def toFinite(fareyVector, N):
'''
Return the finite vector corresponding to the Farey vector provided for a given modulus/length N
and the multiplicative inverse of the relevant Farey angle
Wraps function from Farey module
'''
return farey.toFinite(fareyVector, N)
def toDRT(projections, angles, N, P, Q, center=False):
'''
Convert the Mojette (asymetric) projection data to DRT (symetric) projections.
Use the iFRT to reconstruct the image. Requires N+1 or N+N/2 projections if N is prime or dyadic respectively.
Returns the resulting DRT space as a 2D array
'''
size = int(N + N/2)
dyadic = True
if N % 2 == 1: # if odd, assume prime
size = int(N+1)
dyadic = False
m = 0
frtSpace = np.zeros( (size,N) )
if dyadic:
print("Dyadic size not tested yet.")
#for each project
'''for index, proj in enumerate(projections):
p, q = farey.get_pq(angles[index])
m, inv = farey.toFinite(angles[index], N)
frtSpace[m][:] = finiteProjection(proj, angles[index], P, Q, N, center)'''
else: #prime size
for index, proj in enumerate(projections):
p, q = farey.get_pq(angles[index])
m, inv = farey.toFinite(angles[index], N)
frtSpace[m][:] = finiteProjection(proj, angles[index], P, Q, N, center)
return frtSpace
def finiteProjection(projection, angle, P, Q, N, center=False):
'''
Convert a Mojette projection taken at angle into a finite (FRT) projection.
'''
dyadic = True
if N % 2 == 1: # if odd, assume prime
dyadic = False
shiftQ = int(N / 2.0 + 0.5) - int(Q / 2.0 + 0.5)
shiftP = int(N / 2.0 + 0.5) - int(P / 2.0 + 0.5)
finiteProj = np.zeros(N)
p, q = farey.get_pq(angle)
m, inv = farey.toFinite(angle, N)
# print "p:", p, "q:", q, "m:", m, "inv:", inv
translateOffset, perp = farey.finiteTranslateOffset(angle, N, P, Q)
angleSign = p * q
if dyadic:
for translate, bin in enumerate(projection):
if angleSign >= 0 and perp: #Reverse for perp
translateMojette = translateOffset - translate
else:
translateMojette = translate - translateOffset
translateFinite = (inv * translateMojette) % N
if center:
translateFinite = (translateFinite + shiftQ + m *
(N - shiftP)) % N
finiteProj[translateFinite] += bin
else:
for translate, bin in enumerate(projection):
if angleSign >= 0 and perp: #Reverse for perp
translateMojette = int(translateOffset) - int(translate)
else:
translateMojette = int(translate) - int(translateOffset)
if translateMojette < 0:
translateFinite = (N - (inv * abs(translateMojette)) % N) % N
else:
translateFinite = (inv * translateMojette
) % N #has issues in C, may need checking
if center:
translateFinite = (translateFinite + shiftQ + m *
(N - shiftP)) % N
finiteProj[translateFinite] += bin
return finiteProj
def angleSetSliceCoordinates(angles, P, Q, N, center=False):
'''
Compute the 2D coordinates of each translate (in NxN DFT space) of every projection having angle in angles.
