Merge pull request #10 from marysia/3d-conv

3D GrouPy

Author: tscohen

groupy/garray/C4h_array.py

@@ -0,0 +1,148 @@
+import numpy as np
+from groupy.garray.finitegroup import FiniteGroup
+from groupy.garray.matrix_garray import MatrixGArray
+from groupy.garray.C4ht_array import C4htArray
+from groupy.garray.Z3_array import Z3Array
+
+"""
+Implementation of the finite group C4h, the cyclic group of rectangular cuboid symmetry, consisting of 8 elements in total:
+    - Identity element
+    - one 4-fold axis over opposing square faces (90 degree rotations)
+    - two 2-fold axes over opposing rectangular faces (180 degree rotations)
+    - two 2-fold axes over opposing long edges (180 degree rotations)
+As C4h is abelian, each transformation can be described in terms of number of rotations around the various fold axes.
+However, all elements from C4h can also be generated by two elements: 180 degree rotation over the y-axis and 90 
+degree rotation over the z-axis. 
+
+The int parameterization is in the form of (y, z) where y represents the number of 180 degree rotations 
+over y (in {0, 1}) and z the number of 90 degree rotations over z (in {0, 1, 2, 3}) 
+"""
+
+
+class C4hArray(MatrixGArray):
+    parameterizations = ['int', 'mat', 'hmat']
+    _g_shapes = {'int': (2,), 'mat': (3, 3), 'hmat': (4, 4)}
+    _left_actions = {}
+    _reparameterizations = {}
+    _group_name = 'C4h'
+
+    def __init__(self, data, p='int'):
+        data = np.asarray(data)
+        assert data.dtype == np.int
+
+        # classes C4hArray can be multiplied with
+        self._left_actions[C4hArray] = self.__class__.left_action_hmat
+        self._left_actions[C4htArray] = self.__class__.left_action_hmat
+        self._left_actions[Z3Array] = self.__class__.left_action_vec
+
+        super(C4hArray, self).__init__(data, p)
+        self.elements = self.get_elements()
+
+    def mat2int(self, mat_data):
+        '''
+        Transforms 3x3 matrix representation to int representation.
+        To handle any size and shape of mat_data, the original mat_data
+        is reshaped to a long list of 3x3 matrices, converted to a list of
+        int representations, and reshaped back to the original mat_data shape.
+
+        mat-2-int is achieved by taking the matrix, looking up the index in the
+        element list, and converting that index to two numbers: y and z. The index
+        is the result of (y * 4) + z.
+        '''
+
+        input = mat_data.reshape((-1, 3, 3))
+        data = np.zeros((input.shape[0], 2), dtype=np.int)
+        for i in xrange(input.shape[0]):
+            index = self.elements.index(input[i].tolist())
+            z = int(index % 4)
+            y = int((index - z) / 4)
+            data[i, 0] = y
+            data[i, 1] = z
+        data = data.reshape(mat_data.shape[:-2] + (2,))
+        return data
+
+    def int2mat(self, int_data):
+        '''
+        Transforms integer representation to 3x3 matrix representation.
+        Original int_data is flattened and later reshaped back to its original
+        shape to handle any size and shape of input.
+
+        The element is located in the list at index (y * 4) + z.
+        '''
+        y = int_data[..., 0].flatten()
+        z = int_data[..., 1].flatten()
+        data = np.zeros((len(y),) + (3, 3), dtype=np.int)
+
+        for j in xrange(len(y)):
+            index = (y[j] * 4) + z[j]
+            mat = self.elements[index]
+            data[j, 0:3, 0:3] = mat
+
+        data = data.reshape(int_data.shape[:-1] + (3, 3))
+        return data
+
+    def _multiply(self, element, generator, times):
+        element = np.array(element)
+        for i in range(times):
+            element = np.dot(element, np.array(generator))
+        return element
+
+    def get_elements(self):
+        '''
+        Elements are stored as lists rather than numpy arrays to enable
+        lookup through self.elements.index(x) and sorting.
+
+        All elements are found by multiplying the identity matrix with all
+        possible combinations of the generators, i.e. 0 or 1 rotations over y
+        and 0, 1, 2, or 3 rotations over z.
+        '''
+        # specify generators
+        mode = 'zyx'
+        if mode == 'xyz':
+            g1 = np.array([[-1, 0, 0], [0, 1, 0], [0, 0, -1]])  # 180 degrees over y
+            g2 = np.array([[0, -1, 0], [1, 0, 0], [0, 0, 1]])  # 90 degrees over z
+        elif mode == 'zyx':
+            g1 = np.array([[-1, 0, 0], [0, 1, 0], [0, 0, -1]])  # 180 degrees over y
+            g2 = np.array([[1, 0, 0], [0, 0, -1], [0, 1, 0]])  # 90 degrees over z
+
+
+        element_list = []
+        element = np.array([[1, 0, 0], [0, 1, 0], [0, 0, 1]])
+        for i in range(0, 2):
+            element = self._multiply(element, g1, i)
+            for j in range(0, 4):
+                element = self._multiply(element, g2, j)
+                element_list.append(element.tolist())
+        return element_list
+
+
+class C4hGroup(FiniteGroup, C4hArray):
+    def __init__(self):
+        C4hArray.__init__(
+            self,
+            data=np.array([[i, j] for i in xrange(2) for j in xrange(4)]),
+            p='int'
+        )
+        FiniteGroup.__init__(self, C4hArray)
+
+    def factory(self, *args, **kwargs):
+        return C4hArray(*args, **kwargs)
+
+C4h = C4hGroup()
+
+def rand(size=()):
+    '''
+    Returns an C4hArray of shape size, with randomly chosen elements in int parameterization.
+    '''
+    data = np.zeros(size + (2,), dtype=np.int)
+    data[..., 0] = np.random.randint(0, 2, size)    # rotations over y
+    data[..., 1] = np.random.randint(0, 4, size)    # rotations over z
+    return C4hArray(data=data, p='int')
+
+def identity(p='int'):
+    '''
+    Returns the identity element: a matrix with 1's on the diagonal.
+    '''
+    li = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
+    e = C4hArray(data=np.array(li, dtype=np.int), p='mat')
+    return e.reparameterize(p)

