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Transformations

- We want to be able to make changes to the image
- larger/smaller
- rotate
- move
- This can be efficiently achieved through mathematical operations known as transformations

Transformations

- We will transform the endpoints only
- If we then draw the (new) lines between the transformed endpoints, we get the transformed image
- This only works for certain types of transformations known as affine transformations
- Such transformations preserve lines and distances and relative proportions
- i.e., points on the same line before remain on the same line after an affine transformation

Transformations

- Three transformations that fall into this category are
- Scaling
- Rotation
- Translation

But First…

- We’re going to need a bit of math…
- … just enough to get the general idea

Matrices

- Matrix
- 2 dimensional array (of numbers)
- m x n matrix
- m rows
- n columns

Matrices

- Matrix
- 2 dimensional array (of numbers)
- m x n matrix
- m rows
- n columns
- xij is the entry at row I, column j

2 rows

3 columns

Matrix Multiplication

- In matrix multiplication, elements in the result matrix are obtained by taking the sums of the products of the elements of a row of the first with a column of the second
- Calculating the sum of products of the ith row with the jth column produces the element at location [i][j]

Matrix Multiplication

- In order to calculate a sum of products, the length of a row of the first matrix must be equal to the length of a column in the second matrix
- length of a row = # columns
- length of a column = # rows

3 columns

row

column

3 rows

Matrix Multiplication

- Can therefore only multiply m x k matrix with a k x n matrix
- # of columns of first operand = # rows of second operand
- Results in an m x n matrix

3 rows

2 rows

3 rows

4 columns

2 columns

4 columns

An Example

(1 1) + (2 5) = 1 + 10 = 11

An Example

(1 2) + (2 6) = 2 + 12 = 14

An Example

(1 3) + (2 7) = 3 + 14 = 17

An Example

(1 4) + (2 8) = 4 + 16 = 20

An Example

(3 1) + (4 5) = 3 + 20 = 23

An Example

(3 2) + (4 6) = 6 + 24 = 30

An Example

(3 3) + (4 7) = 9 + 28 = 37

An Example

(3 4) + (4 8) = 12 + 32 = 44

An Example

(5 1) + (6 5) = 5 + 30 = 35

An Example

(5 2) + (6 6) = 10 + 36 = 46

An Example

(5 3) + (6 7) = 15 + 42 = 57

An Example

(5 4) + (6 8) = 20 + 48 = 68

The Algorithm

multiply(a, b) // a = M x K b = K x N

result = new Matrix(m, n)

for i = 1, M // M rows in a

for j = 1, N // N columns in b

result[i][j] = 0

for k = 1, K // K columns in a, rows in b

result[i][j] += a[i, k] * b[k, j]

return result

What’s this got to do with us?

- Matrices are a convenient and powerful way of expressing transformations
- Allows complex sequences of complex transformations to be easily expressed and calculated
- Let’s look at one simple transformation and see how

Scaling

- Transformation that enlarges or reduces image

Scaling

- Scaling can be done in the x-coordinate …

Scaling

- … in the y-coordinate …

Scaling

- … or in both …

Scaling

- We could simply say that
- To scale in the x-coordinate, multiply by the scaling factor
- that is, to scale to double the size in the x-coordinate, simply multiply the x-coordinate of all endpoints by 2
- Similarly to reduce the size
- Similarly in the y-direction

Simple enough

- The above works and is totally adequate to scale
- Why complicate matters?
- Why even consider doing anything else?

Multiple Transformations

- Will want to
- scale and rotate
- translate, rotate and translate again
- etc,…
- Don’t want to have to apply each transformation individually

Using Matrices

- Let’s represent a point as a 1 x 2 matrix
- We often call a 1 x n matrix a vector
- Let’s reexamine multiplying this vector with a 2 x 2 matrix

Applying Matrix Multiplication

- We can think of the above multiplication taking the point (x, y) and producing a new point (x', y') where
- x' = ax + cy
- y' = bx + dy

Transformation Matrix

- We see that when a 2 x 2 matrix
- is multiplied with a 1 x 2 vector representing a point …
- … a new 1 x 2 vector is produced …
- … that can be though of as representing a new point
- We thus call the 2 x 2 matrix a transformation matrix
- The matrix when applied to the original point transforms it into the new point

