform is used for a easier way to apply different kind of filters to the signal Filters are used for: 1) Removing unwanted frequencies from signal 2) Removing noise from signal 3) Altering signal somehow
- 16. FEW FILTER EXAMPLES Low pass filter = Image blur
- 17. CONT… High pass filter = Edges * Can be used in edge detection
- 18. CONT… Sharpening = boosting high frequency pixels
- 20. THANKS…