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import numpy as sb
pi = sb.pi
# fast discrete cosine transforms of real sequences (using the fft)
# These implement the DCT-II and inverse DCT-II (DCT-III)
# described at http://en.wikipedia.org/wiki/Discrete_cosine_transform
def dct(x,axis=-1):
"""Discrete cosine transform based on the FFT.
For even-length signals it uses an N-point FFT
For odd-length signals it uses a 2N-point FFT.
"""
n = len(x.shape)
N = x.shape[axis]
even = (N%2 == 0)
slices = [None]*4
for k in range(4):
slices[k] = []
for j in range(n):
slices[k].append(slice(None))
if even:
xtilde = 0.0*x
slices[0][axis] = slice(None,N/2)
slices[1][axis] = slice(None,None,2)
slices[2][axis] = slice(N/2,None)
slices[3][axis] = slice(N,None,-2)
else:
newshape = list(x.shape)
newshape[axis] = 2*N
xtilde = sb.empty(newshape,sb.float64)
slices[0][axis] = slice(None,N)
slices[2][axis] = slice(N,None)
slices[3][axis] = slice(None,None,-1)
for k in range(4):
slices[k] = tuple(slices[k])
xtilde[slices[0]] = x[slices[1]]
xtilde[slices[2]] = x[slices[3]]
Xt = sb.fft(xtilde,axis=axis)
pk = sb.exp(-1j*pi*sb.arange(N)/(2*N))
newshape = sb.ones(n)
newshape[axis] = N
pk.shape = newshape
if not even:
pk /= 2;
Xt = Xt[slices[0]]
return sb.real(Xt*pk)
def idct(v,axis=-1):
n = len(v.shape)
N = v.shape[axis]
even = (N%2 == 0)
slices = [None]*4
for k in range(4):
slices[k] = []
for j in range(n):
slices[k].append(slice(None))
k = sb.arange(N)
if even:
ak = sb.r_[1.0,[2]*(N-1)]*sb.exp(1j*pi*k/(2*N))
newshape = sb.ones(n)
newshape[axis] = N
ak.shape = newshape
xhat = sb.real(sb.ifft(v*ak,axis=axis))
x = 0.0*v
slices[0][axis] = slice(None,None,2)
slices[1][axis] = slice(None,N/2)
slices[2][axis] = slice(N,None,-2)
slices[3][axis] = slice(N/2,None)
for k in range(4):
slices[k] = tuple(slices[k])
x[slices[0]] = xhat[slices[1]]
x[slices[2]] = xhat[slices[3]]
return x
else:
ak = 2*sb.exp(1j*pi*k/(2*N))
newshape = sb.ones(n)
newshape[axis] = N
ak.shape = newshape
newshape = list(v.shape)
newshape[axis] = 2*N
Y = zeros(newshape,sb.Complex)
#Y[:N] = ak*v
#Y[(N+1):] = conj(Y[N:0:-1])
slices[0][axis] = slice(None,N)
slices[1][axis] = slice(None,None)
slices[2][axis] = slice(N+1,None)
slices[3][axis] = slice((N-1),0,-1)
Y[slices[0]] = ak*v
Y[slices[2]] = conj(Y[slices[3]])
x = sb.real(sb.ifft(Y,axis=axis))[slices[0]]
return x
def dct2(x,axes=(-1,-2)):
return dct(dct(x,axis=axes[0]),axis=axes[1])
def idct2(v,axes=(-1,-2)):
return idct(idct(v,axis=axes[0]),axis=axes[1])
def dctn(x,axes=None):
if axes is None:
axes = sb.arange(len(x.shape))
res = x
for k in axes:
res = dct(res,axis=k)
return res
def idctn(v,axes=None):
if axes is None:
axes = sb.arange(len(v.shape))
res = v
for k in axes:
res = idct(res,axis=k)
return res
def makeC(N):
n,l = ogrid[:N,:N]
C = sb.cos(pi*(2*n+1)*l/(2*N))
return C
def dct2raw(x):
M,N = x.shape
CM = makeC(M)
CN = makeC(N)
return dot(transpose(CM),dot(x,CN))
def idct2raw(v):
M,N = v.shape
iCM = linalg.inv(makeC(M))
iCN = linalg.inv(makeC(N))
return dot(transpose(iCM),dot(v,iCN))
def makeS(N):
n,k = ogrid[:N,:N]
C = sin(pi*(k+1)*(n+1)/(N+1))
return C
# DST-I
def dst(x,axis=-1):
"""Discrete Sine Transform (DST-I)
