Function idft

Synopse

Inverse Discrete Fourier Transform.

f = iaidft(F)
- f: Image.

F: Image.



In [1]:

    
import numpy as np

def idft(F):
    import ia898.src as ia
    s = F.shape
    if F.ndim == 1: F = F[np.newaxis,np.newaxis,:] 
    if F.ndim == 2: F = F[np.newaxis,:,:] 

    (p,m,n) = F.shape
    A = ia.dftmatrix(m)
    B = ia.dftmatrix(n)
    C = ia.dftmatrix(p)
    Faux = np.conjugate(A).dot(F)
    Faux = Faux.dot(np.conjugate(B))
    f = np.conjugate(C).dot(Faux)/(np.sqrt(p)*np.sqrt(m)*np.sqrt(n))

    return f.reshape(s)

Examples



In [2]:

    
testing = (__name__ == "__main__")

if testing:
    ! jupyter nbconvert --to python idft.ipynb
    import numpy as np
    import sys,os
    import matplotlib.image as mpimg
    ia898path = os.path.abspath('../../')
    if ia898path not in sys.path:
        sys.path.append(ia898path)
    import ia898.src as ia









    



[NbConvertApp] Converting notebook idft.ipynb to python
[NbConvertApp] Writing 2245 bytes to idft.py

Example 1



In [3]:

    
if testing:
    f = np.arange(24).reshape(4,6)
    F = ia.dft(f)
    g = ia.idft(F)
    print(np.round(g.real))









    



[[  0.   1.   2.   3.   4.   5.]
 [  6.   7.   8.   9.  10.  11.]
 [ 12.  13.  14.  15.  16.  17.]
 [ 18.  19.  20.  21.  22.  23.]]



In [4]:

    
if False: #testing:
    import matplotlib.image as mpimg
    
    f = mpimg.imread('../data/cameraman.tif')
    F = ia.dft(f)
    print(F.shape)
    H = ia.circle(F.shape, 50,[F.shape[0]/2,F.shape[1]/2] )
    H = ia.normalize(H,[0,1])
    FH = F * ia.idftshift(H)
    print(ia.isdftsym(FH))
    g= ia.idft(FH)
    ia.adshow(f)
    ia.adshow(ia.dftview(F))
    ia.adshow(ia.normalize(H,[0,255]))
    ia.adshow(ia.dftview(FH))
    ia.adshow(ia.normalize(abs(g)))

Equation

$$ \begin{matrix} f(x) &=& \frac{1}{N}\sum_{u=0}^{N-1}F(u)\exp(j2\pi\frac{ux}{N}) \\ & & 0 \leq x < N, 0 \leq u < N \\ \mathbf{f} &=& \frac{1}{\sqrt{N}}(A_N)^* \mathbf{F} \end{matrix} $$

$$ \begin{matrix} f(x,y) &=& \frac{1}{NM}\sum_{u=0}^{N-1}\sum_{v=0}^{M-1}F(u,v)\exp(j2\pi(\frac{ux}{N} + \frac{vy}{M})) \\ & & (0,0) \leq (x,y) < (N,M), (0,0) \leq (u,v) < (N,M) \\ \mathbf{f} &=& \frac{1}{\sqrt{NM}} (A_N)^* \mathbf{F} (A_M)^* \end{matrix} $$

Contribution