KR20060133847A - Editing graphical objects using noise editing - Google Patents

Abstract

A method for editing graphical objects by noise editing is provided to calculate result values for user's noise restriction demands while keeping statistic characteristics of noise by optimization for minimizing a difference between an ideal statistic value and a statistic value of gradient value of a real noise function. In a method for editing graphical objects by noise editing, M 3D PRN unit vectors are composed to be stored in a G table, which has M size, and M-1 random number permutations from P to O are stored in another table having the same size. 3 integer intervals[qi0,qi1](i=1,2,3) for each dimension of given coordinates are obtained, wherein qi0=[xi]mod M and qi0=qi1+1. The 3 integer intervals generate a single regular tetrahedron. 8 gradient vectors are obtained for each point of the calculated regular tetrahedron: gjkl=G[P[P[Pq1j]+q2k+q3l]], wherein jkl represent all permutations from 0 to 1 having a length of 3. A result value of noise is calculated by calculation of 8 inner products, that is, a weighted sum of gjklÀ[vjkl-(x1,x2,x3)], wherein the weighted sum is calculated by ease curve formula represented by s(t)=6t5-15t4+10t3.

Description

Editing Graphical Objects using Noise Editing}

Figure 1 is a one-dimensional noise function with different random number distribution

Fig. 2 is a structural diagram of noise algorithm and hashing (one-dimensional case)

Figure 3 is an example of the problem of passing through zero and the result of one-dimensional noise

4 is a diagram of the Voxkar and approximation functions.

5 shows the change in chi-square statistic as control is applied to the noise function.

Figure 6 is a result of FBM (Fractional Brownian Motion) control

Figure 7 is an animation path control result

8 shows the result of the flame animation control

Fig. 9 shows the control result of the progressive texture

10 is a randomized terrain control diagram.

<Description of the symbols for the main parts of the drawings>

Figure 1. (a): One-dimensional noise function with uniform random distribution.

Figure 1. (b): One-dimensional noise function with a skewed distribution.

Figure 1. (c): 500 times of control using the control method of the present invention was applied.

Figure 4. (a): Original Voxkar function.

Figure 4. (b): Approximate boxcar using approximation curve.

Figure 7. (a): When animation path collides with an external object by noise

Figure 7. (b): Result of resolving collision through noise constraint

Figure 8. (a): Flame animation using noise

Figure 8. (b): Result animation after control is made in the direction of the arrow

Figure 9. (a): Wood pattern using noise

Figure 9. (b): Control result in the direction of arrow

Figure 9. (c): difference map between (a) and (b)

Figure 9. (d): Result of noise scaling applied

Figure 10. (a): Randomized terrain

Figure 10. (b): Terrain under control.

Since Perlin introduced the noise function, it has been used to create progressive textures, to represent various natural shapes such as water or fire, and to make moving motion more natural. Noise has been used in almost every process of simulating natural phenomena and is sometimes used as an alternative to complex physics-based simulations. Recently, noise algorithms have also been implemented in hardware for real-time graphics. Spectral synthesis has been proposed as another algorithm to represent natural phenomena, but Perlin's algorithm is becoming more preferred in computer graphics due to lack of fluidity in control.

There has been a lot of research on the application of noise, but very little research controls it. Previous studies have focused only on the fundamental variables for fractal sums, which consist of noise or sums of noises. Perlin recently proposed a new algorithm that solves two problems of the existing noise algorithm. He proposed a new interpolation method to satisfy C ² continuity in all domains, and enabled fast calculations while avoiding skewed results through 16 fixed bias vectors. We did not use any of these two improvements to fix the bias vector. Because our algorithm itself is to obtain the desired result through control of the bias vector. Some researchers have studied how to control the shape or specific phenomena of objects through stochastic models. Through various studies, we have proposed a method that can be easily implemented in general rendering software through the use of a multiple scale stochastic model, which requires less storage space. In the study of Texture Synthesis, many have attempted to use Approximated Statistical Fields to produce large images similar to a given small image.

Based on the theory of multi-step modeling, our theory is the last step for fine-tuning the animation. In this study, we focused on controlling Perlin's bias noise algorithm, the most frequently used application.