Where Matrix Multiplication Comes In

- Looking at the above we can get a sense of how the 2 x 2 matrix transforms the point:

y'

x'

b: the effect of the original x-value on the new y-value

a: the effect of the original x-value on the new x-value

d: the effect of the original y-value on the new y-value

c: the effect of the original y-value on the new x-value

An Trivial Example

- Following this line of thought, the matrix:

b: the original x-value has no effect on the new y-value

a: the original x-value has an identity effect on the new x-value

d: the original y-value has an identity effect on the new y-value

c: the original y-value has no effect on new x-value

should transform the original point back to itself

Applying this to Scaling

- Using this approach, let’s try to produce some transformation matrices for scaling
- Let’s first scale the x-coordinate alone
- We’d like
- the new (transformed) x-value
- to be a factor of the original x-value
- not be affected by the original y-value
- the new (transformed) y-value
- to be identical to the original x-value
- (not be affected by the original x-value)

Doubling the Size in the x-Direction

- As an example, to double the x-value
- We’d like
- the new (transformed) x-value
- to be 2 times the original x-value
- not be affected by the original y-value
- the new (transformed) y-value
- to be identical to the original x-value
- (not be affected by the original x-value)

The Effect of the Transformation Matrix

- By recalling how the transformation matrix affects the original point, we can come up with the following ‘educated’ guess

b: the effect of the original x-value on the new y-value

a: the effect of the original x-value on the new x-value

d: the effect of the original y-value on the new y-value

c: the effect of the original y-value on the new x-value

Other Scaling Matrices

- The same line of reasoning produces
- The general transformation matrix for scaling in the x-direction alone
- The general transformation matrix for scaling in the y-direction alone
- The general transformation matrix for scaling in both directions

For practice, verify these by doing the matrix multiplications!!

Applying Multiple Transformations

- If we multiply the ‘scale x’ matrix and the ‘scale y’ matrix, we obtain the scale matrix for both

Applying Multiple Transformations

- Similarly, if we multiply the ‘double size’ matrix and the ‘half size’ matrix, we obtain the identity matrix

Applying Multiple Transformations

- Although we won’t prove it, it can be shown that multiplying two transformation matrices produces a transformation matrix whose effect is the first transformation followed by the second!
- This result extends to three or more as well

Applying Multiple Transformations

- This is a valuable result because it means we can achieve the effect of several transformation by applying a single matrix to our image rather than having to perform a sequence of transforms.

Rotations About the Origin

- The next transformation involves rotating the endpoints (and therefore the line) about the point (0, 0)

Rotations About the Origin

- Again, we will try to derive the transformation matrix
- This one is a bit more involved and requires some trigonometry and geometry

y axis

- We view the point (x, y) as the endpoint of a line segment whose other end is the origin
- The line segment forms some angle-- call it θ -- with the x-axis

(x, y)

θ

x axis

(0, 0)

- the rotation involves rotating the endpoint (x, y) around the origin to a new point (x', y').
- the other endpoint remains the origin,
- the length of the line remains the same.

y axis

B

(x', y')

A

(x, y)

x axis

O

Rotation About the Origin

- We’d like to define the value of the new (transformed) point (x',y') in terms of the original point(x, y)
- If we can do that, we can come up with a transformation matrix!
- And, again, as we said before, this will require a bit of math

y axis

- we can think of the rotation as ‘increasing’ the original angle of the line, θ,by an additional amount,

(x', y')

(x, y)

θ

x axis

The Sum of Two Angles

- Given two angles and :

sin( + ) = cos sin + sin cos

cos( + ) = cos cos - sin sin

- We’re not going to derive these formulae

Remember SOHCAHTOA?

θ

θ

θ

- For right triangles
- sine = opposite / hypotenuse
- cosine = adjacent / hypotenuse
- tangent = opposite / adjacent (we won’t be using this)

y axis

- Recall, the length of the line, L, stays the same
- The angle of the line ending at (x2y2) is θ+

(x2, y2)

(x1, y1)

L

L

θ

x axis

y axis

From the diagram:

(x2, y2)

and we stated before that:

so…

(x1, y1)

L

y2

θ

x axis

Through the Magic of Algebraic Manipulation

And since

We get

We have defined y2 in terms of x1 and y1– exactly what we were looking for!!!

