Labeling via majority vote of connected clusters in python3 - python

I have a tensor with three dimensions and three classes (0: background, 1: first class, 2: second class). I would like to find connected clusters and assign outlier's labels by performing a majority vote. A 2D example:
import numpy as np
data = np.array([[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0, 0],
[1, 1, 1, 0, 0, 1, 2],
[1, 2, 0, 0, 2, 2, 2],
[0, 1, 0, 0, 0, 2, 0],
[0, 0, 0, 0, 0, 0, 0],])
should be changed to
data = np.array([[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0, 0],
[1, 1, 1, 0, 0, 2, 2],
[1, 1, 0, 0, 2, 2, 2],
[0, 1, 0, 0, 0, 2, 0],
[0, 0, 0, 0, 0, 0, 0],])
It is enough to see connected regions as one cluster an count the appearence of the labels. I am not looking for any machine learning method.

You can use scipy.ndimage.measurements.label to find the connected components and then use np.bincount for the counting
from scipy.ndimage import measurements
lbl,ncl = measurements.label(data)
lut = np.bincount((data+2*lbl).ravel(),None,2*ncl+3)[1:].reshape(-1,2).argmax(1)+1
lut[0] = 0
lut[lbl]
# array([[0, 0, 0, 0, 0, 0, 0],
# [0, 0, 0, 0, 0, 0, 0],
# [0, 1, 0, 0, 0, 0, 0],
# [1, 1, 1, 0, 0, 2, 2],
# [1, 1, 0, 0, 2, 2, 2],
# [0, 1, 0, 0, 0, 2, 0],
# [0, 0, 0, 0, 0, 0, 0]])

Related

Convert array of indices to array of zeros where the index is one

I have the following numpy array representing indexes:
array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3,
3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4,
4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5])
How can I convert this to an array of zeros where only the given index is a one?
Without numpy, this works:
a = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3,
3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4,
4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5,
5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5]
for i in range(len(a)):
idx = a[i]
a[i] = [0 for i in range(6)]
a[i][idx] = 1
print(a)
>>> [[1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1]]
However, I am looking for a numpy function (preferably no iterating).
Maybe you could use np.identity:
>>> import numpy as np
>>> np.identity(6, np.int64)
array([[1, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0],
[0, 0, 1, 0, 0, 0],
[0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 0, 1]])
Use your "index" array to index into the identity matrix:
In [2]: indices = np.array([0, 0, 0, ...]) # provided by OP
In [3]: np.identity(6, int)[indices]
Out[3]:
array([[1, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 0, 0],
.
.
.
[0, 0, 0, 0, 0, 1],
[0, 0, 0, 0, 0, 1],
[0, 0, 0, 0, 0, 1],
[0, 0, 0, 0, 0, 1],
[0, 0, 0, 0, 0, 1],
[0, 0, 0, 0, 0, 1]])
You can use advanced indexing:
b = np.zeros((len(a), 6), dtype=a.dtype)
b[range(len(a)), a.tolist()] = 1
output:
array([[1, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 0, 0],
...
[0, 0, 1, 0, 0, 0],
[0, 0, 0, 1, 0, 0],
[0, 0, 0, 1, 0, 0],
[0, 0, 0, 1, 0, 0],
[0, 0, 0, 1, 0, 0],
[0, 0, 0, 1, 0, 0],
[0, 0, 0, 1, 0, 0],
[0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 1, 0],
...
[0, 0, 0, 0, 0, 1],
[0, 0, 0, 0, 0, 1],
[0, 0, 0, 0, 0, 1],
[0, 0, 0, 0, 0, 1],
[0, 0, 0, 0, 0, 1],
[0, 0, 0, 0, 0, 1]])

