Multidimensional transform

Multi-dimensional transforms are used to analyze the frequency content of multi-dimensional signals (signal domain has a dimensional >= 2).

M-D Fourier Transform

One most popular m-D transform is the Fourier Transform, which converts the signal from a time/spatial domain representation to a frequency domain representation. Give equation like below.

            $F(w_{1},w_{2},...,w_{m})=\sum _{n_{1}=-\infty }^{\infty }\sum _{n_{2}=-\infty }^{\infty }...\sum _{n_{m}=-\infty }^{\infty }f(n_{1},n_{2},...,n_{m})e^{-jw_{1}n_{1}-jw_{2}n_{2}...-jw_{m}n_{m}}$ 
                       F stands for m-D Fourier transform. m stands for m-Dimension.

Such that, the m-D Inverse Fourier Transform is shown below.

        $f(n_{1},n_{2},...,n_{m})={({\frac {1}{2\pi }})}^{m}\int _{-\pi }^{\pi }...\int _{-\pi }^{\pi }F(w_{1},w_{2},...,w_{m})e^{jw_{1}n_{1}+jw_{2}n_{2}...+jw_{m}n_{m}}\,dw_{1}...\,dw_{m}$

Also ,we can have m-D Fourier Transform for continuous signals:

      F(\Omega_1,\Omega_2,...,\Omega_m) = \int_{-\infty}^{\infty} \int_{-\infty)^{\infty} f(t_1,t_2,...,t_m) e^{-j \Omega_1 n_1-j \Omega_2 n_2 ... -j \Omega_m n_m} \, dt_1 ... \,dt_m

And, multidimensional DFT. The multidimensional DFT can be seen as two things: it is a Fourier Transform for finite-extent sequences, what’s more, it’s also a Fourier transform for multidimensional periodic sequences.

      $F(k_{1},k_{2},...,k_{n})=\sum _{n_{1}=0}^{N_{1}-1}...\sum _{n_{m}}^{N_{m}-1}x(n_{1},n_{2},...,n_{N})e^{-j{\frac {2\pi }{N_{1}}}n_{1}k_{1}-j{\frac {2\pi }{N_{2}}}n_{2}k_{2}...-j{\frac {2\pi }{N_{m}}}n_{m}k_{m}}$ 
                                      for 0≤k_1,...,k_m≤N_(1,...,m)  - 1

And the Inverse DFT equation is

         $x(n_{1},n_{2},...,n_{m})={\frac {1}{(N_{1}...N_{m})}}\sum _{k_{1}=0}^{N_{1}-1}...\sum _{k_{m}}^{N_{m}-1}X(k_{1},k_{2},...,k_{m})e^{j{\frac {2\pi }{N_{1}}}n_{1}k_{1}+j{\frac {2\pi }{N_{2}}}n_{2}k_{2}...+j{\frac {2\pi }{N_{m}}}n_{m}k_{m}}$ 
                                     for  0≤n_(1,)…,n_m≤N_(1,…,m)  – 1

M-D Discrete Cosine Transform

The discrete cosine transform really been used for a wide range of application like, data compression, feature extraction, Image reconstruction, multi-frame detection and so on. which can be calculated by polynomial transform algorithm. And the m-D DCT can be shown as below,

           $X(k_{1},k_{2},...,k_{r})=\sum _{n_{1}=0}^{N_{1}-1}\sum _{n_{2}=0}^{N_{2}-1}...\sum _{n_{r}=0}^{N_{r}-1}x(n_{1},n_{2},...,n_{r})\cos {\frac {\pi (2n_{1}+1)k_{1}}{2N_{1}}}\cos {\frac {\pi (2n_{r}+1)k_{r}}{2N_{r}}}$ 
                        k_i=0,1,…,N_i-1;i=1,2,…,r.

Satellite Communication

The Satellite Communication is going bigger in he communication market. Such that, more research has been done to complete this topic. One usual multidimensional application is in the environment simulation. Examples, Space-Time Rain Rate Field Generator.

File:Rain Rate in 3hour.png

fft for 2image

Figure1. 3hour rain rate by NASS

This technology using multidimensional Stochastic Differential Eqations(SDEs) to simulate rain attenuation. The basal form of the multidimensional SDE will be the following formula :

      $dR_{t}=F(R_{t})dt+Z(R_{t})dW_{t}$

where $R_{t}={[R_{t}^{1},...,R_{t}^{n}]}^{T}$ is rain rate vector and F(R_t) is drift N*1 matrix, Z(R_t) is N*N diffusion matrix. After that, the specific rain attenuation(A_0) can be calculated on every point:

         $A_{0}(x,y)=aR{(x,y)}^{b}$

a and b depend on the polarization, frequency and elevation angle of the link. Rain attenuation with slant path L at given condition can be computed by following equation:

          $A=\int _{0}^{L}A_{0}z={(x^{2}+y^{2})}^{0.5}\,dz$

     Such that, we can calculate the rain attenuation and simulate the rain condition, satellite communication. Through the numerical result compared with ITU-R. P. 618-10. This technology is much better than ITU.
     Through this reference paper, the methodology based on m-D SDEs for generation of time series of two-dimension rain rate fields produces accurately first order statistics of rain rate, which is funded by EU(Europe Union) and GSRT(Greek national funds program).