More Magic of Algebraic Manipulation

And again, since

We get

We have similarly defined x2 in terms of x1 and y1– again exactly what we were looking for!!!

A Rotation-Around-the-Origin Matrix

- Given

and

we can clearly see the effects of x1 and y1 on x2 and y2

- x1 affectsx2 via cos
- x1 affectsy2 via sin
- y1 affectsx2 via -sin
- y1 affectsy2 via cos

A Rotation Around the Origin Matrix

- This results in the transformation matrix for a rotation about the origin of angle

the effect of the original x-value on the new y-value

the effect of the original x-value on the new x-value

the effect of the original y-value on the new x-value

the effect of the original y-value on the new y-value

Translation

- Moving the image a fixed amount in either
- the x-direction
- x2 = x1 + Tx Tx is the fixed amount to move in the x-direction
- the y-direction
- y2 = y1 + Ty Ty is the fixed amount to move in the y-direction
- both

Translation

- Sounds easy
- add the x translation amount to the x coordinate
- add the y translation amount to the y coordinate
- But we’d like to have a matrix
- Would like to combine our various transformations
- OTOH, is that really all that important?

Rotation About an Arbitrary Point

- We’d like to rotate around points other than the origin

Rotation About an Arbitrary Point

- We can accomplish this by
- Translating the desired rotation point to the origin…

Rotation About an Arbitrary Point

- …rotating about the origin…

Rotation About an Arbitrary Point

- …translating back to the original point…

The Problem

- Our transformation matrices till now had entries for
- How the old x affects the new x and y
- How the old y affects the new x and y

b: the effect of the original x-value on the new y-value

a: the effect of the original x-value on the new x-value

d: the effect of the original y-value on the new y-value

c: the effect of the original y-value on the new x-value

The Problem

- In a translation, the changes are fixed
- independent of the original x and y values
- Where would they go in the matrix?

Homogeneous Coordinates

- An approach to incorporating a fixed translation into a transformation matrix
- Using homogeneous coordinates involves…

Homogeneous Coordinates

- Using a 3 x 3 transformation matrix rather than a 2 x 2…
- For example, our scaling matrix becomes
- Not too bad– the extra row/column looks like an identity matrix
- The 0’s and 1 shouldn’t make the sums of products much harder to do
- Similarly for the rotation matrix

Homogeneous Coordinates

- The introduction of an additional ‘dummy’ coordinate, w
- Points are now specified by a 1 x 3 vector
- We can always get x and y back again by dividing by w
- And in any event, don’t get too worried, we’re going to keep w = 1

Homogeneous Coordinates

- Let’s just see the effect of all this
- As an example, we’ll do a scaling
- Which is the correct representation under homogeneous coordinates for the new (scaled) point
- … And similarly for rotation
- You can do the math if you want

A Translation Matrix

- Let’s try to understand this matrix
- The original x and y have an identity effect (the shaded 2 x 2 matrix is the identity matrix) on the new points
- The Tx and Ty will be multiplied by w (if you can’t visualize this, you’ll see it on the next slide) and added into the sum of products
- Dividing the result by w would then produce the fixed translation value
- To see this, let’s do the math

A Translation Matrix

- And again, this is the desired point
- modulo the division by w

Revisiting Rotation about an Arbitrary Point

- Given
- a rotation point of (xc, yc)
- A rotation angle of

(x, y)

(xc, yc)

Revisiting Rotation about an Arbitrary Point

- We first translate (xc, yc) to the origin
- Translation matrix

(xc, yc)

Revisiting Rotation about an Arbitrary Point

- We then perform the rotation (around the origin) of angle …
- Rotation matrix

Revisiting Rotation about an Arbitrary Point

- And finish off with a translation back to (xc, yc)
- Translation matrix

(x, y)

(xc, yc)

A Matrix for Rotation about an Arbitrary Point

- Putting it all together, gives us

and performing the multiplications produces

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