 The present invention can be used when the result value that meets the needs of the user is required while maintaining the nature of the noise. For example, when a particular animation is made in a progressive form, random numbers are added through noise functions or their Fractal Sums for more naturalness. Since the exact form after the random number is applied cannot be predicted, it is difficult to meet exactly the other constraints desired by the user. If, like physics simulation, the incremental calculations that make an animation require complex calculations, you may not want to recalculate the animation paths for constraints. Even if the path is recalculated, there is no guarantee that it will satisfy the constraint because random numbers due to noise are added. Therefore, in this case, there is a need for a method for controlling noise, that is, control of added random numbers, not control of an underlying path. However, if there is an artificial variation of the bias, the source of the noise function, for the particular desired result, then the ideal noise characteristics mentioned above are difficult to maintain. This is because an artificial variation in bias can result in the distribution of random numbers skewed to one side. Therefore, in controlling animation, there is a need for a method capable of controlling random numbers while maintaining a statistical distribution of added noise.

Hereinafter, described in detail by the accompanying drawings as follows.

Our method is based on Perlin's bias noise algorithm. Referring to FIG. 2, a bias noise algorithm is obtained by interpolating a value obtained through hash indexing through a deflection table G composed of a unit deflection vector made of pseudorandom numbers (PRN) and a table P filled with PRN integers. Get the result. Integer Lattice is determined by the dimension of the domain of the noise function. Therefore, N-dimensional noise requires as much as 2 ^N of bias vector to calculate the result. An example algorithm for a three-dimensional noise function is as follows.

Step 1: by constructing a 3D PRN the M unit vectors and stores the G [0, ..., M -1 ] table having a size of as much as M. In addition, a random permutation from 0 to M -1 is stored in another table P with the same M size.

,

] (i=1,2,3)을 얻는다. 여기서

mod M 이고

=

+1을 나타낸다. 각각의 차원에 대한 3개의 정수 간격은 주어진 좌표 (x ₁ ,x ₂ ,x ₃ )를 포함하는 하나의 정사면체를 생성한다. Step 2: Three integer intervals for each dimension of a given coordinate, i.e. [

,

] ( i = 1,2,3). here

mod M

=

Indicates +1. The three integer intervals for each dimension produce one tetrahedron containing the given coordinates ( x ₁ , x ₂ , x ₃ ).

. 여기서 jkl 은 3의 길이를 가지는 0에서 1까지의 가능한 모든 수열을 뜻한다. Step 3: Obtain eight bias vectors for each point of the tetrahedron computed in Step 2:

. Where jkl is all possible sequences from 0 to 1 with length 3.

의 가중치 합을 통해 계산된다. 여기서 가중치는

로 표현되는 이즈 곡선(ease curve)공식을 통해 계산된다. 우리는 펄린(Perlin)에 의해 개선된 이즈 곡선을 사용하였다[13]. Step 4: The resulting noise results in eight internal products, i.e.

Calculated by the weighted sum of. Where the weight is

It is calculated through the Ease curve formula expressed as We used an ease curve improved by Perlin [13].

1) Chi-square test

Chi-square tests have been used to compare the distribution of observed data with the desired distribution. Assuming that Y = y ₁ , ..., y _m is a set of observed data, we can divide Y 's domain [a, b] into K buckets of constant size. In this case, each separated bucket may be represented by Y _k, ( k = 1, ..., K ). Z represents a desired distribution, assuming they want the Y and Z is interested a similar distribution of chi-square statistic is calculated as follows:

를 |Z _k |로 나누는 것은, K번째 버켓의 기대 값이 크면 클수록 제곱차인 가 D ² 값에 주는 영향이 줄어듦을 의미한다. 버켓의 크기는 엄지손가락의 법칙(rule-of-thumb)을 따라서 각각의 버켓에 최소한 5개의 데이터가 들어갈 수 있는 크기로 결정하였다. 본 연구에선 노 이즈의 256개의 크기를 가지는 노이즈 테이블을 사용하였으므로, 카이-스퀘어 테스트를 위해 50개의 버켓으로 테이블을 나누었다.Assuming |. | Represents the amount of data contained in the bucket, | Z _k | is the expected value to enter the Kth bucket in the distribution of Z. For example, if Z assumes an even distribution of random numbers, then for all K | Z _k | = M / K. As the value of D ² decreases, the distribution of Y becomes similar to that of Z. In equation (1)

To | Dividing by Z _k | means that the larger the expected value of the Kth bucket, the smaller the effect of the squared difference on the value of D ² . The size of the bucket was determined according to the rule-of-thumb, and the size of each bucket could contain at least five pieces of data. In this study, a noise table with 256 magnitudes of noise was used, so the table was divided into 50 buckets for chi-square test.

2) Generation of Constrained Noise Using Optimization Techniques

As described above, the noise function “Noise ( x )” is a formula of the deflection values derived through the coordinate value x of the N dimension and can be expressed as follows.

