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
# In[7]:
if testing:
import matplotlib.image as mpimg
f = mpimg.imread('../data/cameraman.tif')
get_ipython().magic('time F1 = ia.dft(f)')
get_ipython().magic('time F2 = np.fft.fft2(f)')
print('Max difference is:', np.abs(F1 - F2).max())
# ## Equation
if testing:
import numpy as np
import sys, os
ea979path = os.path.abspath('../../')
if ea979path not in sys.path:
sys.path.append(ea979path)
import ea979.src as ia
# ### Example 1
# In[2]:
if testing:
f = np.array([100., 500., 1000.])
g1 = ia.normalize(f, [0, 255])
print(g1)
# In[3]:
if testing:
g2 = ia.normalize(f, [-1, 1])
print(g2)
# In[4]:
if testing:
g3 = ia.normalize(f, [0, 1])
print(g3)
# In[5]:
print('m:\n', m)
print('t:\n', t)
# ### Image examples
# ### Example 1.
# In[3]:
if testing:
f = mpimg.imread('../data/cameraman.tif')
(g, a) = ia.sobel(f)
nb = ia.nbshow(2)
nb.nbshow(ia.normalize(g), title='Sobel')
nb.nbshow(ia.normalize(np.log(g + 1)), title='Log of sobel')
nb.nbshow()
# ### Example 2.
# In[4]:
if testing:
f = ia.circle([200, 300], 90, [100, 150])
m, t = ia.sobel(f)
dt = np.select([m > 2], [t])
nb = ia.nbshow(3)
nb.nbshow(f, title='Image f')
# In[2]:
if testing:
get_ipython().magic('matplotlib inline')
import matplotlib.pyplot as plt
import matplotlib.image as mpimg
import sys, os
ea979path = os.path.abspath('../../')
if ea979path not in sys.path:
sys.path.append(ea979path)
import ea979.src as ia
r, c = np.indices((256, 256))
f = ((r - 129)**2 + (c - 129)**2 < 10**2) * 255
ia.adshow(ia.normalize(f), 'Imagem original')
F = ia.dct(f)
ia.adshow(ia.normalize(np.log(abs(F) + 1)), 'DCT')
# ### Example 3
# Compare with dft
# In[3]:
if testing:
f = mpimg.imread('../data/cameraman.tif')
nb = ia.nbshow(3)
nb.nbshow(f, 'Imagem original')
def dftview(F):
import ea979.src as ia
FM = ia.dftshift(np.log(np.abs(F) + 1))
return ia.normalize(FM).astype(np.uint8)
if testing:
F = ia.ramp([5,7], 3, [4,10])
print(F)
F = ia.ramp((1,5,7),(0,3,0), [0,0,4,10,0,0])
print(F)
F = ia.ramp([1,5,7],[3,0,0], [4,10,0,0,0,0])
print(F)
# - **Image example**
# In[4]:
if testing:
F = ia.ramp([1,200,300], [0,10,0], [0,0,0,255,0,128])
ia.adshow(ia.normalize(F.reshape(200,300)))
# In[6]:
if testing:
F = ia.ramp([200,300], 10, [0,255])
ia.adshow(ia.normalize(F))
# In[7]:
if testing:
F = ia.ramp([1,200,300], [10,0,0], [0,255,0,0,0,0])
ia.adshow(ia.normalize(F.reshape(200,300)))
if testing:
f = np.arange(24).reshape(4, 6)
F = ia.dft(f)
g = ia.idft(F)
print(np.round(g.real))
# 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
#
# $$ \begin{matrix}
# f(x) &=& \frac{1}{N}\sum_{u=0}^{N-1}F(u)\exp(j2\pi\frac{ux}{N}) \\ & & 0 \leq x < N, 0 \leq u < N \\ \mathbf{f} &=& \frac{1}{\sqrt{N}}(A_N)^* \mathbf{F}
# \end{matrix} $$
print("\nVisualiza propriedade A*A'= I:\n", B)
# ### Example 2
# In[2]:
if testing:
get_ipython().magic('matplotlib inline')
import matplotlib.pyplot as plt
import matplotlib.image as mpimg
import sys,os
ea979path = os.path.abspath('../../')
if ea979path not in sys.path:
sys.path.append(ea979path)
import ea979.src as ia
A = ia.dctmatrix(128)
ia.adshow(ia.normalize(A,[0,255]),'DCT 128x128')
# In[ ]:
# In[ ]:
a = ia.conv(f, h)
print(a)
# ### Example 4
# In[5]:
if testing:
get_ipython().magic('matplotlib inline')
import matplotlib.pyplot as plt
import matplotlib.image as mpimg
f = mpimg.imread('../data/cameraman.tif')
h = np.array([[1, 2, 1], [0, 0, 0], [-1, -2, -1]])
g = ia.conv(f, h)
gn = ia.normalize(g, [0, 255])
ia.adshow(f, title='input')
ia.adshow(gn, title='filtered')