Successive zeroing of columns of a numpy array

I have an array a of ones and zeroes (it might be rather big)
a = np.array([[1, 0, 0, 1, 0, 0],
[1, 1, 0, 0, 1, 0],
[0, 1, 1, 0, 0, 1],
[0, 0, 0, 1, 1, 1])
in which the "upper" rows are more "important" in the sense that if there is 1 in any column of the i-th row, then all ones in that columns in the following rows must be zeroed.
So, the desired output should be:
array([[1, 0, 0, 1, 0, 0],
[0, 1, 0, 0, 1, 0],
[0, 0, 1, 0, 0, 1],
[0, 0, 0, 0, 0, 0]])
In other words, there should only be single 1 per column.
I'm looking for a more numpy way to do this (i.e. minimising or, better, avoiding the loops).
Your array:
[[1, 0, 0, 1, 0, 0],
[1, 1, 0, 0, 1, 0],
[0, 1, 1, 0, 0, 1],
[0, 0, 0, 1, 1, 1]]
Transpose it with numpy:
a = np.transpose(your_array)
Now it looks like this:
[[1, 1, 0, 0],
[0, 1, 1, 0],
[0, 0, 1, 0],
[1, 0, 0, 1],
[0, 1, 0, 1],
[0, 0, 1, 1]]
Zero all the non-zero (and "not upper") elements row wise:
res = np.zeros(a.shape, dtype="int64")
idx = np.arange(res.shape[0])
args = a.astype(bool).argmax(1)
res[idx, args] = a[idx, args]
The output of res is this:
#### Output
[[1, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, 1, 0],
[1, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, 1, 0]]
Re-transpose your array:
a = np.transpose(res)
[[1, 0, 0, 1, 0, 0],
[0, 1, 0, 0, 1, 0],
[0, 0, 1, 0, 0, 1],
[0, 0, 0, 0, 0, 0]])
EDIT: Thanks to #The.B for the tip
An alternative solution is to do a forward fill followed by the cumulative sum and then replace all values which are not 1 with 0:
a = np.array([[1, 0, 0, 1, 0, 0],
[1, 1, 0, 0, 1, 0],
[0, 1, 1, 0, 0, 1],
[0, 0, 0, 1, 1, 1]])
ff = np.maximum.accumulate(a, axis=0)
cs = np.cumsum(ff, axis=0)
cs[cs > 1] = 0
Output in cs:
array([[1, 0, 0, 1, 0, 0],
[0, 1, 0, 0, 1, 0],
[0, 0, 1, 0, 0, 1],
[0, 0, 0, 0, 0, 0]])
EDIT
This will do the same thing and should be slightly more efficient:
ff = np.maximum.accumulate(a, axis=0)
ff ^ np.pad(ff, ((1,0), (0,0)))[:-1]
Output:
array([[1, 0, 0, 1, 0, 0],
[0, 1, 0, 0, 1, 0],
[0, 0, 1, 0, 0, 1],
[0, 0, 0, 0, 0, 0]])
And if you want to do the operations in-place to avoid temporary memory allocation:
out = np.zeros((a.shape[0]+1, a.shape[1]), dtype=a.dtype)
np.maximum.accumulate(a, axis=0, out=out[1:])
out[:-1] ^ out[1:]
Output:
array([[1, 0, 0, 1, 0, 0],
[0, 1, 0, 0, 1, 0],
[0, 0, 1, 0, 0, 1],
[0, 0, 0, 0, 0, 0]])
You can traverse through each column of array and check if it is the first one -
If Not: Make it 0
for col in a.T:
f=0
for x in col:
if(x==1 and f==0):
f=1
else:
x=0

