def fp_match(f, f1):
# DFT das imagens
F = np.fft.fft2(f)
F1 = np.fft.fft2(f1)
# Idetificação do ângulo de rotação
F = ia.dftview(F)
F1 = ia.dftview(F1)
Fp = ia.polar(F,F.shape)
F1p = ia.polar(F1,F1.shape)
Pc = ia.phasecorr(Fp,F1p)
peak = np.unravel_index(np.argmax(Pc), Pc.shape)
ang = (float(peak[1])/Pc.shape[1])*360
#Correção de rotação
t1 = np.array([
[1,0,-f1.shape[0]/2.],
[0,1,-f1.shape[1]/2.],
[0,0,1]]);
t2 = np.array([
[1,0,f1.shape[0]/2.],
[0,1,f1.shape[1]/2.],
[0,0,1]]);
theta = np.radians(ang)
r1 = np.array([
[np.cos(theta),-np.sin(theta),0],
[np.sin(theta),np.cos(theta),0],
[0,0,1]]);
T = t2.dot(r1).dot(t1)
f1c = ia.affine(f1,T,0)
#Máxima correlação eentre f e f1
g = ia.phasecorr(f,f1c)
idx = np.argmax(g)
row,col = np.unravel_index(idx,g.shape)
t = np.array(f.shape) - np.array((row,col))
f1rt = ia.ptrans(f1c,-t)
return [g[row,col],f1rt]
FskiResz1 = np.fft.fft2(skimage_resize1) FskiResz2 = np.fft.fft2(skimage_resize2) FsciResz1 = np.fft.fft2(scipy_imresize1) FsciResz2 = np.fft.fft2(scipy_imresize2) FsciZoom1 = np.fft.fft2(scipy_zoom1) FsciZoom2 = np.fft.fft2(scipy_zoom2) # In[12]: # Plotando espectros nb = ia.nbshow(1) #nb.nbshow(f, 'Imagem original') nb.nbshow(ia.dftview(F), 'Espectro original') nb.nbshow() nb = ia.nbshow(2) #nb.nbshow(ia.normalize(skimage_rescale1),'skiResc x%s'%(scale1)) #nb.nbshow(ia.normalize(skimage_resize1),'skiResz x%s'%(scale1)) #nb.nbshow(ia.normalize(scipy_imresize1),'sciResz x%s'%(scale1)) #nb.nbshow(ia.normalize(scipy_zoom1),'sciZoom x%s'%(scale1)) nb.nbshow(ia.dftview(FskiResc1), 'skimage_rescale x%s' % (scale1)) nb.nbshow(ia.dftview(FskiResz1), 'skimage_resize x%s' % (scale1)) nb.nbshow(ia.dftview(FsciResz1), 'scipy_imresize x%s' % (scale1)) nb.nbshow(ia.dftview(FsciZoom1), 'scipy_Zoom x%s' % (scale1)) nb.nbshow()
sys.path.append(ia898path)
import ia898.src as ia
# ### Example 1
# In[2]:
if testing:
import matplotlib.image as mpimg
import numpy.fft as FFT
f = mpimg.imread('../data/cameraman.tif')
ia.adshow(f, "Original 2D image - Cameraman")
F = FFT.fft2(f)
Fv = ia.dftview(F)
ia.adshow(Fv, "Cameraman DFT optical spectrum")
# ## Equation
#
#
# $$ \begin{matrix}
# Gaux &=& \log(|F_{xc,yc}| + 1)\\xc &=& \lfloor W/2 \rfloor \\yc &=& \lfloor H/2 \rfloor\\ G &=& Gaux|_0^{255}
# \end{matrix} $$
# In[3]:
if testing:
print('testing dftview')
# 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
#
# $$ \begin{matrix}
# f(x) &=& \frac{1}{N}\sum_{u=0}^{N-1}F(u)\exp(j2\pi\frac{ux}{N}) \\ & & 0 \leq x < N, 0 \leq u < N \\ \mathbf{f} &=& \frac{1}{\sqrt{N}}(A_N)^* \mathbf{F}
# \end{matrix} $$
# $$ \begin{matrix}
# f(x,y) &=& \frac{1}{NM}\sum_{u=0}^{N-1}\sum_{v=0}^{M-1}F(u,v)\exp(j2\pi(\frac{ux}{N} + \frac{vy}{M})) \\ & & (0,0) \leq (x,y) < (N,M), (0,0) \leq (u,v) < (N,M) \\
# \mathbf{f} &=& \frac{1}{\sqrt{NM}} (A_N)^* \mathbf{F} (A_M)^*
# \end{matrix} $$
import matplotlib.image as mpimg
ia898path = os.path.abspath('../../')
if ia898path not in sys.path:
sys.path.append(ia898path)
import ia898.src as ia
# ### Example 1
# In[6]:
if testing:
f = ia.circle([120, 150], 6, [60, 75])
F = ia.dft(f)
Fs = ia.dftshift(F)
ia.adshow(ia.dftview(F))
ia.adshow(ia.dftview(Fs))
# In[7]:
if testing:
F = np.array([[10 + 6j, 20 + 5j, 30 + 4j], [40 + 3j, 50 + 2j, 60 + 1j]])
Fs = ia.dftshift(F)
print('Fs=\n', Fs)
