## Introduction

2D image transformation in .NET has been very much simplified by the `Matrix`

class in the `System.Drawing.Drawing2D`

namespace. In this article, I would like to share with the reader on the use of `Matrix`

class for 2D image transformation.

## Background

The `Matrix`

class takes 6 elements arranged in 3 rows by 2 cols. For example, the default matrix constructed by the default constructor has value of ( 1,0,0,1,0,0 ). In matrix representation:

This is a simplification of:

The last column is always:

Thus a translation transformation of movement of 3 in x-axis and 2 in the y-axis would be represented as:

but internally is represented as

An important thing to note is that the transformation matrix is post multiplied to the image vectors. For example, we have an image with 4 points: (1,1) ( 2,3) (5,0) (6 7). The image vectors would be represented as a 4 rows by 2 columns matrix:

but internally is represented as

When the transformation matrix is operated on the image matrix, the transformation matrix is multiplied on the right of the image matrix.

The last column of the resulting matrix is ignored. Thus the resulting image would have points (4,3) (5,5) (8,2) and (9,9).

A composite transformation is made up of the product of two or more matrices. Take for example, a scaling matrix with factor 2 in x-axis and 3 in y-axis.

internally represented as

When we have a composite transformation of a translation followed by a scaling, the scaling matrix would be multiplied to the right of the translation matrix:

Likewise, if we have a composite matrix of a scaling followed by a translation, the translation matrix would be multiplied to the right of the scaling matrix.

Multiplying to the right is also known as appending, and to the left as prepending. Matrices on the left are always operated first.

## Matrix Transformation

In this article, I would focus only on the following transformations:

- Rotation
- Translation
- Stretching (Scaling)
- Flipping (Reflection)

To create a `Matrix`

object:

Matrix m=new Matrix();

To initialize the values of the matrix at creation:

Matrix m=new Matrix(1,2,3,4,5,6);

The `Matrix`

class implements various methods:

`Rotate`

`Translate`

`Scale`

`Multiply`

To create a composite matrix, first create a identity matrix. Then use the above methods to append/prepend the transformation.

Matrix m=new Matrix();
m.Translate(200,200);
m.Rotate(90,MatrixOrder.Prepend);

In the above code, since the rotation transformation is prepended to the matrix, the rotation transformation would be performed first.

In matrix transformations, the order of operation is very important. A rotation followed by a translation is very different from a translation followed by a rotation, as illustrated below:

## Using the Matrix Object

The following GDI+ objects make use of the `Matrix`

object:

`Graphics`

`Pen`

`GraphicsPath`

Each of these has a `Transform`

property which is a `Matrix`

object. The default `Tranform`

property is the identity matrix. All drawing operations that involve the `Pen`

and `Graphics`

objects would perform with respect to their `Transform`

property.

Thus, for instance, if a 45 deg clockwise rotation matrix has been assigned to the `Graphics`

object `Transform`

property and a horizontal line is drawn, the line would be rendered with a tilt of 45 deg.

The operation of matrix transformation on a `GraphicsPath`

is particularly interesting. When its `Transform`

property is set, the `GraphicsPath`

's `PathPoints`

are changed to reflect the transformation.

One use of this behavior is to perform localized transformation on a `GraphicsPath`

object and then use the `DrawImage`

method to render the transformation.

Graphics g=Graphics.FromImage(pictureBox1.Image);
GraphicsPath gp=new GraphicsPath();
Image imgpic=(Image)pictureBoxBase.Image.Clone();
if(cbFlipY.CheckState ==CheckState.Checked)
gp.AddPolygon(new Point[]{new Point(0,imgpic.Height),
new Point(imgpic.Width,imgpic.Height),
new Point(0,0)});
else
gp.AddPolygon(new Point[]{new Point(0,0),
new Point(imgpic.Width,0),
new Point(0,imgpic.Height)});
gp.Transform(mm1);
PointF[] pts=gp.PathPoints;
g.DrawImage(imgpic,pts);

## Flipping

Unfortunately, there is no flipping method for the `Matrix`

class. However, the matrix for flipping is well known. For a flip along the x-axis, i.e., flipping the y-coordinates, the matrix is (1,0,0,-1,0,0). For flipping the x-coordinates, the matrix is (-1,0,0,1,0,0).

## Affined Transformation

The transformation on an image using the `Matrix`

object is not just a simple point to point mapping.

Take for instance the rectangle with the vertices (0,0) (0,1) (1,1) (1,0). If the units are in pixels, then there are only four pixels making up the whole rectangle. If we subject the rectangle to a scaling matrix of factor 2 in both x and y-axis about the origin (0,0), the resulting rectangle would have vertices (0,0) (0,2) (2,2) (2,0). It would also contain other points within these vertices. How could a 4 points figure be mapped to a figure with more than 4 points?

The answer is that the transformation operation generated those other points that have no direct mapping by interpolation (estimation of unknown pixel values using known mapped pixels).

## Using the Transformation Tester

The use of the demo program is quite intuitive. At startup, the CodeProject beloved iconic figure is loaded. The axes are based on graph-paper coordinate system. There is a check box to unflip the y-coordinates to reflect the computer coordinate system. The origin is set at (200,200) relative to the picture box.

Adjust the tracker for each of the transformation operations, and order the transformation as shown in the list box by using the + and - buttons. Click Go to start the operation.

Thanks to leppie (a reader) for his comment, I have added a new checkbox to allow the user to see real time transformation. After the Real Time checkbox is checked, all adjustments to the trackers and reordering of the transformation will cause the transformation to be performed immediately. This gives the effect of a real time update.

## Conclusion

The code is quite adequately commented. I hope that the reader would benefit from this article and the codes, and start to unlock the power of the `Matrix`

object in .NET.