Noise ( x ) = c ₁ g ₁ + c ₂ g ₂ + ... + c _j g _j

Where J = 2N and g _j , ( j = 1,2 , ... , J ) are values obtained from the deflection vector table G obtained through the x coordinate. The coefficient c _j is obtained through a noise calculation algorithm. Controlling noise means assigning the desired value H to the value of noise to be derived at a specific x coordinate (ie Noise ( x ) = H ). In this case the bias vector g _j must be changed for the given constraint. In short, you can solve the above equation to get g _j and put it on the table. However, if continuous constraints are applied to noise, there is a risk that the quality of pseudorandom numbers will be lost and the table will lose the ideal statistical features that the noise function should have. In order to preserve the distribution of the bias table, we must go in the direction that the new table that is changed has the most uniform random number distribution.

For the sake of simplicity, suppose that the one-dimensional noise function satisfies a given constraint Noise ( x ) = H. We will describe the extension to N dimension later. In one dimension, Noise ( x ) = c _u g _u + c _v g _v . Using the hashing mechanism, we can obtain two bias vectors g _u and g _v from the table.

If there are K buckets for the chi-square test, and the total distribution of the data has a uniform random number distribution, the expected value of each bucket can be defined as Z = M / K. Through the transformation of g _u and / or g _v , the amount of data in the Kth bucket is transformed into r _k for all k . Thus, the chi-square statistic for our problem can be expressed by the formula

.

.

Therefore, we constructed the following optimization problem to minimize the chi-square statistics.

subject to

But we need to fix the optimization problem above. This is because the expression representing the constraint is an expression of the value of the deflection value, while the objective function D ² is an expression of the number of deflection values in the bucket. g _u and g _v size of the bucket u, v is was before entering may be represented by [b _u, b _{u +1]} and _{[b v, b v +1]} . There is no assumption that u and v are necessarily different buckets. When gu and gv transition from h _u and h _v , the amount of deflection in the bucket u will change from m _u to r _u . That is, it can be defined as follows.

,

,

Where B _u ( h ) is a Boxcar function defined as

,

,

Similarly, r _{v is} also the amount to enter bucket v after the bias changes. Based on the contents so far, the objective function can be redefined as follows.

To calculate the multidimensional noise function, we use the j = 2 ^N deflection vectors g _j , ( j = 1, ..., J ) in the table. The new deflection vector h _j to be calculated for the control of the desired noise can be represented by Noise ( x ) = c ₁ h ₁ + ... + c _j h _j = H. The objective function for the N- dimensional extended optimization problem is defined as

Where I ( g _j ) is the index of the bucket to which the bias vector g _j belonged before the mutation. The constraint expression of the above objective function is as follows.

,

,

는 벡터의 크기를 뜻한다.here

Is the size of the vector.

4) Approximation Optimization Function

Since it is necessary to differentiate to obtain the minimum value of the objective function, as shown in FIG. Any point on the approximated curve exists inside the original Vaxkar function. In this study, approximation is implemented by simple quadratic curve for simple calculation.

5 shows the change in chi-square statistical values when a large number of controls are applied to the noise having the original pseudorandom number distribution. As can be seen from the figure, the ideal Z value is used as the amount of deflection to enter the bucket, indicating that the Chi-Square statistical value decreases almost constant. We use the sequential equality constrained quadratic programming method and the active set technique proposed by Spellucci to solve the optimization problem.

5) Fractal Consensus Control

The "fractal sum" is often used as an extension of the low-period noise function. Through the fractal sum, pure noise such as the problem of always passing zeros in the integer lattice and the uniform form of the oscillation of a given period Natural phenomena are also made easier to express because they consist of sums of phenomena with various periodicities.

,

,

Where U is the number of noise functions to be added (called 'layer' or 'octave'), a _j is the weight of the amplitude, and b _j is the weight that controls the period of each noise function to be added. . The two weight values generally follow the following rules. If we define two weights in j = 1, ..., U as a _j < a _j +1, b _j < b _j +1, a low-period noise function has a high amplitude and a high-period noise The function will have a low amplitude. By controlling these weights or adding certain mathematical functions to the weights, you can create many different effects. Among them, FBM (Fractional Brownian Motion) and Turbulence are the most used methods. To control the result of the fractal sum, we propose two methods. The first method is used when there is a need to change the pattern of fractal consensus in a specific area, and the second method is used to fix the result of fractal consensus of a specific part without considering additional variation applied to the outside.