SymPy rref() returning an identity matrix for a singular matrix

import numpy
import sympy
n = 7
k = 3
X = numpy.random.randn(n,k)
Px = X#numpy.linalg.inv(numpy.transpose(X)#X)#numpy.transpose(X) #X(X'X)^(-1)X'
print(sympy.Matrix(Px).rref())
As you may verify yourself, Px is singular. However, sympy.rref() returns this:
(Matrix([[1, 0, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0, 0],
[0, 0, 1, 0, 0, 0, 0],
[0, 0, 0, 1, 0, 0, 0],
[0, 0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 0, 0, 1]]), (0, 1, 2, 3, 4, 5, 6))
Why doesn't it return the real rref? I read somewhere I could pass simplify=True, however it didn't make any difference.
In [49]: Px
Out[49]:
array([[ 0.5418898 , 0.44245552, 0.04973693, -0.06834885, -0.19086119,
-0.07003176, 0.06325021],...
[ 0.06325021, -0.11080081, 0.21656224, -0.07445145, -0.28634725,
0.06648907, 0.19199866]])
In [50]: np.linalg.det(Px)
Out[50]: 2.141647537907433e-67
In [51]: np.linalg.inv(Px)
Out[51]:
array([[-7.18788695e+15, 4.95655702e+15, 7.52738018e+15,
-4.40875311e+15, -1.64015565e+16, 2.63785320e+15,
-3.03465003e+16],
[ 1.59176426e+16, ....
[ 3.31636798e+16, -3.39094560e+16, -3.60287970e+16,
-1.27160460e+16, 2.14338015e+16, 3.32345350e+15,
3.60287970e+16]])
Your Px is close to singular, but not exactly so. Contrast that with
In [52]: M = np.arange(9).reshape(3,3)
In [53]: np.linalg.det(M)
Out[53]: 0.0
In [55]: np.linalg.inv(M)
LinAlgError: Singular matrix
In [56]: sympy.Matrix(M).rref()
Out[56]:
(Matrix([
[1, 0, -1],
[0, 1, 2],
[0, 0, 0]]), (0, 1))
Numerically speaking your Px is not singular, just close:
In [57]: sympy.Matrix(Px).rref()
Out[57]:
(Matrix([
[1, 0, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0, 0],
[0, 0, 1, 0, 0, 0, 0],
[0, 0, 0, 1, 0, 0, 0],
[0, 0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 0, 0, 1]]), (0, 1, 2, 3, 4, 5, 6))
But with a custom iszerofunc:
In [58]: sympy.Matrix(Px).rref(iszerofunc=lambda x: abs(x)<1e-16)
Out[58]:
(Matrix([
[1, 0, 0, 0.647383887198708, -1.91409951634531, -1.43377991000974, 0.578981680134581],
[0, 1, 0, -0.839184067893959, 1.88998490600173, 1.43367640627271, -0.611620902311026],
[0, 0, 1, -0.962221703397948, 0.203783478612254, 1.45929622452135, 0.404548167005728],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0]]),
(0, 1, 2))

What is an efficient way of counting groups of ones on 2D grid in python? [Figures below]

I have a few hundred 2D numpy arrays. They contain zeros and ones. A few examples with plots, yellow indicates ones, purple indicates zeros:
grid1=np.array([[1, 1, 0, 0, 1, 1, 0, 0],
[1, 1, 0, 1, 1, 1, 0, 0],
[1, 1, 0, 1, 1, 1, 0, 0],
[1, 0, 0, 0, 1, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0]])
plt.imshow(grid1)
grid2=np.array([[1, 1, 0, 0, 0, 0, 0, 0],
[1, 1, 1, 1, 1, 0, 0, 0],
[1, 1, 1, 1, 1, 0, 0, 0],
[1, 0, 0, 1, 1, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0]])
plt.imshow(grid2)
grid3=np.array([[1, 1, 0, 0, 1, 0, 0, 1],
[0, 1, 0, 1, 1, 0, 1, 1],
[0, 1, 0, 1, 1, 0, 0, 0],
[0, 0, 0, 0, 1, 1, 1, 0],
[1, 1, 1, 0, 0, 0, 0, 0]])
plt.imshow(grid3)
I'm looking for an efficient way to count the number of yellow blobs on the images. 2, 1 and 4 blobs on the images above, from top to bottom.
Is there an easy way to do this or I have to check each yellow bit to be in the same blob as all the other yellows, and write a script for that? (That looks very painful.)
scipy.ndimage.measurements.label does what you need.

Create boolean mask on TensorFlow

Suppose I have a list
x = [0, 1, 3, 5]
And I want to get a tensor with dimensions
s = (10, 7)
Such that the first column of the rows with indexes defined in x are 1, and 0 otherwise.
For this particular example, I want to obtain the tensor containing:
T = [[1, 0, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0]]
Using numpy, this would be the equivalent:
t = np.zeros(s)
t[x, 0] = 1
I found this related answer, but it doesn't really solve my problem.
Try this:
import tensorflow as tf
indices = tf.constant([[0, 1],[3, 5]], dtype=tf.int64)
values = tf.constant([1, 1])
s = (10, 7)
st = tf.SparseTensor(indices, values, s)
st_ordered = tf.sparse_reorder(st)
result = tf.sparse_tensor_to_dense(st_ordered)
sess = tf.Session()
sess.run(result)
Here is the output:
array([[0, 1, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0]], dtype=int32)
I slightly modified your indexes so you can see the x,y format of the indices
To obtain what you originally asked, set:
indices = tf.constant([[0, 0], [1, 0],[3, 0], [5, 0]], dtype=tf.int64)
Output:
array([[1, 0, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0]], dtype=int32)

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