def filtroidealdemo(f,U,V):
'''
Inputs:
f - input image
U - vertical radius
V - horizontal radius
Output:
fh - filtered image
'''
F = np.fft.fft2(f)
H,W = F.shape
Fh = np.zeros_like(F)
r,c = np.indices((H,W))
Fh[( ((r-H//2)/U)**2 + ((c-W//2)/V)**2)<1]=1
Fh = ptr.ptransfat(Fh,(H//2,W//2))
fh = np.fft.ifft2(F * Fh)
plt.figure(1,figsize=(12,12))
plt.subplot(221)
plt.imshow(f,cmap='gray')
plt.title("f - imagem original")
plt.subplot(222)
plt.imshow(ptr.ptransfat(np.log(1+ np.abs(F)),(H//2,W//2)),cmap='gray')
plt.title("F - espectro da imagem")
plt.subplot(223)
plt.imshow(ptr.ptransfat(np.log(1+ np.abs(F*Fh)),(H//2,W//2)),cmap='gray')
plt.title("F*Fh - espectro filtrado")
plt.subplot(224)
plt.imshow(fh.real,cmap='gray')
plt.title("fh - imagem filtrada")
return fh
def dftrotationproperty(f, scale, angle):
'''
Inputs:
f - image
scale - image scale. It should be less than 1.
angle - angle in rads.
'''
r, c = f.shape
T = np.eye(3)
T = T * 0.5
T[0, 2] = r // 8
T[1, 2] = c // 8
T1 = np.array([[1, 0, r // 2], [0, 1, c // 2], [0, 0, 1]])
T2 = np.array([[np.cos(angle), -np.sin(angle), 0],
[np.sin(angle), np.cos(angle), 0], [0, 0, 1]])
T3 = np.array([[1, 0, -r // 2], [0, 1, -c // 2], [0, 0, 1]])
Tr = T1.dot(T2.dot(T3))
g = aff2.affine2(f, T)
gr = aff2.affine2(g, Tr)
F = np.fft.fft2(f)
G = np.fft.fft2(g)
GR = np.fft.fft2(gr)
fig, axes = plt.subplots(nrows=2, ncols=3, figsize=(12, 5))
im = axes[0, 0].imshow(f, cmap="gray")
im = axes[0, 1].imshow(g, cmap="gray")
im = axes[0, 2].imshow(gr, cmap="gray")
im = axes[1,
0].imshow(np.log(np.abs(ptr.ptransfat(F, (r // 2, c // 2))) + 1),
cmap='gray')
im = axes[1,
1].imshow(np.log(np.abs(ptr.ptransfat(G, (r // 2, c // 2))) + 1),
cmap='gray')
im = axes[1, 2].imshow(
np.log(np.abs(ptr.ptransfat(GR, (r // 2, c // 2))) + 1), cmap='gray')
fig.colorbar(im, ax=axes.ravel().tolist())
def dftscaleproperty(froi,scale):
Froi = np.fft.fft2(froi) # F is the DFT of f
r,c = froi.shape
froix = np.zeros(scale*np.array(froi.shape)) # size is 4 times larger
froix[::scale,::scale] = froi
Froix = np.fft.fft2(froix) # Fx4 is the DFT of fx4 (expanded f)
r2,c2 = froix.shape
plt.figure(1,figsize=(2,2))
plt.subplot(121)
plt.imshow(froi, cmap="gray")
plt.title('froi')
plt.subplot(122)
plt.imshow(np.log(np.abs(ptr.ptransfat(Froi,(r//2,c//2)))+1),cmap='gray')
plt.title('Froi')
plt.figure(2,figsize=(2*scale,2*scale))
plt.subplot(121)
plt.imshow(froix, cmap="gray")
plt.title('froix{}'.format(scale))
plt.subplot(122)
plt.imshow(np.log(np.abs(ptr.ptransfat(Froix,(r2//2,c2//2)))+1),cmap='gray')
plt.title('Froix{}'.format(scale))
def isccsymAlt(F):
H, W = F.shape
U = (H // 2)
V = (W // 2)
fq = F[:U + 1, :V + 1]
Ft = ptr.ptransfat(F, (-1, -1))
if (H % 2 == 0):
Ux = U - 1
else:
Ux = U
if (W % 2 == 0):
Vy = V - 1
else:
Vy = V
tq = np.conjugate(np.flipud(np.fliplr(Ft[Ux:, Vy:])))
sq = F[1:U + 1, -V:]
qq = np.conjugate(np.flipud(np.fliplr(F[-U:, 1:V + 1])))
