def tv(value):
"""Total variation of a vector or matrix.
Uses L1 norm of discrete gradients for vectors and
L2 norm of discrete gradients for matrices.
Parameters
----------
value : Expression or numeric constant
The value to take the total variation of.
Returns
-------
Expression
An Expression representing the total variation.
"""
value = Expression.cast_to_const(value)
rows, cols = value.size
if value.is_scalar():
raise ValueError("tv cannot take a scalar argument.")
# L1 norm for vectors.
elif value.is_vector():
return norm(value[1:] - value[0:max(rows, cols) - 1], 1)
# L2 norm for matrices.
else:
row_diff = value[0:rows - 1, 1:cols] - value[0:rows - 1, 0:cols - 1]
col_diff = value[1:rows, 0:cols - 1] - value[0:rows - 1, 0:cols - 1]
return sum_entries(norm2_elemwise(row_diff, col_diff))
def l1_aniso_2d(value1, value2):
"""\sum \sqrt{value1^2 + value2^2}
Parameters
----------
value : Expression or numeric constant
The value to take the total variation of.
Returns
-------
Expression
An Expression representing the total variation.
"""
value1 = Expression.cast_to_const(value1)
value2 = Expression.cast_to_const(value2)
len = value1.size[0]
return sum_entries(cvxnorm(value1 + value2, p='1'))
def tv(value, *args):
"""Total variation of a vector, matrix, or list of matrices.
Uses L1 norm of discrete gradients for vectors and
L2 norm of discrete gradients for matrices.
Parameters
----------
value : Expression or numeric constant
The value to take the total variation of.
args : Matrix constants/expressions
Additional matrices extending the third dimension of value.
Returns
-------
Expression
An Expression representing the total variation.
"""
# Accept single list as argument.
if isinstance(value, list) and len(args) == 0:
args = value[1:]
value = value[0]
value = Expression.cast_to_const(value)
rows, cols = value.size
if value.is_scalar():
raise ValueError("tv cannot take a scalar argument.")
# L1 norm for vectors.
elif value.is_vector():
return norm(value[1:] - value[0:max(rows, cols)-1], 1)
# L2 norm for matrices.
else:
args = list(map(Expression.cast_to_const, args))
values = [value] + list(args)
diffs = []
for mat in values:
diffs += [
mat[0:rows-1, 1:cols] - mat[0:rows-1, 0:cols-1],
mat[1:rows, 0:cols-1] - mat[0:rows-1, 0:cols-1],
]
length = diffs[0].size[0]*diffs[1].size[1]
stacked = vstack(*[reshape(diff, 1, length) for diff in diffs])
return sum_entries(norm(stacked, p='fro', axis=0))
def tv(value, *args):
"""Total variation of a vector, matrix, or list of matrices.
Uses L1 norm of discrete gradients for vectors and
L2 norm of discrete gradients for matrices.
Parameters
----------
value : Expression or numeric constant
The value to take the total variation of.
args : Matrix constants/expressions
Additional matrices extending the third dimension of value.
Returns
-------
Expression
An Expression representing the total variation.
"""
# Accept single list as argument.
if isinstance(value, list) and len(args) == 0:
args = value[1:]
value = value[0]
value = Expression.cast_to_const(value)
rows, cols = value.size
if value.is_scalar():
raise ValueError("tv cannot take a scalar argument.")
# L1 norm for vectors.
elif value.is_vector():
return norm(value[1:] - value[0:max(rows, cols)-1], 1)
# L2 norm for matrices.
else:
args = map(Expression.cast_to_const, args)
values = [value] + list(args)
diffs = []
for mat in values:
diffs += [
mat[0:rows-1, 1:cols] - mat[0:rows-1, 0:cols-1],
mat[1:rows, 0:cols-1] - mat[0:rows-1, 0:cols-1],
]
length = diffs[0].size[0]*diffs[1].size[1]
stacked = vstack(*[reshape(diff, 1, length) for diff in diffs])
return sum_entries(norm(stacked, p='fro', axis=0))
def tvnorm2d(value, Dx, Dy):
"""Total variation of a vector, matrix, or list of matrices.
Uses L1 norm of discrete gradients for vectors and
L2 norm of discrete gradients for matrices.
Parameters
----------
value : Expression or numeric constant
The value to take the total variation of.
Returns
-------
Expression
An Expression representing the total variation.
"""
value = Expression.cast_to_const(value)
len = value.size[0]
diffs = [Dx * value, Dy * value]
stack = vstack(*[reshape(diff, 1, len) for diff in diffs])
return sum_entries(cvxnorm(stack, p='fro', axis=0))
def tv(value, *args):
"""Total variation of a vector, matrix, or list of matrices.
Uses L1 norm of discrete gradients for vectors and
L2 norm of discrete gradients for matrices.
Parameters
----------
value : Expression or numeric constant
The value to take the total variation of.
args : Matrix constants/expressions
Additional matrices extending the third dimension of value.
Returns
-------
Expression
An Expression representing the total variation.
"""
value = Expression.cast_to_const(value)
rows, cols = value.size
if value.is_scalar():
raise ValueError("tv cannot take a scalar argument.")
# L1 norm for vectors.
elif value.is_vector():
return norm(value[1:] - value[0:max(rows, cols)-1], 1)
# L2 norm for matrices.
else:
args = list(map(Expression.cast_to_const, args))
values = [value] + list(args)
diffs = []
for mat in values:
diffs += [
mat[0:rows-1, 1:cols] - mat[0:rows-1, 0:cols-1],
mat[1:rows, 0:cols-1] - mat[0:rows-1, 0:cols-1],
]
return sum_entries(norm2_elemwise(*diffs))
def tv(value, *args):
"""Total variation of a vector, matrix, or list of matrices.
Uses L1 norm of discrete gradients for vectors and
L2 norm of discrete gradients for matrices.
Parameters
----------
value : Expression or numeric constant
The value to take the total variation of.
args : Matrix constants/expressions
Additional matrices extending the third dimension of value.
Returns
-------
Expression
An Expression representing the total variation.
"""
value = Expression.cast_to_const(value)
rows, cols = value.size
if value.is_scalar():
raise ValueError("tv cannot take a scalar argument.")
# L1 norm for vectors.
elif value.is_vector():
return norm(value[1:] - value[0:max(rows, cols) - 1], 1)
# L2 norm for matrices.
else:
args = map(Expression.cast_to_const, args)
values = [value] + list(args)
diffs = []
for mat in values:
diffs += [
mat[0:rows - 1, 1:cols] - mat[0:rows - 1, 0:cols - 1],
mat[1:rows, 0:cols - 1] - mat[0:rows - 1, 0:cols - 1],
]
return sum_entries(norm2_elemwise(*diffs))
def l1_anisotropic_2d(signal):
return sum_entries(cvxnorm(signal, 2, axis=1))
def tvnorm_anisotropic_2d(signal, Dx, Dy):
magnitudes = pnorm(signal, 2, axis=1)
diffs = [Dx * magnitudes, Dy * magnitudes]
stack = vstack(*[reshape(diff, 1, magnitudes.size[0]) for diff in diffs])
return sum_entries(pnorm(stack, 2, axis=0))