## A512_x

'''
1. Complete L507 by defining D4T(f,r=1) that implements
the Daub4 = D4 Transform using D4V and D4W (goto L155)
'''
from cdi import *
from math import *
import numpy as np
import matplotlib.pyplot as plt

# Synonyms
Id = np.eye
sqrt = np.sqrt
array = np.array
stack = np.vstack
splice= np.hstack
dot = np.dot
ls  = np.linspace
zeros=np.zeros
mat=np.matrix

cos = np.cos
sin = np.sin
array = np.array
stack = np.vstack
splice= np.hstack
dot = np.dot
ls  = np.linspace
zeros=np.zeros
mat=np.matrix
transpose=np.transpose
det = np.linalg.det
inv = np.linalg.inv

canvas = plt.figure
axis=plt.axis
show = plt.show
view = plt.imshow
close = plt.close
plot = plt.plot
xlim = plt.xlim

# Basic constants
nd = 4
r2=sqrt(2); r3=sqrt(3);
a1=(1+r3)/(4*r2); a2=(3+r3)/(4*r2); 
a3=(3-r3)/(4*r2); a4=(1-r3)/(4*r2);
[b1,b2,b3,b4] = [a4,-a3,a2,-a1]

'''
I) D4trend, D4fluct, D4
'''

def D4trend_rec(f, r=1):
    N = len(f)
    f = list(f)
    if r == 0: return array(f)
    if N % 2**r: 
        return "D4trend_rec: %d is not divisible by 2**%d " % (N, r)
    if r == 1:
        f = f + f[:2]
        return \
            array([a1*f[2*j]+a2*f[2*j+1]+a3*f[2*j+2]+a4*f[2*j+3] \
              for j in range(N//2)])
    else: return D4trend_rec( D4trend_rec(f,1),r-1)
        
def D4fluct_rec(f,r=1):
    N = len(f)
    f = list(f)
    if r == 0: return zeros(N)
    if N % 2**r: 
        return "D4fluct_rec: %d is not divisible by 2**%d " % (N, r)
    if r == 1:
        f = f + f[:2]
        return \
            array([b1*f[2*j]+b2*f[2*j+1]+b3*f[2*j+2]+b4*f[2*j+3] \
            for j in range(N//2)])
    else: return D4fluct_rec(D4trend_rec(f,r-1),1)


def D4trend(f, r=1):
    N = len(f)
    if r == 0: return array(f)
    if N % 2**r: 
        return "D4trend: %d is not divisible by 2**%d " % (N, r)
    while r >= 1:
        N = len(f)
        f = array([a1*f[2*j]+a2*f[2*j+1]+ \
              a3*f[(2*j+2)%N]+a4*f[(2*j+3)%N] \
              for j in range(N//2)])
        r -= 1
    return f
        
def D4fluct(f,r=1):
    N = len(f)
    if r == 0: return zeros(N)
    if N % 2**r: 
        return "D4fluct: %d is not divisible by 2**%d " % (N, r)
    a = D4trend(f,r-1)
    N = len(a)
    d = array([b1*a[2*j]+b2*a[2*j+1]+b3*a[(2*j+2)%N]+b4*a[(2*j+3)%N] \
            for j in range(N//2)])
    return d

def D4(f,r=1):
    N = len(f)
    f = list(f)
    if r == 0: return array(f)
    if N % 2**r: return "D4: %d is not divisible by 2**%d " % (N, r)
    d = []
    while r>= 1:
        a = D4trend(f)
        d = splice([D4fluct(f),d])
        f = a
        r -=1
    return splice([f,d])


'''
II) Daub4 scaling and wavelet arrays
'''

# To construct the array of D4 level r scaling vectors
# from the array V of D4 level r-1 scaling vectors
def D4V(V):   # analogous to HaarV(V)
    N = len(V)
    X = a1*V[0,:]+a2*V[1,:]+a3*V[2%N,:]+a4*V[3%N,:]
    for j in range(1,N//2):
        v = a1*V[2*j,:]+a2*V[2*j+1,:]+ \
            a3*V[(2*j+2)%N,:]+a4*V[(2*j+3)%N,:] 
        X = stack([X,v])
    return X

