1. Sisteme de programe pentru timp real Universitatea “Politehnica” din Bucuresti 2007-2008 Adina Magda Florea http://turing.cs.pub.ro/sptr_08 si curs.cs.pub.ro

2. Curs Nr. 7 & 8 Programare genetica Generarea imaginilor cu sistemul GAIA 2

3. 1 Programare genetica Aplicarea algoritmilor genetici asupra unei populatii de programe Indivizii = nu gene cu dim fixe ci programe Op genetici: –crossover (2 parinti) –reproducere (1 parinte) –mutatie –duplicare gene –stergere gene 3

4. Programare genetica Programele = arbori sintactici max(x * x, x + 3 * y) Noduri interne = functii Frunze = terminale Programe = compuse din rutine Set de arbori (rutine) grupate intr-o radacina Nr, tip, structura rutine = arhitectura Arborii – notatie prefixata (S-expression) (max (* x x) (+ x (* 3 y))) 4

5. Pasi pregatitori Descriere de nivel inalt 1.Set de terminale (vars, funct cu 0 args, const) 2.Set de functii (aritmetice, conditionale) 3.Fitness (explicit sau implicit) 4.Parametrii de control al AG 5.Conditia de terminare si metoda de alegere a rez Ex: Tragerea la tinta Aspersoare Robot care se deplaseaza 5

6. Algoritmul genetic 1.Populatie generata aleator cu programe compuse din functii si terminale 2.Repeta pentru mai multe generatii Executa fiecare program si determina fitness Selectioneaza cf schemei de selectie Creaza noi indivizi prin aplicarea op gen: –Reproductie: copiaza individ –Crossover: creeaza 2 descendenti din 2 parinti sau 1 parinte –Mutatie: asupra unui individ existent in populatie –Operatii de alterare a arhitecturii – asupra unui individ 3. Testeaza cond de terminare 6

7. Algoritmul genetic 7

8. Metoda de initializare Populatie generata aleator cu programe compuse din functii si terminale Construieste aleator arborele pana la o adancime maxima 1.Metoda “Full” – numai functii pana la nivelul frunzelor 2.Metoda “Grow” – functii si terminale pe orice nivel 8

9. Cum se genereaza gen_rnd_expr(func_set, term_set, max_d, method) if max_d = 0 or method = “Grow” and random_digit = 1 then expr = choose_random_element(term_set) else func = choose_random_element(func_set) for i=1 to arity(func) do arg_i = gen_rnd_expr(func_set, term_set, max_d-1, method) expr=(fun, arg_1, arg_2,..) return expr end 9

10. Full si Grow 10

11. Operatori genetici Crossover 2 parinti = parinti diferiti Crossover 1 parinte = parinti identici (reproducere) Crossover point = 90% nodui interne, 10% frunze Mutatie = crossover intre un program existent si unul generat aleator Toate programele obtinute sunt valide din punct de vedere sintactic 11

12. Crossover 12

13. Crossover 13

14. Mutatie 14

15. Evaluare Compilare separata Interpretare cod pe masina virtuala eval(expr) if expr este lista then proc=expr(1) value value = proc(eval(expr(2)), eval(expr(3)),..) else if exp este var sau const value then value =expr value else value = expr(0) value return value end 15

16. Evaluare Fitness eroarea intre iesirea actuala si iesirea dorita cat de bine recunoaste anumite sabloane castig in jocuri, etc. Valori calculate si efecte laterale Executat pe seturi de valori – fitness cases 16

17. Exemplu Program genetic pt calculul radacinilor ecuatiei x 2 + x + 1 T (terminal set)={x,R}, R = (-5, 5) F (function set) = {+, -, *, /}, cu / modif pt 0 Fitness =  (-1,+1) |(x 2 + x + 1)- (v 2 + v + 1)| In practica se aproximeaza – mai multe valori pt x Selectie: turneu, proportionala cu fitness Crossover – 90 % din populatie Reproductie - 8% Mutatie - 2% Fitness < 0.01 Grow 17

