chapter 14 image warping. why do image warping? looks cool! can correct for optical distortion (i.e....
DESCRIPTION
Homogenious All the same. Homogenious point processing means that all the points are processed the same. Homogenious coordinate transforms means that all coordinates are transformed the same way.TRANSCRIPT
Chapter 14
Image Warping
Why do image warping?
• Looks cool!• Can correct for optical distortion (i.e.
keystoning).• Remote Sensing (matching together
multiple images).• Entertainment value (morphing)• Special Effect• Looking for lost people.
Homogenious
• All the same.• Homogenious point processing means that
all the points are processed the same.• Homogenious coordinate transforms means
that all coordinates are transformed the same way.
HCT’s
• homogenous transforms include scaling, translation, rotation and shear which, collectively are special cases of affine transforms.
Translation
11001001
1''
yx
tt
yx
y
x
x' x tx
y' y ty
setting a translation matrix
• given:
222120
121110
020100
aaaaaaaaa
A
setting up xlation
• public void setTranslation(double tx, double ty) {
• a[0][0] = 1;• a[1][1] = 1;• a[2][2] = 1;• a[0][2] = tx;• a[1][2] = ty;• }
scaling
• setting up to scale:
11000000
1''
yx
ss
yx
y
x
x' sx xy'syy
Simple Imlementation
• public void setScaling(double sx, double sy) {
• a[0][0] = sx;• a[1][1] = sy;• a[2][2] = 1;• }
To Scale about any point
x 'y '1
1 0 tx
0 1 ty
0 0 1
sx 0 00 sy 00 0 1
1 0 tx
0 1 ty
0 0 1
xy1
How does this simplify?
x 'y '1
sx 0 tx sxtx
0 sy ty syty
0 0 1
xy1
after working it out…
x 'sx x tx sxtx sx(x tx) tx
y 'sy y ty syty sy(y ty) ty
Concating the matrix• public static void main(String args[]) {• Mat3 tr1 = new Mat3();• Mat3 tr2 = new Mat3();• Mat3 sc = new Mat3();• Mat3 at ;• tr1.setTranslation(1,1);• sc.setScale(2,2);• tr2.setTranslation(-1,-1);• at = tr1.multiply(sc);• at = at.multiply(tr2);• at.print();• }
Rotation
x 'y '1
cos sin 0sin cos 0
0 0 1
xy1
Euler’s identity
e i cos i sin
)21(12 iii eee
Where does rotation come from?
• Multiply a complex number, times another complex number…what do you get?
• Use
Euler’s identity
Power Law of Exponents
a b a be e e
How do we get Euler?
cos sinire r ri
cosr X
Y
r sinr
SOHCAHTOA
Euler Rotation
2sin112cos2sin12cos1
'
2sin2cos22)2112(2121'
)22)(11('
)21(12
xyyx
p
iiyxyxyxiyyxxp
iyxiyxpeee iii
Derivation of Rotation
1 2 1 2( )1 1 2 2
1 2 1 2 1 2 1 2
1 2 1 2 1 2 1 2
2 2 1
2 2 1
cos sin
( )(cos sin )cos sin cos sincos sin ( sin cos )
' cos sin 0' sin cos 0
1 0 0 1 1
i
i i i
e i x iy
e e e x iy ix x i iy iy ix y i x y
x xy y
2 2
2 2
' cos sin' sin cos
x x yy x y
matrix form.
