Digital image processing

Transcription

1 Digital image processing The two-dimensional discrete Fourier transform and applications: image filtering in the frequency domain Introduction Frequency domain filtering modifies the brightness values of the pixels in the image depending on their periodic patterns and on the spatial distributions of brightness variations in the image. The high pass filters tend to preserve the sharp brightness variation patterns and to remove the smooth variations in the image (along with the DC component, i.e. with the average brightness which is set to zero). The low pass filters, on the opposite, preserve the smooth brightness variations in the image, and remove the sharp edges (sharp brightness variations) in the image. To apply a frequency domain filtering procedure, the most typical approach is to first apply a Fourier transform to the image, to obtain its frequency domain representation; the frequency domain representation of the image through its Fourier transform provides information about the spectral components of the image. The frequency domain filtering process simply modifies certain spectral components in the image and afterwards, by applying the inverse Fourier transform on the frequency domain filtered image represented in the frequency domain, provides the filtered image representation in the spatial domain. Filtering Inverse FFT FFT f(x,y) F(u,v) G(u,v) g(x,y) Figure The architecture of a frequency domain filtering system In a digital image, the details and the sharp boundaries are associated to high spatial frequencies, since they introduce sharp brightness transitions. On the opposite, the approximately uniform surfaces and background areas are associated to the low spatial frequencies, since they correspond to the smooth variations of the brightness or even to no spatial variations of the brightness. A digital image can exhibit, e.g., a periodic noise in the form of some periodical lines, resulting from an imperfect digitization. In the spatial frequency domain, this noise will introduce spectral components in the high frequency range, with a good localization due to its periodicity. If one filters out these spectral components by setting the corresponding coefficients of the Fourier transform to zero, and afterwards applies the inverse Fourier transform on the modified spectrum, the result is an image without noise and typically without any noticeable loss in details. The Discrete Fourier Transform The discrete Fourier transform is one of the most important transforms used in signal processing and image processing. For a -D periodical sequence of samples {u(n), n=,...,-}, its discrete Fourier transform is given by:

2 7. The -D DFT and applications: image filtering in the frequency domain - kn n= v(k)= u(n), k =,.., - with: -jπ = exp( ) The inverse discrete Fourier transform of this sequence is: u(n) = - kn v(k) ; n =,,..., - k= The two-dimensional discrete Fourier transform The -D discrete Fourier transform of an image of size M pixels, U[Mx]={u(m,n)}, is a separable transform defined by the expression: v(k,l)= u(m,n), k M - l - M M- - km ln M ; 3 m= n= The inverse -D discrete Fourier transform is defined as: M - - u(m,n)= v( k, l) km ln M, m M - ; n - 4 k= l= Let s consider the computation of the -D DFT of an image of size 3 3 pixels, whose isometric (3-D) representation is given in Figure (left) the image is a square of size pixels, centred in the image; the background of the image is black, and the square is white (and for convenience, the brightness is normalized in the range [;]). The -D DFT amplitude spectrum of this image is shown in Figure in the centre; the right diagram in Figure represents the amplitude spectrum shifted by mirroring to give the DC component in the centre and the high frequencies on the corners. The digital image represented in 3D The amplitude spectrum Shifted amplitude spectrum Figure Exemplification of the -D DFT Interpretation of the DFT Using trigonometry relations, one can re-write the inverse -D DFT in the following form: / π u f ( x) = F () + F ( u ) cos( + phase ( F ( u )) 5 u = This expression shows that the original signal can be expressed as a weighted sum of cosine basis functions. Approximating this sum by only the first two largest terms, we get the following approximation of the signal:

3 Digital image processing π + o π 3 f ( x) cos, phase : 9 +. cos, phase : 9 o 6 The horizontal profile of the reconstructed signal is the following: Fast algorithms for the discrete Fourier transform I. Rows and columns decomposition Since the DFT is extensively used in signal processing, there was a great interest in developing algorithms for its efficient computation (also taking into account that DFT may be used in computing other signal/image transforms, such as DCT). To estimate the efficiency of the computation of -D DFT of a digital image by the rows and columns decomposition method, let us first examine the computational complexity of the direct computation of the -D DFT for a complex -D sequence of points, denoted by x(n,n ). The -D DFT for this sequence is denoted by X ( k, k ). The graphical representation of the rows and columns decomposition is given in Figure 3: n ( -) k (-) D-DFT in points ( -) n ( -) k 3