# ## Equation
#
# $$ \begin{matrix}
# HS &=& H_{xo,yo} \\xo &=& \lfloor W/2 \rfloor \\yo &=& \lfloor H/2 \rfloor
# \end{matrix} $$
# ## Identificação do ângulo de rotação # ### Calcular a transformada polar da visualização da DFT # In[12]: F = np.fft.fft2(f) F1 = np.fft.fft2(f1) #nb = ia.nbshow(2) #nb.nbshow(ia.dftview(F),'F') #nb.nbshow(ia.dftview(F1),'F1') #nb.nbshow() F = ia.dftview(F) F1 = ia.dftview(F1) #H,W = F.shape #U,V = (120,120) #F = F[H//2-U:H//2+U,W//2-V:W//2+V] #F1 = F1[H//2-U:H//2+U,W//2-V:W//2+V] nb = ia.nbshow(2) nb.nbshow(F,'F') nb.nbshow(F1,'F1') nb.nbshow() Fp = ia.polar(F,F.shape) F1p = ia.polar(F1,F1.shape)
print(ia.isccsym(np.fft.fft2(np.random.rand(101,101))))
# ### Image Example: circular filter
# In[14]:
if testing:
img = mpimg.imread('../data/cameraman.tif')
F = ia.dft(img)
imgc = 1 * ia.circle(img.shape, 50, [img.shape[0]/2, img.shape[1]/2])
imgct = ia.ptrans(imgc, np.array(imgc.shape)//2)
ia.adshow(ia.normalize(imgct),'circular filter')
res = F * imgct
ia.adshow(ia.dftview(res))
print('Is this filter symmetric?', ia.isccsym(res))
# ### Image Example 2: retangular filter
# In[17]:
if False: # testing:
mquadra = ia.rectangle(img.shape, [50,50], [img.shape[0]/2, img.shape[1]/2])
ia.adshow(mquadra,'RETANGULO')
mquadra = ia.ptrans(mquadra, array(mquadra.shape)/2)
ia.adshow(ia.normalize(mquadra),'retangular filter')
mfiltrada = F * mquadra
g = imrsz.imresize(f, scale)
g2 = scipy.misc.imresize(f,scale)
print('f shape:', f.shape)
print('imresize shape:', g.shape)
print('scipy.imresize:', g2.shape)
nb = ia.nbshow(3)
nb.nbshow(f, 'Imagem original')
nb.nbshow(g, 'imresize' )
nb.nbshow(g2, 'scipy.imresize')
nb.nbshow()
nb = ia.nbshow(4)
nb.nbshow(ia.dftview(np.fft.fft2(f)) , 'Espectro original')
nb.nbshow(ia.dftview(np.fft.fft2(g)), 'imresize')
nb.nbshow(ia.dftview(np.fft.fft2(g2)), 'scipy.imresize')
nb.nbshow()
# In[ ]:
if testing:
# Testing integer
scale = 50
g = imrsz.imresize(f, scale)
g2 = scipy.misc.imresize(f,scale)
F = ia.dft(f) # proposed dft
F1 = np.fft.fftn(f) # numpy dft
print('ia.dft:', '\n', F.round(2), '\n')
print('fft.fftn:', '\n', F1.round(2), '\n')
print('Equal Results? (max error)', abs(F1 - F).max())
# ### Image example: 2d circle
# In[5]:
if testing:
f = ia.circle([256, 256], 10, [129, 129])
ia.adshow(f)
F = ia.dft(f)
Fv = ia.dftview(F)
ia.adshow(Fv)
# ### Image example: 3d square
# In[6]:
if False: #testing:
f = np.zeros((25, 30, 40))
f[10:15, 20:26, 21:27] = 1
F = ia.dft(f)
ia.adshow(ia.normalize(ia.mosaic(f, 5)), 'Original Image')
ia.adshow(ia.mosaic(ia.dftview(F), 5), 'Fourier Transformation')
# ### Comparison with other implementations
import sys,os
ia898path = os.path.abspath('../../')
if ia898path not in sys.path:
sys.path.append(ia898path)
import ia898.src as ia
# ### Example 1
# In[2]:
if testing:
H2_10 = ia.bwlp([100,100],2,2) # cutoff period: 2 pixels, order: 10
ia.adshow(ia.dftview(H2_10))
# In[3]:
if testing:
H4_1 = ia.bwlp([100,100],4,1) # cutoff period: 4, order: 1
ia.adshow(ia.dftview(H4_1))
# In[4]:
if testing:
H8_100 = ia.bwlp([100,100],8,100) # cutoff period: 8, order: 100
# Será necessário a função `comb` para poder fazer esta demonstração. # # Será necessário também ver outras propriedades da convolução e do teorema da convolução para poder fazer esta demonstração. Recomendo ler livros ou textos sobre esta explicação da interpretação no domínio da frequência da redução via interpolação. # In[15]: c = ia.comb((f.shape), (4, 4), (0, 0)) ia.adshow(ia.normalize(c)) # In[16]: C = np.fft.fft2(c) F = np.fft.fft2(f) G2 = ia.pconv(F, C) nb = ia.nbshow(3) nb.nbshow(ia.dftview(F), 'Espectro da imagem original') nb.nbshow(ia.dftview(C), 'Espectro do trêm de impulsos') nb.nbshow(ia.dftview(G2), 'Espectro resultante da convolução') nb.nbshow() # In[17]: g2 = np.fft.ifft2(G2) gReal = ia.normalize(g2.real) gMinified = gReal[::4, ::4] nb.nbshow(gReal, 'Imagem resultante') nb.nbshow(gMinified, 'Imagem reduzida') nb.nbshow() # In[18]:
