Image Enhancement in the Frequency Domain

Transcription

1 Image Enhancement in the Frequency Domain Jesus J. Caban Outline! Assignment #! Paper Presentation & Schedule! Frequency Domain! Mathematical Morphology %&

2 Assignment #! Questions?! How s OpenCV?! You have another 48hrs. Keep working! Paper Presentation & Schedule! Check the schedule online for the date you are presenting! Check the reading list for your paper assignment! If your name is not there, please let me know! Let me know about errors #&

3 Presentation: Evaluation Form. Delivery 2. Visual Aids & Organization 3. Topic & Papers Possible change to the syllabus! Instead of reading and writing a question for each paper, do you want to only read one paper? '&

4 Image Enhancement in the Frequency Domain Recall: Point Processing! Based on the intensity of a single pixel only as opposed to a neighborhood or region (&

5 Moving Window Transform: Example original 3x3 average Moving Window Transform: Example original 3x3 average )&

6 Spatial Filtering: Smoothing / 9 / 9 / 9 / 9 / 9 / 9 / 9 / 9 / 9 / 6 2 / 6 / 6 2 / 6 4 / 6 2 / 6 / 6 2 / 6 / 6 Spatial Filtering: Edge Enhancement *&

7 Jean Baptiste Joseph Fourier! Claim (807): Any periodic function can be rewritten as a weighted sum of sines and cosines of different frequencies! Published his work in 822: La Théorie Analitique de la Chaleur! His work was translated into English over 55+ years later (878): The Analytic Theory of Heat! Researchers didn t pay too much attention to his work when it was first published! Fourier Theory then became one of the most important mathematical theories in modern engineering The General Idea = Any function that periodically repeats itself can be expressed as a sum of sines and cosines of different frequencies each multiplied by a different coefficient Resulting collection is called Fourier series +&

8 Frequency t 0 t t 2 Frequency is the number of occurrences of a repeating event per unit time Amplitude The amplitude is the height of the wave.,&

9 Phase shifts The phase shift describes how far to the left or right the wave slides. The same wave after a central section underwent a phase shift (e.g. passing through glass) Frequency Example! g(t) = sin(2!f t) + (/3)sin(2! (3f) t) = +!&

10 Fourier Series - Example First 60 Fourier approximations Fourier Transform: Definition Sum of sine and cosine basis functions!! Changes in u and v obtain different basis functions of different frequencies! Point in the new image F(u,v) represent the contribution of that frequency to the original image %$&

11 Fourier Transform: Real & Imaginary Part! General concept f(x) Fourier Transform F(w) For every w from 0 to inf, F(!) holds the amplitude A and phase " of the corresponding sine 2D Domain! Any image can be captured in a single Fourier term that encodes! the spatial frequency! the magnitude (positive or negative)! the phase x u y v %%&

12 Frequency Domain: Illustration Frequency Domain: Illustration %#&

13 Extension to 2D (a-jb) u (a+jb) v Frequency Domain: Properties %'&

14 Properties of the Fourier Transform! Rotation: rotated image is equivalent to rotated DFT! Scaling: a*f(x,y) is equivalent to a*f(u,v)! F(0,0) = average value of f(x,y)! Others: symmetry, distributive, etc The Discrete Fourier Transform (DFT). Given an image f(x,y) or size MxN x = 0,, 2 M- y = 0,,2 N- Origin M 2. Discrete Fourier Transform of f(x, y) is given by N u = 0,, 2 M- v = 0,, 2 N- j= sqrt(-) %(&

15 The Inverse DFT! The key feature of a Fourier Transform is that it s completely reversible! The inverse DFT is given by: x = 0,, 2 M- y = 0,, 2 N- The Inverse DFT %)&

16 ! Concept of convolution Recall: Basics of Spatial Filtering! Filter image by moving the convolution kernel. Convolution Theorem! Convolution in the spatial domain is equivalent to multiplication in the frequency domain! Convolution of large kernels to images can be performed by a simple multiplication %*&

17 Origin Recall: Smoothing Spatial Filtering x * / 9 / 9 / 9 / 9 / 9! / 9 / 9 / 9 / 9 Simple 3*3 Neighbourhood 04 / 9 00 / 9 08 / 9 / / 3*3 Smoothing 9 98 / 9 / 9 / 9 / 9 Filter y Image f (x, y) Original Image Pixels Filter e = / 9 *06 + / 9 *04 + / 9 *00 + / 9 *08 + / 9 *99 + / 9 *98 + / 9 *95 + / 9 *90 + / 9 *85 = The DFT and Image Processing To filter an image in the frequency domain:. Compute F(u,v) the DFT of the image 2. Multiply F(u,v) by a filter function H(u,v) 3. Compute the inverse DFT of the result %+&

18 Frequency Domain: Filters. A low-pass filter: attenuates high frequencies 2. A high-pass filter: attenuates low frequencies 3. A band-pass filter: attenuates very low and very high frequencies. Low-pass Filters! Attenuates high frequencies and retains low frequencies unchanged.! high frequencies correspond to sharp intensity changes! The result in the spatial domain is equivalent to that of a smoothing filter! fine-scale details and noise in the spatial domain image are filtered %,&

19 Ideal Low Pass Filter Simply cut off all high frequency components that are a specified distance D 0 from the origin of the transform changing the distance changes the behaviour of the filter Ideal Low Pass Filter The transfer function for the ideal low pass filter can be given as: %!&

