Chapter 14

1. Chapter 14 Image Warping

2. 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.

3. 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.

4. HCT’s • homogenous transforms include scaling, translation, rotation and shear which, collectively are special cases of affine transforms.

5. Translation

6. setting a translation matrix • given:

7. setting up xlation • publicvoid 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; • }

8. scaling • setting up to scale:

9. Simple Imlementation • publicvoid setScaling(double sx, double sy) { • a[0][0] = sx; • a[1][1] = sy; • a[2][2] = 1; • }

10. To Scale about any point

11. How does this simplify?

12. after working it out…

13. Concating the matrix • publicstaticvoid 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(); • }

14. Rotation

15. Euler’s identity

16. Where does rotation come from? • Multiply a complex number, times another complex number…what do you get? • Use Euler’s identity

17. Power Law of Exponents

18. How do we get Euler? SOHCAHTOA Y r X

19. Euler Rotation

20. Derivation of Rotation

21. matrix form.

22. 2x2*2x1

23. Without Euler

24. implementation of rotation in mat3 • publicvoid 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; • }

25. Using Java2d • AffineTransform atr = new AffineTransform(); • atr.setToTranslation(x1, y1); • atr.scale(sx, sy); • atr.translate(-x1, -y1); • Shape transformedShape = atr.createTransformedShape(gp);

26. 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);

27. 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 • }

28. 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)); • }

29. 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)); • } • }

30. We can thankEuler’s identity!

31. 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.

32. shear

33. setShear • publicvoid 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; • }

34. The AffineFrame

35. rotation

36. scaling

37. shear

38. 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]; • } • }

39. rotation

40. scale

41. shear in x

42. 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);

43. Use Matrix 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.

44. GUI • Main Menu>AffineTransformMenu • RunMenuItems: • Translate… • Rotate…, Scale…, Shear… • RotateAbout…, ScaleAbout…, ShearAbout… • RotateScaleShearAbout… • Use OK and Cancel RunButtons

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