# ## Limitations
#
# Both image and kernel are internally converted to double.
# ## Equation
#
# $$ \begin{matrix}
# (f \ast h)(r,c) &=& \sum_{i=0}^{H-1} \sum_{j=0}^{W-1} f_{e}(i,j) h_{e}(r-i, c-j) \\
# f_{e}(r,c) &=& \left\{ \begin{array}{llcl} f(r,c), & 0 \leq r \leq H_{f}-1 & and & 0 \leq r \leq W_f-1 \\
# 0, & H_f \leq r \leq H-1 & or & W_f \leq c \leq W-1 \end{array}\right.\\
# h_{e}(r,c) &=& \left\{ \begin{array}{llcl} f(r,c), & 0 \leq r \leq H_{h}-1 & and & 0 \leq r \leq W_h-1 \\
# 0, & H_h \leq r \leq H-1 & or & W_h \leq c \leq W-1 \end{array}\right.\\
print('image dimensions = ', s)
print('center of function = ', mu)
print('spread factor =', sigma)
print('Laplacian of Gaussian image : \n', F.round(2))
# #### Generating a image 2D 128x128, centered at 64x64 and sigma 4:
# In[5]:
if testing:
s, mu, sigma = [128, 128], [64, 64], 4
F = ia.log(s, mu, sigma)
print('image dimensions = ', s)
print('center of function = ', mu)
print('spread factor =', sigma)
ia.adshow(ia.normalize(F), 'Laplacian of Gaussian')
# #### Generating a image 2D 256x256, centered at 128x128 and sigma 20
# In[6]:
if testing:
s, mu, sigma = [256, 256], [128, 128], 20
F = ia.log(s, mu, sigma)
print('image dimensions = ', s)
print('center of function = ', mu)
print('spread factor =', sigma)
ia.adshow(ia.normalize(F), 'Laplacian of Gaussian')
# ## Measuring time:
ea979path = os.path.abspath('../../')
if ea979path not in sys.path:
sys.path.append(ea979path)
import ea979.src as ia
get_ipython().magic('matplotlib inline')
import matplotlib.pyplot as plt
# ### Example 1 - Numeric 2-dimensional
# In[2]:
if testing:
f = ia.gaussian((8, 4), np.transpose([[3, 1]]), [[1, 0], [0, 1]])
print('f=\n', np.array2string(f, precision=4, suppress_small=1))
g = ia.normalize(f, [0, 255]).astype(np.uint8)
print('g=\n', g)
# ## Example 2 - one dimensional signal
# In[3]:
# note that for 1-D case, the tuple has extra ,
# and the covariance matrix must be 2-D
if testing:
f = ia.gaussian((100, ), 50, [[10 * 10]])
g = ia.normalize(f, [0, 1])
plt.plot(g)
plt.show()
# ### Example 3 - two-dimensional image
print('Is this function symmetric?')
print(ia.isccsym(np.fft.fft2(np.random.rand(100,
100)))) # dimension variation
print(ia.isccsym(np.fft.fft2(np.random.rand(101, 100))))
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
import matplotlib.image as mpimg
get_ipython().magic('matplotlib inline')
# ### Example 1
# In[3]:
if testing:
f = mpimg.imread('../data/cameraman.tif')
F = ia.hadamard(f)
g = ia.ihadamard(F)
print(sum(sum(abs(f.astype(float) - g.astype(float)))))
nb = ia.nbshow(3)
nb.nbshow(f, 'Original')
nb.nbshow(ia.normalize(F), 'hadamard of f')
nb.nbshow(ia.normalize(g), 'ihadamard of F')
nb.nbshow()
# ## Measuring time:
# In[4]:
if testing:
f = mpimg.imread('../data/cameraman.tif')
F = ia.hadamard(f)
print('Computational time is:')
get_ipython().magic('timeit ia.ihadamard(F)')
# In[ ]:
import ea979.src as ia
# ### Example 1
# In[2]:
if testing:
F = ia.rectangle([7, 9], [3, 2], [3, 4])
print(F)
# - **Example 2**
# In[3]:
if testing:
F = ia.rectangle([200, 300], [90, 120], [70, 120])
ia.adshow(ia.normalize(F))