Returns a list of u, v coordinate arrays [[u_0[...],v_0[...]], [u_1[...],v_1[...]], ...] per angle
'''
coords = []
translateOffset = 0
translateMojette = 0
translateFinite = 0
m = 0
offset = 0.0
if center:
offset = N / 2.0
for index, angle in enumerate(angles):
u = []
v = []
coordinateList = []
p = int(angle.imag)
q = int(angle.real)
angleSign = p * q
m, inv = farey.toFinite(angle, N)
translateOffset, perp = farey.finiteTranslateOffset(angle, N)
B = projectionLength(angle, P, Q)
for translate in range(0, B):
if angleSign >= 0 and perp: #Reverse for perp
translateMojette = translateOffset - translate
else:
translateMojette = translate - translateOffset
translateFinite = (inv * translateMojette
) % N #has issues in C, may need checking
# frtSpace[m][translateFinite] += bin
u.append((translateFinite + offset) % N)
v.append((m * translateFinite + offset) % N)
coordinateList.append(u)
coordinateList.append(v)
coords.append(coordinateList)
return coords
def computeLines(kSpace, angles, centered=True, twoQuads=False):
'''
compute finite lines coordinates
Returns a list or list of slice 2-tuples and corresponding list of m values
'''
p, s = kSpace.shape
lines = []
mValues = []
for angle in angles:
m, inv = farey.toFinite(angle, p)
u, v = radon.getSliceCoordinates2(m, kSpace, centered, p)
lines.append((u, v))
mValues.append(m)
#second quadrant
if twoQuads:
if m != 0 and m != p: #dont repeat these
m = p - m
u, v = radon.getSliceCoordinates2(m, kSpace, centered, p)
lines.append((u, v))
mValues.append(m)
return lines, mValues
def createFractal4(reduction, N, proportion):
#parameters
M = 2*N
twoQuads = True
angles, lengths = mojette.angleSet_Symmetric(N,N,1,True,50)
perpAngle = farey.farey(1,0)
angles.append(perpAngle)
powerSpect = np.zeros((M,M))
#compute lines
centered = True
mLines = []
sLines = []
mValues = []
sValues = []
pValues = []
qValues = []
for angle in angles:
m, s, p, q, inv = farey.toFinite(angle, M)
pValues.append(p)
qValues.append(q)
if m not in mValues and m < M:
u, v = radon.getSliceCoordinates2(m, powerSpect, centered, M)
mLines.append((u,v))
mValues.append(m)
#second quadrant
if twoQuads:
if m != 0 and m != M: #dont repeat these
m = M-m
if m not in mValues and m < M:
u, v = radon.getSliceCoordinates2(m, powerSpect, centered, M)
mLines.append((u,v))
mValues.append(m)
i = (len(mValues)+len(sValues))/float(M)
if i >= 1:
break
ss = []
ss.append(M)
ss.extend(range(0, N/2))
maxLines = len(ss)
for i,s in enumerate(ss):
u, v = radon.getSliceCoordinates2(s, powerSpect, centered, M)
sLines.append((u,v))
sValues.append(s)
i = (len(mValues)+len(sValues))/float(M)
if i >= 1.5:
break
length = 0
fig, ax = plt.subplots(nrows=1, ncols=2, figsize=(16, 8))
plt.gray()
plt.tight_layout()
maxLines = len(sLines+mLines)
i = 0
ax[0].imshow(powerSpect)
ax[1].imshow(powerSpect)
color=iter(cm.jet(np.linspace(0,1,maxLines+1)))
fareyImage = np.zeros_like(powerSpect)
fareyImage1 = np.zeros_like(powerSpect)
for i, sLine in enumerate(sLines):
u, v = sLine
ax[1].plot(u, v, '.w',markersize=1)
fareyImage[u,v] = 1
length = length + 1
i = np.count_nonzero(fareyImage)/float((M*M))
if i >= reduction*proportion:
break
maxLines = len(mLines)
for i, mLine in enumerate(mLines):
u, v = mLine
ax[0].plot(u, v, '.r', markersize=1)
ax[1].plot(u, v, '.r',markersize=1)
fareyImage[u,v] = 1
fareyImage1[u,v] = 1
length = length + 1
i = np.count_nonzero(fareyImage)/float((M*M))
if i >= reduction:
break
print("Proportion of M:", (length/float(M)))
print("Non-zero elements with holes: ", np.count_nonzero(fareyImage1)/float((M*M)) * 100)
print("Non-zero elements without holes: ", np.count_nonzero(fareyImage)/float((M*M)) * 100)
print("Absolute difference percentage extra filled in is ", (np.count_nonzero(fareyImage)- np.count_nonzero(fareyImage1))/float((M*M)) *100)
withHoles = np.count_nonzero(fareyImage1)/float((M*M)) * 100
withoutHoles = np.count_nonzero(fareyImage)/float((M*M)) * 100
percentage = (withoutHoles - withHoles)/float(withHoles) * 100
print("Percentage difference percentage extra filled in is ", percentage)
ax[0].set_title('Sampling (colour per line) for dyadic size:'+str(M))
ax[1].set_title('Sampling (same colour per line) for dyadic size:'+str(M))
imageio.imsave("farey_image_"+str(M)+"_"+".png", fareyImage)
plt.show()
lines = mLines + sLines
return fareyImage, lines
powerSpect = np.zeros((M,M))
#np.set_printoptions(threshold=np.nan)
#compute lines
print("Computing Finite lines...")