20 Ideal Low Pass Filter (cont ) Above we show an image, it s Fourier spectrum and a series of ideal low pass filters of radius 5, 5, 30, 80 and 230 superimposed on top of it Ideal Low Pass Filter (cont ) Original image Result of filtering with ideal low pass filter of radius 5 Result of filtering with ideal low pass filter of radius 5 Result of filtering with ideal low pass filter of radius 30 Result of filtering with ideal low pass filter of radius 80 Result of filtering with ideal low pass filter of radius 230 #$&

21 Ideal Low Pass Filter (cont ) Result of filtering with ideal low pass filter of radius 5 Ideal Low Pass Filter (cont ) Result of filtering with ideal low pass filter of radius 5 #%&

22 Butterworth Lowpass Filters The transfer function of a Butterworth lowpass filter of order n with cutoff frequency at distance D 0 from the origin is defined as: Butterworth Lowpass Filter (cont ) Original image Result of filtering with Butterworth filter of order 2 and cutoff radius 5 Result of filtering with Butterworth filter of order 2 and cutoff radius 5 Result of filtering with Butterworth filter of order 2 and cutoff radius 30 Result of filtering with Butterworth filter of order 2 and cutoff radius 80 Result of filtering with Butterworth filter of order 2 and cutoff radius 230 ##&

23 Gaussian Lowpass Filters The transfer function of a Gaussian lowpass filter is defined as: #'&

24 Gaussian Lowpass Filters (cont ) Original image Result of filtering with Gaussian filter with cutoff radius 5 Result of filtering with Gaussian filter with cutoff radius 5 Result of filtering with Gaussian filter with cutoff radius 30 Result of filtering with Gaussian filter with cutoff radius 85 Result of filtering with Gaussian filter with cutoff radius 230 Lowpass Filters Compared Result of filtering with ideal low pass filter of radius 5 Result of filtering with Butterworth filter of order 2 and cutoff radius 5 Result of filtering with Gaussian filter with cutoff radius 5 #(&

25 Lowpass Filtering Examples A low pass Gaussian filter is used to connect broken text Lowpass Filtering Examples #)&

26 Lowpass Filtering Examples Different lowpass Gaussian filters used to remove blemishes in a photograph. High-pass Filters! Attenuates high frequencies and retains low frequencies unchanged.! high frequencies correspond to sharp intensity changes! The result in the spatial domain is equivalent to that of a smoothing filter! fine-scale details and noise in the spatial domain image are filtered #*&

27 High-pass Filters A highpass filter yields edge enhancement or edge detection in the spatial domain Edges contain mostly high frequencies while other areas of the image are rather constant gray level (i.e. low frequencies) which are suppressed. Ideal High Pass Filters The ideal high pass filter is given as: where D 0 is the cut off distance as before #+&

28 Ideal High Pass Filters (cont ) Results of ideal high pass filtering with D 0 = 5 Results of ideal high pass filtering with D 0 = 30 Results of ideal high pass filtering with D 0 = 80 Butterworth High Pass Filters The Butterworth high pass filter is given as: where n is the order and D 0 is the cut off distance as before #,&

29 Butterworth High Pass Filters (cont ) Results of Butterworth high pass filtering of order 2 with D 0 = 5 Results of Butterworth high pass filtering of order 2 with D 0 = 80 Results of Butterworth high pass filtering of order 2 with D 0 = 30 Gaussian High Pass Filters The Gaussian high pass filter is given as: where D 0 is the cut off distance as before #!&

30 Gaussian High Pass Filters (cont ) Results of Gaussian high pass filtering with D 0 = 5 Results of Gaussian high pass filtering with D 0 = 80 Results of Gaussian high pass filtering with D 0 = 30 Highpass Filter Comparison Results of ideal high pass filtering with D 0 = 5 '$&

31 Highpass Filter Comparison Results of Butterworth high pass filtering of order 2 with D 0 = 5 Highpass Filter Comparison Results of Gaussian high pass filtering with D 0 = 5 '%&

32 Highpass Filter Comparison Results of ideal high pass filtering with D 0 = 5 Results of Butterworth high pass filtering of order 2 with D 0 = 5 Results of Gaussian high pass filtering with D 0 = 5 Laplacian In The Frequency Domain Laplacian in the frequency domain 2-D image of Laplacian in the frequency domain '#&

33 Frequency Domain Laplacian Example Original image Laplacian filtered image Laplacian image scaled Enhanced image Some Basic Frequency Domain Filters Low Pass Filter High Pass Filter ''&

34 3. Band-pass Filters A bandpass filter attenuates very low and very high frequencies, but retains a middle range band of frequencies. Bandpass filtering is used to enhance edges (suppressing low frequencies) while reducing the noise at the same time (attenuating high frequencies). Band-pass Filters: Example '(&

35 Band-pass Filters: Example Take-home message! Frequency Domain (Properties)! Image filtering in Fourier Domain! Filters:! Low-pass! High-pass! Band-pass ')&

36 Basic of filtering: Frequency Domain! How to filter in the frequency domain:. Multiply the input image by (-) x+y to center the transform 2. Compute F(u,v) (The DFT of the image) 3. Multiply F(u,v) by a filter function H(u,v) 4. Compute the invert DFT of the resulting image 5. Get the real part of the complex image 6. Multiply the resulting image by (-) x+y Fast Fourier Transform! The reason that Fourier based techniques have become so popular is the development of the Fast Fourier Transform (FFT) algorithm! Allows the Fourier transform to be carried out in a reasonable amount of time! Reduces the amount of time required to perform a Fourier transform by a factor of times! '*&

37 Summary In this lecture we examined image enhancement in the frequency domain! The Fourier series & the Fourier transform! Image Processing in the frequency domain! Image smoothing! Image sharpening! Fast Fourier Transform Acknowledgements! Some of the images and diagrams have been taken from the Gonzalez et al, Digital Image Processing book. '+&