# To construct the array of D4 level r wavelet vectors
# from the array V of D4 level r-1 scaling vectors
def D4W(V):  # analogous to HaarW(V)
    N = len(V)
    Y = b1*V[0,:]+b2*V[1,:]+b3*V[2%N,:]+b4*V[3%N,:]
    for j in range(1,N//2):
        w = b1*V[2*j,:]+b2*V[2*j+1,:]+ \
            b3*V[(2*j+2)%N,:]+b4*V[(2*j+3)%N,:] 
        Y = stack([Y,w])
    return Y
    
# To construct the pair formed with the array V 
# of all D4 scale vectors and the array W of all
# D4 wavelet vectors.
def D4VW(N):  # analogous to HaarVW(V)
    V = Id(N)
    X = [V]
    Y = []
    while N > 2:
        W = D4W(V)
        V = D4V(V)
        X += [V]
        Y += [W]
        N = len(V)
    W = D4W(V)
    V = D4V(V)
    X += [[V]]
    Y += [[W]]
    return (X, Y)

# Orthogonal projection in orthonormal basis
def proj(f,V):
    x = zeros(len(V[0]))
    for v in V:
        x = x + dot(f,v)*v
    return x  

# Projection coefficients
def proj_coeffs(f,V):
    return array([dot(f,v) for v in V])

# The Daub4 = D4 Transform using D4V and D4W    
def D4T(f,r=1):
    N = len(f)
    V,W= D4VW(N)
    x = proj_coeffs(f,V[r])
    
    for j in range(r,0,-1):
        d = proj_coeffs(f,W[j-1])
        x = splice([x,d])
        
    return x


'''
2. Define funtions that implement the filters ...
'''


#auxiliary functions 

def up_sample(a):
    x = []
    for t in a:
        x += [t,0]
    return array(x)
    
    
U_ = up_sample

def dual(h):
    s = 1
    hd = []
    for t in reversed(h):
        hd += [s*t]
        s = -s
    return hd
    
    
a_ = [a1,a2,a3,a4]
b_ = dual(a_)


def filter(h,x):
    m = len(h)
    n = len(x)
    y = zeros(m+n-1)
    for l in range(m+n-1):
        a = max(0,l-m+1)
        b = min(l+1,n)
        s = sum(h[l-j]*x[j] for j in range(a,b))
        y[l] = s 
    return y
    
def H4(x): return filter(a_,x)
def G4(x): return filter(b_,x)


def HF4(f,r=1):
    if r == 0: return array(f)
    a = D4trend(f,r)
    for _ in range(r):
        a = H4(U_(a))
    return array(a)
    
def LF4(f,r=1):
    if r == 0: return zeros(len(f))
    d = G4(U_(D4fluct(f,r)))
    for _ in range(r-1):
        d = H4(U_(d))
    return array(d)
    
        

#Example

def write(X, T, ha='center', va='center', fs='16', rot=0.0, **kwargs):
    return plt.text(X[0], X[1], T, horizontalalignment=ha, verticalalignment=va, fontsize=fs, rotation=rot, **kwargs)
# get sample from signal f
n = 9; N = 2**n
F = lambda x: 20*(-x)**2 * (1-x)**2 * cos(124*x)
f = sample(F,N-1)

# create new canvas (A)
close('all')
canvas('HF4(f)')
xlim(0,N)

# plot original signal for reference
plot(f)
myargs = {'color':'b', 'size':'x-large'}
write((50, 0.75), r'f(x)', **myargs)
cls = ['g','r','c','m']

# plot A^r for different values of r
for r in [2,4,6,8]:
    X = HF4(f,r)
    offset = 1.25*r
    plot([x+offset for x in X])
    
    myargs['color'] = cls[r//2-1]; 
    write((50, 0.75+offset), r'$HF4^'+str(r)+'(f)$', **myargs)

# create new canvas (D)
canvas('LF4(f)')
xlim(0,N)

# plot original signal for reference
plot(f)
myargs = {'color':'b', 'size':'x-large'}
write((50, 0.75), r'f(x)', **myargs)

# plot D^r for different values of r
for r in [2,4,6,8]:
    X = LF4(f,r)
    offset = 1.25*r
    plot([x+offset for x in X])
    
    myargs['color'] = cls[r//2-1]; 
    write((50, 0.75+offset), r'$LF4^'+str(r)+'(f)$', **myargs)