# ## Equation
#
# \begin{equation}
# g(x,y)=\begin{cases}
# 1, & \text{if } x_\text{min} \leq x < x_\text{max} \text{ and } y_\text{min} \leq y < y_\text{max}.\\
# 0, & \text{otherwise}.
# \end{cases}
# \end{equation}
# ## Contributions
#
# Lucas de Vasconcellos Teixeira, 1st semester 2017
print("\n Image Output - (G): ")
print(result)
# ## Example with Image 2D
# In[6]:
if testing:
f = mpimg.imread('../data/cameraman.tif')
ia.adshow(f, title='a) - Original Image')
h = np.array([[-1, -1, -1], [0, 0, 0], [1, 1, 1]])
g = ia.pconv(f, h)
print("\nPrewitt´s Mask")
print(h)
gn = ia.normalize(g, [0, 255])
ia.adshow(gn, title='b) Prewitt´s Mask filtering')
ia.adshow(ia.normalize(abs(g)),
title='c) absolute of Prewitt´s Mask filtering')
# ## Equation
#
# $$ f(i) = f(i + kN), h(i)=h(i+kN)$$
#
# $$ mod(i,N) = i - N \lfloor \frac{i}{N} \rfloor $$
#
# $$ (f \ast_W h) (col) = \sum_{cc=0}^{W-1} f(mod(col-cc,W)) h(cc)$$
#
# $$ (f \ast_{(H,W)} h) (row,col) = \sum_{rr=0}^{H-1} \sum_{cc=0}^{W-1} f(mod(row-rr,H), mod(col-cc,W)) h(rr,cc)$$
#
get_ipython().system(' jupyter nbconvert --to python cos.ipynb')
import numpy as np
import sys, os
ea979path = os.path.abspath('../../')
if ea979path not in sys.path:
sys.path.append(ea979path)
import ea979.src as ia
# ### Example 1
# In[2]:
if testing:
f = ia.cos([128, 256], 100, np.pi / 3, 0)
ia.adshow(ia.normalize(f, [0, 255]))
# ## Equation
#
# $$ \begin{matrix}
# f(r,c) & = & cos( 2\pi (\frac{1}{T_r}r + \frac{1}{T_c}c) + \phi) \\
# f(r,c) & = & cos( 2\pi (\frac{v}{H}r + \frac{u}{W}c) + \phi) \\
# T_r & = & \frac{T}{sin(\theta)} \\
# T_c & = & \frac{T}{cos(\theta)} \\
# u & = & \frac{W}{T_c} \\
# v & = & \frac{H}{T_r}
# \end{matrix} $$
#
# - $\theta$ is the direction of the cosine wave.
# - $T$ is the wave period, in number of pixels.
# - $T_r$ and $T_c$ are the period or wave length in the vertical and horizontal directions, respectively, in number of pixels.
s = ia.sat(f)
print('f (input):\n', f)
print('s (output):\n', s)
a = ia.satarea(s, (0, 0), (3, 8))
print('area:', a)
# ### Image example
#
# In[3]:
if testing:
f = mpimg.imread('../data/lenina.pgm')[::2, ::2]
nb = ia.nbshow(2)
nb.nbshow(f, 'Original Image')
nb.nbshow(ia.normalize(ia.sat(f)), 'Integral Image')
nb.nbshow()
# ### Calculating a rectangle area with SAT (Summed Area Table)
# In[4]:
if testing:
f = mpimg.imread('../data/lenina.pgm')[::2, ::2]
H, W = f.shape
s = ia.sat(f)
a0 = ia.satarea(s, (0, 0), (H - 1, W - 1))
atopleft = ia.satarea(s, (0, 0), (H // 2 - 1, W // 2 - 1))
abotleft = ia.satarea(s, (H // 2, 0), (H - 1, W // 2 - 1))
atopright = ia.satarea(s, (0, W // 2), (H // 2 - 1, W - 1))
abotright = ia.satarea(s, (H // 2, W // 2), (H - 1, W - 1))
# ## Examples
# In[1]:
testing = (__name__ == "__main__")
if testing:
get_ipython().system(' jupyter nbconvert --to python isolines.ipynb')
import numpy as np
import sys, os
ea979path = os.path.abspath('../../')
if ea979path not in sys.path:
sys.path.append(ea979path)
import ea979.src as ia
# ### Example 1
# In[2]:
if testing:
f = ia.normalize(ia.bwlp([150, 150], 4, 1), [0, 255])
f = f.astype('uint8')
g = ia.isolines(f, 10, 3)