4 7. The -D DFT and applications: image filtering in the frequency domain n ( -) k ( -) D-DFT in points ( -) n ( -) k Figure 3 Illustration of the rows and columns decomposition in computing DFT The result of the application of this algorithm on some digital image is shown in Figure 4: Original input image DFT applied on the rows DFT applied on the columns Figure 4 Example of applying the DFT D by the rows and columns decomposition method According to the above considerations, the -D DFT sequence X ( k, k ) may be computed by a total of -D DFTs applied on a series of D-DFTs. Assuming the computation of the -D DFTs according to the Equation above (which implies multiplications and additions), the total number of arithmetic operations involved in the computation of the sequence X is (+) multiplications and additions. Therefore the number of operations is much smaller as compared to the direct computation given by the Equation 3. II. The RADIX- FFT algorithm One of the most efficient algorithms in the computation of the DFT is based on the divide and conquer technique. This algorithm is only applicable when the length of the sequence,, is not prime, i.e. when can be decomposed as a product of integers =r*r*...*rx, where {rj} are primes. A particular case appears for r=r=...=rx=r; in this case = r x, and the DFT of the sequence may be decomposed in a set of DFTs applied to some sequences of length r. The value r is called the radix of the FFT algorithm. Let us consider the computation of the DFT of a sequence of length = v samples through the divide and conquer method. If one denotes by M=/ and L=, the initial sequence of samples may be divided into two vectors, corresponding to the even and odd indices in the sequence. The two sub-sequences are obtained by the decimation of the initial sequence x(n) by a factor of ; therefore the FFT algorithm in this case is called time decimation FFT algorithm. 4

5 Digital image processing f ( n) = x( n) f ( n) = x( n + ) n =,... 7 ow one can express the DFT of the sequence of length in terms of the DFTs of the decimated sequences as follows: X ( k ) = = = n= x( n) n= par / m= kn x( n) kn x(m) + mk k =,,..., n= impar + x( n) / m= kn x(m + ) (m+ ) k But - jπ = exp ( ), therefore =, which leads to the following / reformulation of the above expression: / / mk k X ( k) = f( m) / + f( m) m= m= k = F ( k) + F ( k) where F ( k ) and F k ( ) are the DFTs of the two (even and odd) sub-sequences of length /: f and f. If F (k) and F (k) are periodical with the period /, then F ( k + / ) = F ( k ) and F ( k + / ) = F ( k ) ; furthermore, considering that + / =, one can write: X( k) = F ( k) + X( k + ) = F ( k) k F ( k) k F ( k) mk / k k k =,,..., k =,,..., The direct computation of F ( k ) requires /4 complex multiplications (and the same number is needed for F ( k ) ). Other / complex multiplications are needed for the term k F ( k). Therefore the computation of X(k) requires + = + complex multiplications. Thus, even after this first step, one can estimate a computational complexity reduction, from to only +, which is approximately equivalent to a reduction by a factor of (approximately) for large. Denoting by: G(k) = F ( k) k =,,..., k G (k) = F ( k) k =,,..., 9 5

6 7. The -D DFT and applications: image filtering in the frequency domain the above algorithm can be repeated in a recursive fashion, leading to the radix FFT. To illustrate the operation of this algorithm, let us consider the particular case of a sequence of length =. In general, each butterfly implies one complex multiplication and two complex additions. For an = ν, / butterflies and v = log stages are needed; these will require a number of operations derived above. A butterfly implies one multiply and two additions between the pair of complex numbers (a,b) to give as output the pair (A,B), and there is no need to store the input pair (a,b). Therefore the output result of the butterfly computation (A,B) may be stored in the same location as the input (a,b), which is very efficient from the memory requirements point of view. The basic butterfly corresponding to the time decimation is: a A=a+ r *b - b B=a - r *b Basic butterfly To understand the workflow of the algorithm, one should examine the way the data is updated in the memory, after each decimation stage. This is illustrated in Figure 5: x() x() x() x(3) x(4) x(5) x(6) x(7) x() x() x(4) x(6) x() x(3) x(5) x(7) decimation Data reordering decimation x() x(4) x() x(6) x() x(5) x(3) x(7) Figure 5 The decimation algorithm The previous figure suggests that, whereas the input data are in their normal order, the output data are in a reversed order. This issue must be taken into account in the frequency domain filtering algorithms, in order to appropriately filter the desired spectral components. The entire FFT procedure is presented in Figure 6. This graph is useful to understand the disadvantages of the algorithm for large values of the most significant being the following: - the transform coefficients for large values of take large values as well, therefore their handling is difficult (in respect to the memory requirements); - considering the large number of multiplications needed, it becomes clear that the errors accumulation cannot be neglected for large values of ; these errors will reflect on the reconstructed image, even at no truncation of the transform coefficients. 6