18. Exemplu 18 - + 0 x 1 + 1* xx 2 + 0 * -2 -x (a) x+1(b) x 2 +1(c) 2(d) x 0.671.01.672.67

19. 19 - + 0 x 1+1 2 +** x x0 -2 -x-+0x1+/x-0+xx0 x+ *1 ReprodMutatieCrossover 1x (a) x+1, 0.67(b) x 2 +1, 1.0(c) 2, 1.67(d) x, 2.67 (a) x+1, 0.67(b) 1, 1.0 (c) x, 2.67 (d) x 2 +x+1, 0

20. 2 GA pentru generare de imagini Programul Gaia – generarea de noi imagini Fiecare imagine este generata prin evaluarea unei formule matematice Sa se gaseasca formule care prin evaluare sa produca imagini interesante. GA pt a produce astfel de formule Se genereaza aleator o formula si se genereaza variatii ale acesteia Utilizatorul selecteaza cele mai interesante imagini Formulele selectate devin noi generatii; procesul se repeta Conceptul de evolutie este utilizat pentru a genera noi formule din cele existente 20

21. GA pentru generare de imagini 21

22. GA pentru generare de imagini 22 (lerp #(0.524 0.389 -0.394) (- (triwave (RAD)) 0.590) (PHY)) (color_grad "earth" (gradient (log (+ (&& (lerp (PHY) #(-0.080 0.408 0.254) (RAD)) (RAD)) #(-0.098 -0.277 - 0.840))))) (color_noise (mod (warped_color_noise (X) -0.003 -0.296 #(-0.359 0.020 - 0.790)) (Y)) (color_noise (X) (invert (Y))))

23. GA pentru generare de imagini 23

24. GA pentru generare de imagini 24 (lerp (* (vector -0.422 (warped_bw_noise (RAD) #(-0.468 - 0.375 -0.624) (Y) (RAD)) (RAD)) (X)) (RAD) (bw_noise (PHY) (log 0.529))) (triwave (- (^ (& (RAD) (PHY)) (PHY)) #(-0.176 0.738 -0.928))) (mynoise (triwave (|| (RAD) (PHY))) (RAD))

25. GA pentru generare de imagini 25

26. GA pentru generare de imagini 26 (lerp (* (X) (X)) (/ (/ #(0.204 0.166 0.711) (RAD)) (RAD)) (bw_noise (RAD) (RAD))) (triwave (lerp (min (lerp (PHY) (lerp (PHY) 0.033 (RAD)) #(0.050 0.137 - 0.586)) (PHY)) (IRAD) (RAD))) (color_noise (cos (+ (bw_noise (mod (X) (Y)) #(-0.419 -0.415 0.673)) (Y))) 0.296)

27. GA pentru tranzitii de imagini Programul poate genera de asemenea tranzitii intre imagini generate Utilizatoruls electeaza sursa si desinatia si programul gaseste secventa de imagini care face tranzitia Daca formulele nu au nimic in comun, tranzitiile sunt pure amestecuri ale imaginilor Daca formulele au similaritati se obtin tranzitii interesante 27

28. GA pentru tranzitii de imagini 28 Source: (triwave (abs (RAD))) Target: (triwave (abs (X)))

29. GA pentru tranzitii de imagini 29 Source: (triwave (abs (X))) Target: (triwave (&& (PHY) (PHY)))

30. GA pentru tranzitii de imagini 30 Source: (/ (& (Y) (X)) (RAD)) Target: (/ (& (Y) (RAD)) (RAD))

31. 2.1 Genotip Gaia codifica o imagine printr-o formula si un domeniu Ambele elemente sunt genotipul solutiei Formulele = set de operatori, variabile si constante Formula – S-expression Domeniul indica unde se evalueaza formula Domeniul este o regiune in planul real specificata de limite Domeniul implicit este [-1..1] x [-1..1]. 31

32. Genotip Formulele pot fi oricat de lungi Operatorii cu argumente – noduri interne Operatorii fara argumente – frunze S-Expression Domeniu (+ (+ (X) (Y)) (Y)) [0..1] x [0..1] (gradient (invert (- (0.5) (RAD)))) [-1..1] x [-1..1] (abs (lerp (mynoise (/ (triwave (mod (PHY) -0.190)) (RAD)) (min (RAD) -0.873)) (Y) (X))) [-1..1] x [-1..1] 32