x 'y '1
cos sin 0sin cos 0
0 0 1
xy1
2x2*2x1
1 2 1 21 2
1 2 2 1
2 2 1
2 2 1
cos sinsin cos
cos sinsin cos
x yp p
x y
xy
Without Euler
1 1 1
2 2 2
1 2 1 21 2 1 1 2 2
1 2 2 1
2 2 2 2
1 2 1 21 2
1 2 2 1
(cos ,sin ) ( , )cos sinsin cos
p x iyp x iy
x x y yp p x iy x iy
x y x yby
x yx y
p px y
implementation of rotation in mat3
• public void setRotation(double theta) {• theta = theta * Math.PI/180;• double cas = Math.cos(theta);• double sas = Math.sin(theta);• a[0][0] = cas;• a[1][1] = cas;• a[0][1] = -sas;• a[1][0] = sas;• }
Using Java2d
• AffineTransform atr = new AffineTransform();
• atr.setToTranslation(x1, y1);• atr.scale(sx, sy);• atr.translate(-x1, -y1);• Shape transformedShape =
atr.createTransformedShape(gp);
Using mat3 to scale and rotate• Mat3 tr1 = new Mat3();• Mat3 tr2 = new Mat3();• Mat3 rt = new Mat3();• Mat3 sc = new Mat3();
• tr1.setTranslation(getCentroidX(), getCentroidY());• sc.setScale(1, 1);• rt.setRotation(0);• tr2.setTranslation(-getCentroidX(), -getCentroidY());• at = tr1.multiply(rt);• at = at.multiply(sc);• at = at.multiply(tr2);
J2d, lets rotation occur about any point
• public void drawRotateGraphics(Graphics g) {• final int xc = getCentroidX();• final int yc = getCentroidY();• Graphics2D g2d = (Graphics2D) g;• AffineTransform saveAt = g2d.getTransform();• for (float theta = 0; theta <= 360; theta += 10f) {• g2d.setTransform(AffineTransform.getRotateInstance(• theta * PI_ON_180,• xc, yc));• g2d.draw(p);• }• g2d.setTransform(saveAt);• // This leaves the g2d back on 0 degrees of rotation• }
Rotate with a new shape• public void drawTransformedShape(Graphics g) {• final int xc = getCentroidX();• final int yc = getCentroidY();• Graphics2D g2d = (Graphics2D) g;• for (float theta = 0; theta <= 360; theta += 10f) {• final AffineTransform at =
AffineTransform.getRotateInstance(theta *• PI_ON_180,• xc, yc);• g2d.draw(at.createTransformedShape(p));• }
Or use Mat3 to Draw• public void drawMat3(Graphics g) {• final int xc = getCentroidX();• final int yc = getCentroidY();• tr1.setTranslation(xc, yc);• tr2.setTranslation(-xc, -getCentroidY());• for (float theta = 0; theta < 360; theta += 10f) {
• rt.setRotation(theta);• at = tr1.multiply(rt);• at = at.multiply(tr2);• drawPolygon(g, at.transform(p));• }• }
We can thankEuler’s identity!
e i cos i sin
)21(12 iii eee
Who was Euler?
• Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced "oiler"). Lived to be 76.
• first to use the term "function" (defined by Leibniz - 1694) to describe an expression involving various arguments; ie: y = F(x).
• A mathematical child prodigy. • professor of mathematics in Saint Petersburg, and Berlin, • Most prolific mathematician of all time, 75 volumes. • blind for the last seventeen years of his life, during which
time he produced almost half of his total output.
shear
x 'y '1
1 shx 0shy 1 00 0 1
xy1
setShear
• public void setShear(double shx, double shy) {• a[0][0] = 1;• a[1][1] = 1;• a[2][2] = 1;• a[0][1] = shx;• a[1][0] = shy;• }
The AffineFrame
rotation
scaling
shear
destination scanning• transform = transform.invert();• for (int x = 0; x < w; x++)• for (int y=0; y < h; y++) {• p=transform.multiply(x,y);• xp = (int) p[0];• yp = (int) p[1];• if ((xp < w) && (yp < h) && (xp >= 0) && (yp >= 0)) {• rn[x][y] = r[xp][yp];• gn[x][y] = g[xp][yp];• bn[x][y] = b[xp][yp];• }• }
rotation
scale
shear in x
Create the combinations
– Image scaleAbout(image, tx, ty,sx,sy);– Image rotateAbout(image, tx, ty, theta);– Image shearAbout(image, tx, ty, shx, shy);– Image rotateShearScale(image, theta, shx,shy,
sx, sy);Image rotateShearScaleAbout(image, tx,ty,
theta, shx,shy, sx, sy);
UseMatrix concatenation
• Use matrix concatenation for everything.• Only a single 3x3 matrix will result when
we are done.• Use the AffineTransform Class, as
described on pp. 135 of the handout.
GUI
• Main Menu>AffineTransformMenu• RunMenuItems:
– Translate…– Rotate…, Scale…, Shear…– RotateAbout…, ScaleAbout…, ShearAbout…– RotateScaleShearAbout…– Use OK and Cancel RunButtons
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