Occasionally, the user needs to bring a minimal variation outside of that space in controlling the result of the fractal sum. Specifically, in the case of animation that changes continuously over time, there is a need to prevent sudden global changes in controlling the animation of a specific frame. However, since the fractal sum is composed of several noise functions that consist of the same bias vector table, when all noise functions are divided by constraints, global variation occurs, not local variation. The reason for this phenomenon is that if a bias vector is shifted to control one noise function, all other noises referring to it are affected. Therefore, it is difficult to accurately predict the result of the variation through the control of fractal consensus. For the method of pattern control to change only the pattern of a specific area, we present a method of controlling only one noise function with the largest amplitude (and therefore the lowest period). In this way, local variation occurs with minimal external variation.

For the sake of simplicity, I will illustrate this with an example of the sum of two noise functions.

,

,

는 둘 중 높은 진폭을 가진 쪽이다. 이 경우 두 개의 노이즈 함수는 아래와 같이 두 개의 편향치를 성분으로 한 일차 식으로 표현 할 수 있다.here

Is the higher amplitude of the two. In this case, the two noise functions can be expressed as a linear equation with two bias values as follows.

와

는 함수를 계산할 때 편향치 값 g ₂를 공유하고 있다고 가정했다. 우리가 원하는 건 가장 큰 진폭을 가지는

만을 변이 시키는 것이므로 최적화 문제의 설계에 있어서 편향치 값 h ₁과 h ₂(g ₁과g ₂가 변이될 예상 값)만이 변수로 고려되어야 한다. 식으로 표현하면 F _sum (x)=H로 제어할 때, 아래와 같이 상세한 식으로 서술할 수 있다.In the above expression, we

Wow

Assumes that the bias value g ₂ is shared when computing the function. What we want is to have the largest amplitude

Since only variances are made, only the deflection values h ₁ and h ₂ (the expected values of g ₁ and g ₂ are likely to be shifted) should be considered in the design of the optimization problem. Expressed in the formula, when F _sum ( x ) = H can be described in the following detailed expression.

After a little alignment, we get

Therefore, the constraint equation of fractal sum of multi-dimensional noise with arbitrary number can be summarized as follows.

는 노이즈 함수의 계산에 있어서 변이될 편향치 벡터 g _*에 대한 모든 개수들을 뜻하고,

는 변이되지 않아야 할 노이즈 함수의 계산에 필요한 편향치 벡터와 개수의 합이다. here

Denotes all the numbers for the bias vector g _* to be distorted in the calculation of the noise function,

Is the sum of the deflection vector and the number needed to calculate the noise function that should not be distorted.

Outcome control is assigning a given value to a specific region to a fractal sum. That is, the following general constraints can be used.

In the optimization problem, all the bias vectors of each noise function are considered variables. For example, using the previous pattern control example, the constraint is:

h ₁ , h ₂ , and h ₃ are variables of the optimization problem. As with pattern control, a process of adding the numbers together when using the same deflection vector in different noise functions is required.

6) Results

7 shows a case where control is applied to an animation path. If one-dimensional noise Noise ( t ) is added to the original path P ( t ), there is a risk of moving objects penetrating other objects (Fig. 7 (a)). In order to solve the collision problem, it is possible to solve the problem by applying noise before the collision. We built a system that automatically calculates collisions as objects move and constrains noise in advance.

 The fire animation shown in Figure 8 also includes noise control techniques. Animation used a well-known method of creating three-dimensional textures from free curves with three-dimensional noise. By continuing to control certain parts of the noise, we could simply simulate the effects of wind blowing and fire to one side. This technique eliminates the need to change the free curve to change the shape of the fire.

 In Figure 9 we can see Figure 9 (b), which is the result of the control of the given progressive texture (Figure 9 (a)) in the direction of the white arrow. 9 (c) shows the difference between the texture before control and the texture after control. Finally, Figure 9 (d) is the result of the larger variation by adjusting the noise amplitude limit value.

10 shows the noise control used for terrain control. Noise was applied to the flat plains at high heights to create valley-like bends. If the designer wants to change the height of a particular area, noise control techniques can be used effectively.

 The present invention provides a method for calculating a result value that meets a user's desired constraint while preserving the statistical characteristics of noise. This study is based on an optimization technique that minimizes the difference between the idealized statistical value and the statistical value of the bias of the actual noise function. This gives you finer control over the noiseed animation or shape.

Claims (1)

The present invention, Creating a hash table for the chi-square test of the existing noise table, Designing the problem, optimizing the constraints given by the user To solve the optimization problem. Through these steps, we patent the detailed control method for the noise or animation.

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