groupy/garray/C4ht_array.py

@@ -0,0 +1,151 @@
+import numpy as np
+from groupy.garray.matrix_garray import MatrixGArray
+
+""" 
+Implementation of the non-orientation perserving variant of group C4h -- C4h with translations. 
+The int parameterisation is similar to that of C4h, but with the added 3D translation (u, v, w) to indicate
+translation in Z3. 
+
+4x4 homogeneous matrices (hmat) are used to represent the transformation in matrix format. 
+"""
+
+class C4htArray(MatrixGArray):
+    parameterizations = ['int', 'hmat']
+    _g_shapes = {'int': (5,), 'hmat': (4, 4)}
+    _left_actions = {}
+    _reparameterizations = {}
+    _group_name = 'C4ht'
+
+    def __init__(self, data, p='int'):
+        data = np.asarray(data)
+        assert data.dtype == np.int
+        self._left_actions[C4htArray] = self.__class__.left_action_hmat
+        super(C4htArray, self).__init__(data, p)
+        self.elements = self.get_elements()
+
+    def hmat2int(self, hmat_data):
+        '''
+        Transforms 4x4 matrix representation to int representation.
+        To handle any size and shape of hmat_data, the original hmat_data
+        is reshaped to a long list of 4x4 matrices, converted to a list of
+        int representations, and reshaped back to the original mat_data shape.
+
+        hmat-2-int is achieved by taking the matrix, looking up the index in the
+        element list, and converting that index to two numbers: y and z. The index
+        is the result of (y * 4) + z. u, v, w are retrieved by looking at the last
+        column of the hmat.
+        '''
+
+        input = hmat_data.reshape((-1, 4, 4))
+        data = np.zeros((input.shape[0], 5), dtype=np.int)
+        for i in xrange(input.shape[0]):
+            hmat = input[i]
+            mat = [elem[0:3] for elem in hmat.tolist()][0:3]
+            index = self.elements.index(mat)
+            z = int(index % 4)
+            y = int((index - z) / 4)
+            u, v, w, _ = hmat[:, 3]
+            data[i, 0] = y
+            data[i, 1] = z
+            data[i, 2] = u
+            data[i, 3] = v
+            data[i, 4] = w
+        data = data.reshape(hmat_data.shape[:-2] + (5,))
+        return data
+
+    def int2hmat(self, int_data):
+        '''
+        Transforms integer representation to 4x4 matrix representation.
+        Original int_data is flattened and later reshaped back to its original
+        shape to handle any size and shape of input.
+        '''
+        # rotations over y and z
+        y = int_data[..., 0].flatten()
+        z = int_data[..., 1].flatten()
+
+        # translations
+        u = int_data[..., 2].flatten()
+        v = int_data[..., 3].flatten()
+        w = int_data[..., 4].flatten()
+        data = np.zeros((len(y),) + (4, 4), dtype=np.int)
+
+        for j in xrange(len(y)):
+            index = (y[j] * 4) + z[j]
+            mat = self.elements[index]
+
+            data[j, 0:3, 0:3] = mat
+            data[j, 0, 3] = u[j]
+            data[j, 1, 3] = v[j]
+            data[j, 2, 3] = w[j]