33. 2.2 Formule si operatori 5 clase de operatori: 1.Operatori de domeniu: X,Y,RAD,PHY – depind de domeniul pe care se evalueaza formula. X, Y Returns an image which is the values of the domain in the X- axis or Y-axis directions. The resulting images are horizontal and vertical ramps of luminance. Images obtained with the domain [0..1] x [0..1] 33

34. Formule si operatori RAD The resulting image depends directly on the domain where it is evaluated. The luminance of each pixel in the image is the distance from the coordinates of the pixel in the domain to the origin, typically at the center. IRAD Similar to the operador RAD, the luminance of each pixel is the distance measured to the nearest odd integer in the domain. The black corners represents the four integer coords (-1,-1) (- 1,1) (1,1) (1,-1) of the domain where the image was rendered. 34

35. Formule si operatori PHY The luminance of the resulting image represents the angular coordinate of the pixel, with the zero heading down.There where the value is greater than 1 a white pixel is used. 35

36. Formule si operatori 2. One argument functions: From a single image, returns a new image after applying some function to the pixels values of the argument. Some examples of functions are cos, sin, normalize, gradient, abs, round, triwave,...triwave TRIWAVE This operator is bounded in the interval 0..1 for all the source values. This means that we can feed this operator with any image with any pixel values, that the resulting image will be limited between 0 and 1. 36

37. Formule si operatori 3. Simple operators: Combines two images to return the result of the operator. The combination operator can be as easy as add the images, substract, multitply, or combine logically at pixel level. 4. Complex operators: Need more than two images as arguments and returns a more complex combination of them. They usually use some of them as parameters of the combination. Examples: LERP, noise functions, color grad,... 5.Misc operators: The operators which uses external images imported from other programs or those which use the templates of the program 37

38. Formule si operatori LERP It takes three images as arguments: A,B,C and computes the resulting image as A+(B-A)*Triwave(C). It makes a linear interpolation from A to B, using C as weight of the interpolation. The operator Triwave is included to limit C in the interval 0..1. 38

39. 2.3 Mutatie Evolution is achived by two methods: Mutation and Combination. Mutation Combination Mutations If we are using a long formula to obtain an image, and make some little changes in the formula, the final image will be in general quite similar but different. Mutations are normally applied on a single node, and affects this node and maybe nodes in lower levels. The node is chosen randomly in the tree. 39

40. Mutation New Node: The selected node is substituted by a new simply random node. This type of mutation can achieve important changes when the original tree is big, and the node is near the root of the tree. Adjust Node: The node is modified by another one with the same number of parameters. The tree maintains the same structure, but instead of computing the cos (maybe) we will compute the sin, or maybe we will change a constant with value 0.7 by a new one with value 0.5. This type of mutation makes light changes. Node as arg: The selected node becomes the argument of a new random node which is placed in his previous location in the tree, and the old node hanging from it. If the new node needs more arguments they are generated randomly. 40

41. Mutation Arg as node: The inverse operation. One of the arguments of the node substitutes the node itself. The new tree is simpler than the original. If the original node got more arguments, they are discarded. Node as uncle: The node is substituted by a copy of a node in upper levels of the tree. Reorder arguments: This operator changes the order of the arguments of the selected node (assuming it has more than two). It can be very different to compute X / RAD or RAD / X.X RAD The default number of mutations applied is one, but when formulas become bigger, it is preferable to use two or three. In Gaia, there is only one parent. Children are obtained by performing different types of mutations in different nodes of the same parent. 41