Implemented using 2(N+1)-point FFT
xsym = r_[0,x,0,-x[::-1]]
DST = (-imag(fft(xsym))/2)[1:(N+1)]
adjusted to work over an arbitrary axis for entire n-dim array
"""
n = len(x.shape)
N = x.shape[axis]
slices = [None]*3
for k in range(3):
slices[k] = []
for j in range(n):
slices[k].append(slice(None))
newshape = list(x.shape)
newshape[axis] = 2*(N+1)
xtilde = sb.zeros(newshape,sb.float64)
slices[0][axis] = slice(1,N+1)
slices[1][axis] = slice(N+2,None)
slices[2][axis] = slice(None,None,-1)
for k in range(3):
slices[k] = tuple(slices[k])
xtilde[slices[0]] = x
xtilde[slices[1]] = -x[slices[2]]
Xt = sb.fft(xtilde,axis=axis)
return (-sb.imag(Xt)/2)[slices[0]]
def idst(v,axis=-1):
n = len(v.shape)
N = v.shape[axis]
slices = [None]*3
for k in range(3):
slices[k] = []
for j in range(n):
slices[k].append(slice(None))
newshape = list(v.shape)
newshape[axis] = 2*(N+1)
Xt = sb.zeros(newshape,sb.Complex)
slices[0][axis] = slice(1,N+1)
slices[1][axis] = slice(N+2,None)
slices[2][axis] = slice(None,None,-1)
val = 2j*v
for k in range(3):
slices[k] = tuple(slices[k])
Xt[slices[0]] = -val
Xt[slices[1]] = val[slices[2]]
xhat = sb.real(sb.ifft(Xt,axis=axis))
return xhat[slices[0]]
def dst2(x,axes=(-1,-2)):
return dst(dst(x,axis=axes[0]),axis=axes[1])
def idst2(v,axes=(-1,-2)):
return idst(idst(v,axis=axes[0]),axis=axes[1])
def dstn(x,axes=None):
if axes is None:
axes = sb.arange(len(x.shape))
res = x
for k in axes:
res = dst(res,axis=k)
return res
def idstn(v,axes=None):
if axes is None:
axes = sb.arange(len(v.shape))
res = v
for k in axes:
res = idst(res,axis=k)
return res
def digitrevorder(x,base):
x = sb.asarray(x)
rem = N = len(x)
L = 0
while 1:
if rem < base:
break
intd = rem // base
if base*intd != rem:
raise ValueError, "Length of data must be power of base."
rem = intd
L += 1
vec = r_[[base**n for n in range(L)]]
newx = x[sb.newaxis,:]*vec[:,sb.newaxis]
# compute digits
for k in range(L-1,-1,-1):
newx[k] = x // vec[k]
x = x - newx[k]*vec[k]
# reverse digits
newx = newx[::-1,:]
x = 0*x
# construct new result from reversed digts
for k in range(L):
x += newx[k]*vec[k]
return x
def bitrevorder(x):
return digitrevorder(x,2)
# needs to be fixed
def wht(data):
"""Walsh-Hadamaard Transform (sequency ordered)
adapted from MATLAB algorithm published on the web by
Author: Gylson Thomas
e-mail: gylson_thomas@yahoo.com
Asst. Professor, Electrical and Electronics Engineering Dept.
MES College of Engineering Kuttippuram,
Kerala, India, February 2005.
copyright 2005.
Reference: N.Ahmed, K.R. Rao, "Orthogonal Transformations for
Digital Signal Processing" Spring Verlag, New York 1975. page-111.
"""
N = len(data)
L = sb.log2(N);
if ((L-sb.floor(L)) > 0.0):
raise ValueError, "Length must be power of 2"
x=bitrevorder(data);
k1=N; k2=1; k3=N/2;
for i1 in range(1,L+1): #Iteration stage
L1=1;
for i2 in range(1,k2+1):
for i3 in range(1,k3+1):
i=i3+L1-1; j=i+k3;
temp1= x[i-1]; temp2 = x[j-1];
if (i2 % 2) == 0:
x[i-1] = temp1 - temp2;
x[j-1] = temp1 + temp2;
x[i-1] = temp1 + temp2;
x[j-1] = temp1 - temp2;
L1=L1+k1;
k1 = k1/2; k2 = k2*2; k3 = k3/2;
x = x*1.0/N; # Delete this line for inverse wht
return x
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