7 Digital image processing x() X() x() - X(4) x() - X() x(3) - - X(6) x(4) - X() x(5) - - X(5) x(6) - - X(3) x(7) X(7) Figure 6 The FFT graph in the time decimation algorithm, for = Displaying the Discrete Fourier Transform The following four components describing complex numbers may be used in the visualization of an FFT image: the real part; the imaginary part; the amplitude (absolute value); the phase. The definition of these four components is described by the following expressions: (, ) (, ) (, ) F u v = R u v + ji u v where R(u,v) is the real part, and I(u,v) is the imaginary part, or: j ( u, v) (, ) (, ) F u v = F u v e ϕ 3 where F(u,v) is the amplitude, and ϕ(u,v) is the phase. The amplitude F(u,v) is also called the Fourier spectrum, defined as: (, ) (, ) (, ) = + 4 F u v R u v I u v The square of the Fourier spectrum is called Fourier power spectrum or spectral density. The phase ϕ(u,v) is also called phase angle and is defined as: 7

8 7. The -D DFT and applications: image filtering in the frequency domain (, ) (, ) I u v ϕ ( u, v) = arctan R u v Consider an image with M pixels, and denote by x and y the horizontal and vertical sampling steps. Then the Fourier transform of this image will have the same size, M, and the sampling frequencies u and v will be defined by the expressions: u = ; v = x M y There are two ways to represent the Fourier spectrum of a digital image: the default representation the optical representation The default representation In the case of the standard (default) representation (Figure 7), the high frequencies are grouped in the centre of the Fourier spectrum image, whereas the low frequencies are located on the corners of this image. Thus, the zero frequency (the DC coefficient) is, u, M v located in the four corners of the image. The frequency range is: [ ] [ ] 5 6 A Low frequency B High frequency C D The optical representation Figure 7 The standard (default) representation In the case of the optical representation (Figure ), the low frequencies are located in the centre of the Fourier spectrum image, and the high frequencies are located in the corners of the Fourier spectrum image. Thus, the zero frequency (the DC coefficient) is located in the centre of the Fourier spectrum image. The frequency range is: M M u, u v, v

9 Digital image processing A High frequency B Low frequency C D Figure The optical representation Frequency domain filtering The low pass filtering A frequency domain low pass filter attenuates or sets to zero the high frequency components in the Fourier spectrum. Such a filter suppresses the information introduced by the sharp spatial variations of the brightness in the digital image. As a result of applying such a filter on a digital image, the reconstructed image (through the inverse Fourier transform) has less noise, but the details and the contours of the objects as well as the textures of the surfaces in the image are also attenuated. Low pass attenuation filters In this case, the spectral components are linearly reduced as amplitude (see Figure 9.a)), starting from the DC component (of frequency f ) to the maximum frequency component (of frequency f max ). The algorithm implies the multiplication of all the spectral components by a coefficient C(f) given by the expression: C f f max ( f ) = with C(f )= and C(f max )=. f f max 7 C(f) C(f) f f max f f c f max a) b) Figure 9 The low pass attenuation (a) and low pass truncation (b) 9

