Educational Materials

How Do I Do Basic Image Processing in Mathematica?

Wolfram Research, Inc.

Author

Wolfram Research, Inc.

Title

How Do I Do Basic Image Processing in Mathematica?

Description

This how-to demonstrates how Mathematica can be used to do image analysis tasks. In addition to many common operations, the ListInterpolation function can be used to do new and exciting manipulations and analyses. Also available in Japanese http://library.wolfram.com/howtos/images/index.ja.html

Image Processing in Mathematica 4

This how-to demonstrates how Mathematica 4 can be used to do image analysis tasks. In addition to many common operations, the ListInterpolation function can be used to do new and exiting manipulations and analyses.

Note: To save bandwidth, the original imported picture is not shown in every subsection. Click here to see it

Importing, displaying, and exporting graphics files

The Import command provides Mathematica users with an efficient and intuitive way to read files in most standard graphics formats, including bitmap, JPEG, GIF, TIFF, and numerous platform specific formats. In many cases, Mathematica even automatically detects the correct graphics format; otherwise, users can specify it themselves. Note that one has to use double backslashes in the file path. To interactively select the file, use the GetFilePath command in the Input menu. This command opens a standard file dialog and returns a correctly formatted path to the file at the current text cursor position.

In[]:=

dog=Import["C:\\desk\\40 Demos\\schlapp2.jpg"];

The graphic can now be displayed using Show,

In[]:=

Show[dog]

Out[]=

⁃Graphics⁃

and it can be exported in a different format. For a complete list of import and export formats, consult the documentation for Import and Export.

In[]:=

Export["C:\\windows\\desktop\\dog.tif",dog,"TIFF"]

Out[]=

"C:\\windows\\desktop\\dog.tif"

In[]:=

tifdog=Import["C:\\windows\\desktop\\dog.tif"];

In[]:=

Show[tifdog]

Out[]=

⁃Graphics⁃

Color conversions

Accessing image data

Mathematica handles many different kinds of objects: mathematical formulas, lists, and graphics, to name but a few. Although they often look very different, Mathematica represents all in one uniform way: they are all expressions. The imported graphics files are, for example, stored as a Graphics object.

The object f in an expression f[x, y, … ] is known as the head of the expression. You can extract it using Head. Particularly when you write programs in Mathematica, you will often want to test the head of an expression to find out what kind of thing the expression is.

Both the imported JPEG and TIFF files are, of course, Graphics objects.

In[]:=

Head[dog]

Out[]=

Graphics

In[]:=

Head[tifdog]

Out[]=

Graphics

The easiest way to access the pixel data is to change the head of the expression to a Mathematica list and extract the array of pixel values. The following command replaces the Graphics head with a List head. Explaining the mechanism is beyond the scope of this document. If you are interested, consult the documentation for Replace.

In[]:=

dog=dog/.GraphicsList;

The list contains an array of pixel values and some supplemental information on image size and color model. Short gives an overview.

In[]:=

Short[dog,12]

Out[]=

{Raster[1,{{0,0},{291,302}},{0,255},ColorFunctionRGBColor],AspectRatioAutomatic}

The pixel values are always the first element of the data structure. They can now be treated like any other list in Mathematica. A RGB image, for example, is represented as an array of

{r,g,b}

triples.

f(x,y)=

r(0,0),g(0,0),b(0,0)	...	r(0,N-1),g(0,N-1),b(0,N-1)
r(1,0),g(1,0),b(1,0)	⋯	r(1,N-1),g(1,N-1),b(1,N-1)
⋮	⋱	⋮
r(M-1,0),g(M-1,0),b(M-1,0)	⋯	r(M-1,N-1),g(M-1,N-1),b(M-1,N-1)

The next line extracts the array of pixel values. To learn more about matrix manipulation, consult

Section 1.8.4

. of The Mathematica Book or the Matrix Manipulation how-to.

In[]:=

pixelvalues=dog[[1,1]];

As you can see, the color values are now stored in a 302×291×3array.

In[]:=

Dimensions[pixelvalues]

Out[]=

{302,291,3}

For example, here is a ListDensityPlot of the red channel of the image. All is a new function in Mathematica 4 that simply instructs Mathematica to take all elements in the respective dimension of the array.

In[]:=

ListDensityPlot[pixelvalues[[All,All,1]],MeshFalse];

Simple math

All Mathematica functions that can be applied to lists can also work on the image data. For example, the following two lines are two different ways of inverting a grayscale image.

In[]:=

ListDensityPlot[-pixelvalues[[All,All,1]],MeshFalse];

In[]:=

ListDensityPlot[256-pixelvalues[[All,All,1]],MeshFalse];

The same procedure works for color images, too. Using Apply is a shortcut to convert all three color values in one step. Also, RGBColor expects its values to be between 0 and 1, so pixelvalues gets divided by 256.

In[]:=

Show[Graphics[RasterArray[Apply[(RGBColor[#1,#2,#3]&),1-pixelvalues/256.,{2}]]],AspectRatio1];

Histograms

Histogram operations

Selecting pieces of images

Area selection

Thresholding

Convolution and correlation

The new Mathematica 4 functions ListConvolve and ListCorrelate make a number of advanced image processing tasks easy to do. The following graphic shows the effect of applying a Sobel edge filter to the image by convolving the image with two 3×3 kernels.

In[]:=

ListDensityPlotListConvolve

-1	-2	-1
4	0	5
1	2	1

,ListConvolve

1	0	1
0	0	2
-1	0	1

,pixelvalues[[All,All,1]]256.,MeshFalse;

List interpolation and data recovery

An interesting feature of Mathematica is its ability to rapidly calculate list interpolations of any degree in arbitrary dimensions. This feature opens up new ways to manipulate images and recover information lost by the discretizing of data.

■

The Doghill mountains

■

Data recovery

Let's go back to the small piece of the image selected in "Selecting pieces of images".

righteye=Take[pixelvalues,{150,180},{110,140}];

ListDensityPlot[righteye[[All,All,1]],MeshFalse]

⁃DensityGraphics⁃

As you can see, the image above consists of only 51×31 data points, so the resolution is not terribly high. List interpolation provides a convenient way to recover more features.

The following command creates a list interpolation over the red channel of the image and scales the resulting function between

and 1 in the x and y coordinates.

interpol1=ListInterpolation[righteye[[All,All,1]],{{0,1},{0,1}}];

DensityPlot[interpol1[x,y],{x,0,1},{y,0,1},PlotPoints100,MeshFalse];

■

Mathematical operations

Cite this as: Wolfram Research, Inc., "How Do I Do Basic Image Processing in Mathematica?" from the Notebook Archive (2002), https://notebookarchive.org/2018-10-10r8f3h