42. Mutation examples Parent expression: (triwave (mod (triwave (PHY)) (RAD))) Domain: [-1..1] x [-1..1] 42

43. Mutation examples 43 (triwave (mod (triwave (PHY)) (RAD))) (triwave (mod (lerp (PHY) (Y) (Y)) (RAD)))

44. Mutation examples 1.(triwave (mod (lerp (PHY) (Y) (Y)) (RAD))) 2.(mod (triwave (PHY)) (RAD)) 3.(triwave (mod (^ (triwave (PHY)) (X)) (RAD))) 4.(triwave (mod (PHY) (RAD))) 5.(triwave (lerp (mod (triwave (PHY)) (RAD)) (Y) (RAD))) 6.(triwave (mod (max (triwave (PHY)) (RAD)) (RAD))) 7.(triwave (sin (triwave (PHY)))) 8.(triwave (mod (+ (triwave (PHY)) (RAD)) (RAD))) 9.(triwave (mod (triwave (PHY)) (warped_color_noise (RAD) #(0.653 0.865 -0.369) #(0.683 0.337 -0.625) (X)))) 10.(mod (mod (triwave (PHY)) (RAD)) (PHY)) 11.(triwave (mod (RAD) (triwave (PHY)))) 12.(triwave (rotate (mod (triwave (PHY)) (RAD)) (RAD))) 13.(triwave (mod (gradient (triwave (PHY))) (RAD))) 14.(invert (triwave (mod (triwave (PHY)) (RAD)))) 15.(triwave (* (mod (triwave (PHY)) (RAD)) (X))) 16.(triwave (mod (abs (PHY)) (RAD))) 44

45. 2.4 Combination We want to find a method to generate images that share characteristics of two previously generated images. To accomplish this we will rely on the fact that the relation between formulas and images is basically associated to the sequences of operators used in the genotypes, rather than the individual operators used. Try to maintain sequences of nodes of the parent and mother in the new genotypes. Combination may be applied several times 45

46. Combination 46 Parent Mother Child 1Child 2

47. Combination examples Parent expression: (lerp (* (vector -0.422 (warped_bw_noise (RAD) #(-0.468 -0.375 - 0.624) (Y) (RAD)) (RAD)) (X)) (RAD) (bw_noise (PHY) (log 0.529))) Domain: [-1..1] x [-1..1] Mother expression: (lerp (triwave (PHY)) (* (lerp #(0.524 0.389 -0.394) (- (triwave (RAD)) 0.590) (PHY)) (triwave (mod (triwave (PHY)) (RAD)))) (RAD)) Domain: [-1..1] x [-1..1] 47

48. Combination examples 48

49. Combination examples (lerp (triwave (RAD)) (* (lerp #(0.524 0.389 -0.394) (- (triwave (RAD)) 0.590) (PHY)) (triwave (mod (triwave (PHY)) (RAD)))) (RAD)) (lerp (triwave (PHY)) (* (lerp #(0.524 0.389 -0.394) (bw_noise (PHY) (log 0.529)) (PHY)) (triwave (mod (triwave (PHY)) (RAD)))) (RAD)) (lerp (triwave (PHY)) (* (lerp #(0.524 0.389 -0.394) (- (triwave (RAD)) 0.590) (PHY)) (triwave (mod #(-0.468 -0.375 -0.624) (RAD)))) (RAD)) (* (lerp #(0.524 0.389 -0.394) (- (triwave (RAD)) 0.590) (PHY)) (triwave (mod (triwave (PHY)) (RAD)))) (lerp (triwave (PHY)) (* (lerp #(0.524 0.389 -0.394) (- (triwave (RAD)) 0.590) (X)) (triwave (mod (triwave (PHY)) (RAD)))) (RAD)) (lerp (* (vector -0.422 (warped_bw_noise (RAD) #(-0.468 -0.375 -0.624) (Y) (RAD)) (RAD)) (- (triwave (RAD)) 0.590)) (RAD) (bw_noise (PHY) (log 0.529))) (lerp (triwave (PHY)) (* (lerp #(0.524 0.389 -0.394) (- (triwave (RAD)) 0.590) (PHY)) (triwave (mod (triwave (PHY)) (RAD)))) (bw_noise (PHY) (log 0.529))) (lerp (* (vector -0.422 (warped_bw_noise (RAD) #(-0.468 -0.375 -0.624) (Y) (lerp #(0.524 0.389 -0.394) (- (triwave (RAD)) 0.590) (PHY))) (RAD)) (X)) (RAD) (bw_noise (PHY) (log 0.529))) 49

