Legacy Documentation

Digital Image Processing (2000)

This is documentation for an obsolete product.

Current products and services

Geometric Operations

4.4 Matrix Manipulation

In addition to the typical geometric operations of zooming and rotation/warping, a number of functions for padding and combining matrices are included in the Digital Image Processing package. Edge padding operations occur naturally in the context of any linear or morphological filtering operation. Matrix concatenation and threading functions add to the already impressive library of matrix manipulation functions found in Mathematica.

This loads the package.

In[1]:=

This gives three example arrays.

In[2]:=

Five methods of padding images and multidimensional arrays are available. Arrays may be extended by appending zeros, by appending any user selected constant, by periodic repetition of the elements of the array, by periodic repetition of the elements of the array in reverse order, and by replication of the edge elements.

ConstantPad[img,{b₁, e₁}, ..., c]	pads img by replicating constant c b_i times at the beginning and e_i times at the end of each list at level i
FixedPad[img,{b₁, e₁}, ...]	pads img by replicating b_iand e_i times, respectively,the first and last elements of each list at level i
PeriodicPad[img, {b₁, e₁}, ... ]	pads img by periodically extending each list at level i with b_i elements at the beginning and e_i elements at the end
ReflectPad[img,{b₁, e₁}, ...]	pads img by extending each list at level i with b_i elements at the beginning and e_i elements at the end with its reflected copy
ZeroPad[img, {b₁, e₁}, ... ]	pads img by appending b_i zeros at the beginning and e_izeros at the end of each list at level i

List padding functions.

This demonstrates padding 1D lists with zeros.

In[3]:=

Out[3]=

This contrasts periodic and reflected padding of a 1D list.

In[4]:=

Out[4]=

This shows all the padding methods using a small example matrix.

In[5]:=

Out[5]=

Out[5]=

In addition to padding, a number of matrix manipulation operators are available, which allow for a variety of methods of concatenating and combining arrays.

MatrixJoin[{{mat₁, mat₂, ...}, ...}]	returns a matrix by concatenating the corresponding rows and columns of matrices mat₁, mat₂, ...
MatrixThread[{{mat₁, mat₂, ...}, ...}]	returns a matrix by threading corresponding entries in matrices mat₁, mat₂, ...
ColumnThread[mat₁, mat₂, ...]	returns a matrix by threading the columns in mat₁, mat₂, ...
RowThread[mat₁, mat₂, ...]	returns a matrix by threading the rows in mat₁, mat₂, ...

Matrix combine, join, and thread functions.

Here are examples of threading and concatenating matrices

. First we demonstrate row-only operations.

In[6]:=

Out[6]//MatrixForm=

In[7]:=

Out[7]//MatrixForm=

Here are the results of joining and threading both rows and columns.

In[8]:=

Out[8]=

Reflection is a form of rotation. In the case of a matrix, it is equivalent to a rotation by 180 degrees. For multidimensional arrays it is more conveniently implemented as a reversal of the elements at each level of the array. Reflections naturally occur in convolution operations and many morphological operators.

Reflect function.

Here is an example of the reflection of a matrix.

In[9]:=

Out[9]//MatrixForm=

Enable JavaScript to interact with content and submit forms on Wolfram websites. Learn how »