g = g.astype('uint8')
ia.adshow(f)
ia.adshow(g)
# In[ ]:
if testing:
get_ipython().system(' jupyter nbconvert --to python hadamard.ipynb')
import numpy as np
import sys, os
import matplotlib.image as mpimg
ea979path = os.path.abspath('../../')
if ea979path not in sys.path:
sys.path.append(ea979path)
import ea979.src as ia
# In[3]:
if testing:
f = mpimg.imread('../data/cameraman.tif')
F = ia.hadamard(f)
nb = ia.nbshow(2)
nb.nbshow(f)
nb.nbshow(ia.normalize(np.log(abs(F) + 1)))
nb.nbshow()
# ## Measuring time:
# In[ ]:
if testing:
f = mpimg.imread('../data/cameraman.tif')
print('Computational time is:')
get_ipython().magic('timeit ia.hadamard(f)')
# In[ ]:
# ### Example 1
# In[3]:
if testing:
get_ipython().magic('matplotlib inline')
import matplotlib.pyplot as plt
import matplotlib.image as mpimg
f = mpimg.imread('../data/cameraman.tif')
g07 = ia.logfilter(f, 0.7)
nb = ia.nbshow(3)
nb.nbshow(f, 'Imagem original')
nb.nbshow(ia.normalize(g07), 'LoG filter')
nb.nbshow(g07 > 0, 'positive values')
nb.nbshow()
# ### Example 2
# In[4]:
if testing:
g5 = ia.logfilter(f, 5)
g10 = ia.logfilter(f, 10)
nb = ia.nbshow(2, 2)
nb.nbshow(ia.normalize(g5), 'sigma=5')
nb.nbshow(g5 > 0, 'positive, sigma=5')
nb.nbshow(ia.normalize(g10), 'sigma=10')
if testing:
f = mpimg.imread('../data/cameraman.tif')
g = ia.gshow(f, f>230)
nb = ia.nbshow(2)
nb.nbshow(f,'Original image')
nb.nbshow(g,'Pixels with values above 230 are shown in red')
nb.nbshow()
# In[4]:
if testing:
f = ia.gaussian((256,256), np.transpose([[128,128]]), [[50*50,0],[0,80*80]])
fn = ia.normalize(f, [0,255])
f1 = ia.gshow(fn, fn > 20, fn > 30, fn > 40, fn > 50, fn > 60, fn > 70)
nb = ia.nbshow(2)
nb.nbshow(fn,'Original image')
nb.nbshow(f1,'Overlaid image')
nb.nbshow()
# ## Reference
# - [Function iagshow](http://adessowiki.fee.unicamp.br/adesso/wiki/ia636/iagshow/view/)
# ## Contributions
if testing:
np.set_printoptions(suppress=True, precision=4)
A = ia.haarmatrix(4)
print('Visualiza matriz haar 4x4:\n',A)
B = np.dot(A,np.transpose(A))
print("\nVisualiza propriedade A*A'= I:\n", B)
# ### Example 2
# In[3]:
if testing:
A = ia.haarmatrix(128)
ia.adshow(ia.normalize(A),'Haar matrix 128x128')
# ### Example 3
# In[4]:
if testing:
f = mpimg.imread('../data/cameraman.tif')
A = ia.haarmatrix(f.shape[0])
B = ia.haarmatrix(f.shape[1])
F = np.dot(np.dot(A, f), np.transpose(B))
nb = ia.nbshow(2)
nb.nbshow(f,'Imagem original')
get_ipython().magic('matplotlib inline')
import matplotlib.image as mpimg
# ### Example 1
# Show that the point of maximum correlation for two equal images is the origin.
#
#
# In[4]:
if testing:
# 2D example
f1 = mpimg.imread("../data/cameraman.tif")
noise = np.random.rand(f1.shape[0], f1.shape[1])
f2 = ia.normalize(ia.ptrans(f1, (-1, 50)) + 300 * noise)
g1 = ia.phasecorr(f1, f2)
i = np.argmax(g1)
row, col = np.unravel_index(i, g1.shape)
v = g1[row, col]
print(np.array(f1.shape) - np.array((row, col)))
# In[ ]:
if testing:
print('max at:(%d, %d)' % (row, col))
ia.adshow(ia.normalize(f1), "input image")
ia.adshow(ia.normalize(f2), "input image")
ia.adshow(ia.normalize(g1),
"Correlation peak at (%d,%d) with %d" % (row, col, v))