50. Combination examples (lerp (triwave (PHY)) (* (lerp #(0.524 0.389 -0.394) (- (triwave (RAD)) 0.590) (PHY)) (triwave (mod (triwave (PHY)) (RAD)))) (RAD)) (lerp (triwave (PHY)) (* (lerp #(0.524 0.389 -0.394) (RAD) (PHY)) (triwave (mod (triwave (PHY)) (RAD)))) (RAD)) (lerp (* (vector (PHY) (warped_bw_noise (RAD) #(-0.468 -0.375 -0.624) (Y) (RAD)) (RAD)) (X)) (RAD) (bw_noise (PHY) (log 0.529))) (lerp (* (vector -0.422 (warped_bw_noise (RAD) #(-0.468 -0.375 -0.624) (Y) (RAD)) (RAD)) (X)) (RAD) (bw_noise (PHY) (log (RAD)))) (lerp (triwave (PHY)) (log 0.529) (RAD)) (lerp (* (vector -0.422 (warped_bw_noise (RAD) #(-0.468 -0.375 -0.624) (Y) (RAD)) (RAD)) (X)) (* (lerp #(0.524 0.389 -0.394) (- (triwave (RAD)) 0.590) (PHY)) (triwave (mod (triwave (PHY)) (RAD)))) (RAD)) (lerp (triwave (PHY)) (* (lerp #(0.524 0.389 -0.394) (- (triwave (RAD)) 0.590) (PHY)) (triwave (bw_noise (PHY) (log 0.529)))) (RAD)) (lerp (triwave #(-0.468 -0.375 -0.624)) (* (lerp #(0.524 0.389 -0.394) (- (triwave (RAD)) 0.590) (PHY)) (triwave (mod (triwave (PHY)) (RAD)))) (RAD)) 50

51. 2.5 Image transitions The software is able to make smooth transitions between images The user selects one image as the source, and another as the target, and Gaia will make automatically a smooth transition between both images. Depending on the semblance of both source and target, the transition will be a simple melt if the genotypes have nothing in common, but it can be an interesting transition if there are some similarities in the genotypes. We say that two genotypes are similar if they share the upper nodes of the genotypes (the nodes near the root node). The more nodes they share, the more similar they are. If both expressions have different root nodes, the expressions have nothing in common. 51

52. Image transitions To obtain the animation, Gaia generates a new genotype parametrized in time. This new genotype has the common nodes of source and target images, and interpolates in time the nodes which are different. This means that when the parameter t = 0, the genotype is equivalent to the source genotype, while when t = 1, the genotype match the target. Between, the genotype is a proportional mix of both genotypes. Important here: making a simply melt of the different nodes, but when apply the operators, the melt is visualized as a transformation of forms in the image, and not just as an interpolation of colors. 52

53. Image transitions Source genotypeTarget genotypeNew genotype The blue nodes are common to both source and target genotypes, while the red and green nodes are different. The gray nodes are inserted to interpolate in time the different nodes. The program inserts this special join node every time the nodes of source and target differs. (1-t)*Y+t*(COS(2*PHI)) 53

54. Image transitions Source genotypeTarget genotypeNew genotype Source genotype: (+ (X) (* (RAD) (Y))) = X + RAD*Y Target genotype: (+ (X) (* (RAD) (Cos (* (2) (PHY))))) = X + RAD*(Cos(2*PHY)) New genotype: (+ (X) (* (RAD) (+ (* (1-t) (Y)) (* (t) (Cos (* (2) (PHY))))))) = X + RAD* ((1-t)*(Y) + (t)*(Cos(2*PHY))) 54

55. 55 Source: (triwave (abs (X))) Target: (triwave (&& (PHY) (PHY))) Image transitions

56. 56 Source:(triwave (abs (RAD))) Target:(triwave (abs (X))) Image transitions

57. 57 Source: (lerp (/ (gradient (& (PHY) (RAD))) (RAD)) (Y) #(0.545 0.495 0.981)) Target: (lerp (Y) #(0.545 0.495 0.981) (/ (gradient (& (PHY) (RAD))) (RAD))) Image transitions