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
'''
p, q = get_pq(fareyVector)
coprime = nt.is_coprime(abs(q), N)
prime = 1
if N % 2 == 0: #dyadic
prime = 2
qNeg = q #important, if q < 0, preserve for minverse.
if q < 0:
q += N #necessary for quadrants other than the first
if p < 0:
p += N #necessary for quadrants other than the first
# print("p:", p, "q:", q, "N:", N)
mValue = 0
inv = 1
if coprime:
inv = nt.minverse(qNeg, N)
# identity = (inv*q)%N
mValue = (p * inv) % N
# print("vec:", fareyVector, "m:", mValue, "inv:", inv)
else: #perp projection
inv = nt.minverse(p, N)
# mValue = (q*inv)%N + N
mValue = int((q * inv) % N / prime) + N
# print("perp vec:", fareyVector, "m:", mValue, "inv:", inv)
return mValue, inv
def oversamplingFilter(n, M, p = 2):
'''
Produce the nxn oversampling filter that is needed to exactly filter dyadic DRT etc.
This version returns values as multiplicative inverses (mod M) for use with the NTT
'''
gcd_table = np.zeros(n)
gcdInv_table = np.zeros(n)
gcd_table[0] = n + int(n)/p
gcdInv_table[0] = nt.minverse(gcd_table[0], M)
gcd_table[1] = 1
gcdInv_table[1] = nt.minverse(gcd_table[1], M)
for j in range(2,n):
u, v, gcd_table[j] = nt.extended_gcd( int(j), int(n) )
gcdInv_table[j] = nt.minverse(gcd_table[j], M)
filter = nt.zeros((n,n))
for j in range(0,n):
for k in range(0,n):
if gcd_table[j] < gcd_table[k]:
filter[j,k] = nt.integer(gcdInv_table[j])
else:
filter[j,k] = nt.integer(gcdInv_table[k])
return filter
def remapProjections(p, rotation):
'''
For a given p size, a Farey set is generated and rotated by rotation vector provided.
The new mapping is returned, with their corresponding new Farey vectors.
'''
original_angles = []
rotated_angles = []
ms = []
inverses = []
m_dashs = []
for m in range(p): #add the Farey angles corresponding to m values
ms.append(m)
#~ angle = complex(1.0, m)
angle = complex(m, 1)
original_angles.append(angle)
ms.append(p) #add the pth projection
#~ original_angles.append( complex(0.0, 1.0) )
original_angles.append(complex(1.0, 0.0))
for angle in original_angles:
rotated_angle = angle * rotation
#~ rotated_angle = complex(angle.imag*rotation.imag - angle.real*rotation.real, angle.imag*rotation.real + rotation.imag*angle.real) #Imants' coordinate system
#~ rotated_angle = complex(angle.imag*rotation.real + rotation.imag*angle.real, angle.imag*rotation.imag - angle.real*rotation.real)
#~ rotated_angle = rotation*angle
'''if abs(rotated_angle.real) >= p or abs(rotated_angle.imag) >= p: # only for coomparison, since negative values would be lost otherwise
rotated_angle_rational = complex( int(rotated_angle.real)%p, int(rotated_angle.imag)%p )
else:
rotated_angle_rational = rotated_angle'''
rotated_angle_rational = complex(
int(rotated_angle.real + p) % p,
int(rotated_angle.imag + p) % p)
rotated_angles.append(rotated_angle_rational)
#compute m'
#~ u = nt.minverse(rotated_angle_rational.real, p)
u = nt.minverse(rotated_angle_rational.imag, p)
#~ inverse = (u*int(rotated_angle_rational.real))%p
inverse = (u * int(rotated_angle_rational.imag)) % p
inverses.append(inverse)
if inverse > 1:
print("MInverse Failed")
elif inverse == 0:
m_dash = p
else:
#~ m_dash = ( u * int(rotated_angle_rational.imag) )%p
m_dash = (u * int(rotated_angle_rational.real)) % p
m_dashs.append(m_dash)
#~ print original_angles
#~ print rotated_angles
#~ print inverses
#~ print ms
#~ print m_dashs
return m_dashs, rotated_angles
def oversampling_1D_filter_Integer(n, M, p = 2, norm = False):
'''
The 1D filter for removing oversampling. All values are multiplicative inverses.
To apply this filter multiply with 1D NTT slice
'''
gcd_table = np.zeros(n)
gcdInv_table = nt.zeros(n)
inv = nt.integer(1)
if norm:
inv = nt.integer(nt.minverse(int(n), M))
gcd_table[0] = n + int(n)/p
gcdInv_table[0] = nt.integer(nt.minverse(gcd_table[0], M))
gcd_table[1] = 1
gcdInv_table[1] = nt.integer(nt.minverse(gcd_table[1], M))
for j in range(2,n):
u, v, gcd_table[j] = nt.extended_gcd( int(j), int(n) )
gcdInv_table[j] = (nt.integer(nt.minverse(gcd_table[j], M))*inv)%M
return gcdInv_table
import farey
import numbertheory as nt
import numpy as np
import matplotlib.pyplot as plt
if __name__ == "__main__":
N = 17
a = 1
b = 10
lst_of_multiples = np.zeros((N, 2))
lst_of_thingos = np.zeros((N, 2))
# find grad
m = nt.minverse(a, N) * b
for i in range(0, N):
lst_of_multiples[i, :] = np.array([[(a * i) % N, (b * i) % N]])
lst_of_thingos[i, :] = np.array([[i % N, (m * i) % N]])
img1 = np.zeros((N, N))
img2 = np.zeros((N, N))
for i in range(N):
img1[int(lst_of_multiples[i, 0]), int(lst_of_multiples[i, 1])] = 1
img2[int(lst_of_thingos[i, 0]), int(lst_of_thingos[i, 1])] = 1
# Plot each
fig, axes = plt.subplots(1, 2)