centered = True
lines = []
mValues = []
pValues = []
qValues = []
z=25
for angle in angles:
#m, inv = farey.toFinite(angle, p)
#u, v = radon.getSliceCoordinates2(m, powerSpect, centered, p)
m, p, q, inv = farey.toFinite(angle, M)
u, v = radon.getSliceCoordinates2(m, powerSpect, centered, M)
lines.append((u,v))
mValues.append(m)
pValues.append(p)
qValues.append(q)
#second quadrant
if twoQuads:
#if m != 0 and m != p: #dont repeat these
if m != 0 and m != M: #dont repeat these
#m = p-m
#u, v = radon.getSliceCoordinates2(m, powerSpect, centered, p)
m = M-m
z = M-2*z
u, v = radon.getSliceCoordinates2(m, powerSpect, centered, M)
angles.append(perpAngle)
print("Number of Angles:", len(angles))
print("angles:", angles)
powerSpect = np.zeros((p, p))
#compute lines
print("Computing Finite lines...")
centered = True
lines = []
mValues = []
pValues = []
qValues = []
for angle in angles:
m, p1, q, inv = farey.toFinite(angle, p)
u, v = radon.getSliceCoordinates2(m, powerSpect, centered, p)
lines.append((u, v))
mValues.append(m)
pValues.append(p1)
qValues.append(q)
#second quadrant
if twoQuads:
if m != 0 and m != p: #dont repeat these
m = p - m
u, v = radon.getSliceCoordinates2(m, powerSpect, centered, p)
lines.append((u, v))
mValues.append(m)
mu = len(lines)
print("Number of lines:", len(lines))
print("Proportion of p:", len(lines) / float(p))
#angles = mojette.angleSet_Finite(pDash, 2)
angles, lengths = mojette.angleSet_Symmetric(N, N, 1, True, K)
perpAngle = farey.farey(1, 0)
angles.append(perpAngle)
print("Number of Angles:", len(angles))
print("angles:", angles)
powerSpect = np.zeros((p, p))
#compute lines
print("Computing Finite lines...")
centered = True
lines = []
mValues = []
for angle in angles:
m, inv = farey.toFinite(angle, p)
u, v = radon.getSliceCoordinates2(m, powerSpect, centered, p)
lines.append((u, v))
mValues.append(m)
#second quadrant
if twoQuads:
if m != 0 and m != p: #dont repeat these
m = p - m
u, v = radon.getSliceCoordinates2(m, powerSpect, centered, p)
lines.append((u, v))
mValues.append(m)
mu = len(lines)
print("Number of lines:", len(lines))
print("Proportion of p:", len(lines) / float(p))
print("Proportion of 2D space:", 2.0 - len(lines) / float(p))
print(mValues)
''' Test for the pure independent modules for number theory ''' import _libpath #add custom libs import finitetransform.numbertheory as nt import finitetransform.farey as farey x = 5 m = 641 ########## # test the multiplicative inverse mod m inv = nt.minverse(x, m) print inv identity = (inv * x) % m print identity ########## # test the Farey finite vector function vector = farey.farey(1, 2) m = farey.toFinite(vector, m) print vector, "->", m