10 7. The -D DFT and applications: image filtering in the frequency domain Low pass truncation filters The low pass truncation filters (Figure 9.b)) simply eliminates all the frequency components whose frequencies f are higher than some cut off frequency f c. This is done by multiplying each frequency component f by the coefficient C, taking the values {,}: The high pass filtering ; if f > f C ( f ) = c ; otherwise A frequency domain high pass filter attenuates or sets to zero the low frequency components in the Fourier spectrum. Such a filter suppresses the information introduced by the smooth spatial variations of the brightness in the digital image. As a result of applying such a filter on a digital image, the reconstructed image (through the inverse Fourier transform) has its uniform or quasi-uniform brightness set to zero. High pass attenuation filters In this case, the spectral components are linearly reduced as amplitude with a positive slope (see Figure.a)), starting from the DC component (of frequency f ) to the maximum frequency component (of frequency f max ). The algorithm implies the multiplication of all the spectral components by a coefficient C(f) given by the expression: with C(f )= and C(f max )=. C f f f f ( f ) = max 9 C(f) C(f) f f max f f c f max a) b) Figure The high pass attenuation (a) and the high pass truncation (b) High pass truncation filters The high pass truncation filters (Figure 9.b)) simply eliminates all the frequency components whose frequencies f are lower than some cut off frequency f c. This is done by multiplying each frequency component f by the coefficient C, taking the values {,}: ; dacã f < f C ( f ) = c ; altfel

11 LabView IMAQ Functions for DFT Applications IMAQ GetPalette Selects a display palette. Five predefined palettes are available. To activate a color palette choose a code for Palette umber and connect the Color Palette output to the input Color Palette of IMAQ inddraw. Digital image processing Palette umber (gray) gives you a choice of five predefined palettes. You can choose from the following values: Gray - Grayscale is the default palette. Binary - designed especially for binary images. Gradient Rainbow Temperature Color Palette indicates an array of clusters composed of 56 elements for each of the three color planes. IMAQ FFT Computes the FFT of an image. The FFT is a complex image in which high frequencies are grouped at the center, while low frequencies are located at the corners. Image Src - is the handle of the source image. Image Dst - is the handle of the complex image that contains the resulting FFT image. This input can accept only a complex image. Image Dst Out - is the reference to the destination (output) image that receives the processing results of the VI. IMAQ InverseFFT Computes the inverse FFT of a complex image ( 3-bit floating point). Image Src - is the handle of the source image. This input can accept only a complex image. Image Dst - is the handle of the -bit, 6-bit, or 3-bit floating-point image that contains the resulting spatial image. Image Dst Out - is the reference to the destination (output) image that receives the processing results of the VI. IMAQ ComplexTruncate Truncates the frequencies of a complex image.

12 7. The -D DFT and applications: image filtering in the frequency domain Low pass/high pass (Low pass) - determines which frequencies are truncated. Choose Low pass (F) to remove the high frequencies or High pass (T) to remove the low frequencies. The default is FALSE, which specifies lowpass. Truncation Frequency % - is the percentage of the frequencies that are retained within a Fourier-transformed image. Image Src - is the image reference source. It must be a complex image. Image Dst - is the reference of the image destination. If it is connected, it must be the same type as the Image Src. Image Dst Out - is the reference to the destination (output) image that receives the processing results of the VI. Practice Fourier transform implementation for a greyscale image open a new LabView session create a new project in the diagram window, add the IMAQ functions needed to obtain the following schematic:

13 Digital image processing the user interface should be the following: load different images, perform the FFT and notice the distribution of the FFT coefficients. Modify the color palette used to display the Fourier coefficients and observe the effect on the display of these images. Locate the low frequency, mean frequency and high frequency regions in the Fourier spectrum. Frequency domain filtering for a greyscale image open a new LabView session create a new project in the diagram window, add the IMAQ functions needed to obtain the following schematic: 3

14 7. The -D DFT and applications: image filtering in the frequency domain the user interface should be the following: load different images, perform LPF and HPF for different filtering percentages (see the description of the IMAQ Complex Truncate function). Modify the 4

15 Digital image processing colour palette used for display of the FFT coefficients and observe the effect on the display window draw a block diagram to perform a band pass filtering, where the both thresholds, inferior and superior one, may be modified. Questions and exercises. Implement a diagram that will perform the Fourier transform of a RGB color image.. Modify the first block diagram to obtain a 3D representation of the Fourier spectrum, before and after filtering. 3. Implement a LabView application which will perform highpass attenuation and lowpass attenuation for a grayscale image 5