+            data[j, 3, 3] = 1
+
+        data = data.reshape(int_data.shape[:-1] + (4, 4))
+        return data
+
+    def _multiply(self, element, generator, times):
+        '''
+        Helper function to multiply an _element_ with a _generator_
+        _times_ number of times. Used in self.get_elements()
+        '''
+        element = np.array(element)
+        for i in range(times):
+            element = np.dot(element, np.array(generator))
+        return element
+
+    def get_elements(self):
+        '''
+        Function to generate a list containing elements of group C4ht,
+        similar to get_elements() of BArray.
+
+        These are the base elements in 3x3 matrix notation without translations.
+        '''
+        # specify generators
+        mode = 'zyx'
+        if mode == 'xyz':
+            g1 = np.array([[-1, 0, 0], [0, 1, 0], [0, 0, -1]])  # 180 degrees over y
+            g2 = np.array([[0, -1, 0], [1, 0, 0], [0, 0, 1]])  # 90 degrees over z
+        elif mode == 'zyx':
+            g1 = np.array([[-1, 0, 0], [0, 1, 0], [0, 0, -1]])  # 180 degrees over y
+            g2 = np.array([[1, 0, 0], [0, 0, -1], [0, 1, 0]])  # 90 degrees over z
+
+        element_list = []
+        element = np.array([[1, 0, 0], [0, 1, 0], [0, 0, 1]])
+        for i in range(0, 2):
+            element = self._multiply(element, g1, i)
+            for j in range(0, 4):
+                element = self._multiply(element, g2, j)
+                element_list.append(element.tolist())
+        return element_list
+
+
+def rand(minu=0, maxu=5, minv=0, maxv=5, minw=0, maxw=5, size=()):
+    '''
+    Returns an C4htArray of shape size, with randomly chosen elements in int parameterization.
+    '''
+    data = np.zeros(size + (5,), dtype=np.int64)
+    data[..., 0] = np.random.randint(0, 2, size)    # rotations over y
+    data[..., 1] = np.random.randint(0, 4, size)    # rotations over x
+    data[..., 2] = np.random.randint(minu, maxu, size)  # translation on x
+    data[..., 3] = np.random.randint(minv, maxv, size)  # translation on y
+    data[..., 4] = np.random.randint(minw, maxw, size)  # translation on z
+    return C4htArray(data=data, p='int')
+
+
+def identity(p='int'):
+    '''
+    Returns the identity element: a matrix with 1's on the diagonal.
+    '''
+    li = [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]]
+    e = C4htArray(data=np.array(li, dtype=np.int), p='hmat')
+    return e.reparameterize(p)
+
+
+def meshgrid(minu=-1, maxu=2, minv=-1, maxv=2, minw=-1, maxw=2):
+    '''
+    Creates a meshgrid of all elements of the group, within the given
+    translation parameters.
+    '''
+    li = [[i, m, u, v, w] for i in xrange(2) for m in xrange(4) for u in xrange(minu, maxu) for v in xrange(minv, maxv)
+          for
+          w in xrange(minw, maxw)]
+    return C4htArray(li, p='int')

groupy/garray/D4_array.py

@@ -6,7 +6,6 @@
 
 from groupy.garray.matrix_garray import MatrixGArray
 
-
 class D4Array(MatrixGArray):
 
     parameterizations = ['int', 'mat', 'hmat']