Image Reconstruction

    In many applications like sensing, frame freeze in video, and so on. Demand for higher degree of resolution for image is increasing. But the technology can’t always keep up with the demand. Such that, many research has been design to realize the function
  A high-resolution can be very large to transmit, so we will use low-resolution DCT-based compressed images instead. But, how to reconstruct a high-resolution image from the DCT image is what this section will talk about.
 The desired image size is L₁N₁×L₂N₂  ,where L represent the down sampling vector and each obtained low-resolution image size is N_1×N_2 . The kth  low-resolution frame will be 
                                              $Y_{k}={[y_{k,1},y_{k,2},...,y_{k,m}]}^{T}$

  X remains high-resolution frame.
  Such that,     
                         $y_{m}=\sum _{r=1}^{N}w_{m,r}(S_{k})x_{r}$

 m=1,2,…pM. w is the “contribution” of r-th high resolution pixel.
                   ${\hat {y_{k}}}=W_{k}x+n_{k}$

        W_k is the matrix relationship between Y and X, n_k  is the noise. Hat on Y show that is the estimate of Y.
        Since the quantization information and sub-pixel motion in each low-resolution image, we propose a multi-channel high-resolution algorithm in this part.
        Through the iterative solution we can get all x. which can be resulted in:
         $\sum _{k=1}^{p}[W_{k}^{T}K_{k}^{-1}W_{k}+\alpha _{k}(x)C^{T}BC]x=\sum _{k=1}^{p}W_{k}^{T}K_{k}^{(}-1){\hat {yk}}$

  Where K is the noise covariance matrix in spatial domain, C is the high pass operator, B is the weighting matrix.

File:2 different image.png

a) is low b is high by google image

      Through this way we can reconstruct a high-resolution image from low-resolution images. When comparing the images, ringing and blocking artifacts are removed in the pro posed result, while edge and details are well preserved. Improvements the entire sequence are quantified with the peak signal-to-noise (PSNR) metric.

Image processing for arts surface analysis by FFT

 One easy and cheap tool will be introduced in this section . This tool combines digital photography and 2D FFT image processing for works of arts surface analysis.
 One very important factor is that we must apply a non-destructive method to obtain those rare valuables information about works of art and zero-damage on them.
 We can understand the arts by looking at a color change or by measuring the surface uniformity change. Since the whole image will be very huge, so we use a double raised cosine window to truncate the image:
             $w(x,y)={\frac {1}{4}}[1+\cos {\frac {x\pi }{N}}][1+\cos {\frac {y\pi }{N}}]$

     Where N is the image dimension and x, y are the coordinates from the center of image spans from 0 to N/2.
  The author wanted to compute an equal value for spatial frequency such as:
       $A_{m}{(f)}^{2}=[\sum _{i=-f}^{f}FFT(-f,i)^{2}+\sum _{i=-f}^{f}FFT(f,i)^{2}+\sum _{i=-f+1}^{f-1}FFT(i,-f)^{2}+\sum _{i=-f+1}^{f-1}FFT(i,f)^{2}]$ 
      Where f is the spatial frequency spans from 0 to N/2 – 1.

File:Big camera.png

camera stands for imaging acquisition

      The proposed FFT-based imaging approach is diagnostic technology to ensure a long life and stable to culture arts. This is a simple, cheap which can be used in museums without affecting their daily use. But this method doesn’t allow a quantitative measure of the corrosion rate.

References

[1] Smith,W. Handbook of Real-Time Fast Fourier Transforms:Algorithms to Product Testing ,Wiley_IEEE Press,edition 1,page 73-80,1995
[2]Dudgeon and Mersereau, Multidimensional Digital Signal Processing,2nd edition,1995
[3] Kourogiorgas, C.I. ;Karagiannis, G.A. ; Panagopoulos, A.D. Space-time rain rate
[4] Juxia Ma, Based on the fourier transform and the wavelet transformation of the digital image processing,CSIP 2012International Conference
[5] Yonghong Zeng, Guoan Bi, Abdul Rahim Leyman. New Polynomial Transform Algorithm for Multidimensional DCT IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 10, OCTOBER 2000
[6] Charilaos I. Kourogiorgas, Georgios A. Karagiannis, Athanasios D. Panagopoulos Space-Time Rain Rate Field Generator for Multi Antenna Satellite Communication Applications, 2013 7th European Conference on Antennas and Propagation (EuCAP)
[7] Sung Cheol Park, Moon Gi Kang ; Segall, C.A. ; Katsaggelos, A.K. High-resolution image reconstruction of low-resolution DCT-based compressed images Acoustics, Speech, and Signal Processing (ICASSP), 2002 IEEE International Conference on (Volume:2 ) Page(s):II-1665 - II-1668
[8] Angelini, E., Grassin, S. ; Piantanida, M. ; Corbellini, S. ; Ferraris, F. ; Neri, A. ; Parvis, M. FFT-based imaging processing for cultural heritage monitoring Instrumentation and Measurement Technology Conference (I2MTC), 2010 IEEE

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