A1 = ((abs(fq - tq)) < 10E-4).all()
A2 = ((abs(sq - qq)) < 10E-4).all()
return A1 & A2
# ## Examples
#
# ### Example 1
#
# #### Caso numérico
# In[15]:
if testing:
f = np.arange(25).reshape(5, 5)
t1 = (2, 0)
t2 = (0, 100003)
t3 = (-577, 333)
g1 = ptr.ptransfat(f, t1)
g2 = ptr.ptransfat(f, t2)
g3 = ptr.ptransfat(f, t2)
print("Imagem original:\n", f)
print("\nTranlação de (2,0):\n", g1)
print("\nTranslação de (0,100003):\n", g2)
print("\nTranslação de (-577,333):\n", g3)
# ### Example 2
#
# #### testes com imagens
if testing:
#Create image
f = np.zeros((256, 256))
H, W = f.shape
a, b = (64, 16)
f[H // 2 - a:H // 2 + a, W // 2 - b:W // 2 + b] = 255
f[H // 2 - b:H // 2 + b, W // 2 - a:W // 2 + a] = 255
F = np.fft.fft2(f)
fig, axes = plt.subplots(nrows=1, ncols=2, figsize=(12, 5))
axes[0].set_title('Imagem original')
im = axes[0].imshow(f, cmap="gray")
axes[1].set_title('Transformada de Fourier')
im = axes[1].imshow(np.log(np.abs(ptr.ptransfat(F, (H // 2, W // 2))) + 1),
cmap='gray')
fig.colorbar(im, ax=axes.ravel().tolist())
# In[7]:
if testing:
#rotate image
H, W = f.shape
angle = np.pi / 4
T1 = np.array([[1, 0, H // 2], [0, 1, W // 2], [0, 0, 1]])
T2 = np.array([[np.cos(angle), -np.sin(angle), 0],
[np.sin(angle), np.cos(angle), 0], [0, 0, 1]])
T3 = np.array([[1, 0, -H // 2], [0, 1, -W // 2], [0, 0, 1]])
fig.colorbar(im, ax=axes.ravel().tolist())
# The DFT of the ROI image is taken and its spectrum is displayed
# In[6]:
if testing:
F = np.fft.fft2(froi) # F is the DFT of f
r,c = froi.shape
fig, axes = plt.subplots(nrows=1, ncols=2, figsize=(10, 5))
axes[0].set_title('froi')
axes[1].set_title('spectrum')
im = axes[0].imshow(froi, cmap="gray")
im = axes[1].imshow(np.log(np.abs(ptr.ptransfat(F,(r//2,c//2)))+1),cmap='gray')
fig.colorbar(im, ax=axes.ravel().tolist())
# The image is expanded by 4, but filling the new pixels with 0
# In[7]:
if testing:
fx4 = np.zeros(4*np.array(froi.shape)) # size is 4 times larger
fx4[::4,::4] = froi # filling the expanded image
fig, axes = plt.subplots(nrows=1, ncols=2, figsize=(10, 5))
axes[0].set_title('froi')
axes[1].set_title('fx4')
im = axes[0].imshow(froi, cmap="gray")
sys.path.append(path) import ptrans as ptr # In[36]: Hf = np.zeros_like(F) Hfx = np.zeros_like(F) H, W = F.shape Hf[:H // (4), :W // (4)] = 1 #Ucut = H//(2*4) #frequência de corte em H #Vcut = W//(2*4) #frequência de corte em W #r,c = np.indices((H,W)) #Hfx[( ((r-H//2)/Ucut)**2 + ((c-W//2)/Vcut)**2)<1]=1 #Hfx = ptr.ptransfat(Hfx,(-H//(2),-W//(2))) Hf = ptr.ptransfat(Hf, (-H // (2 * 4), -W // (2 * 4))) Gfh = (F * Hf) gfh = np.fft.ifft2(Gfh) gfh = ia.normalize(gfh.real) gfh4 = gfh[::4, ::4] Gfh4 = np.fft.fft2(gfh4) nb = ia.nbshow(3) nb.nbshow(ia.dftview(F), 'Espectro original') nb.nbshow(ia.dftview(Hf), 'Filtro passa-baixo') nb.nbshow(ia.dftview(Gfh), 'Espectro Filtrado') nb.nbshow(gfh4, 'imagem resultante') nb.nbshow(ia.dftview(Gfh4), 'imagem resultante') nb.nbshow()