PDE and Image Diffusion教学

I ( x, y, 0) I 0 ( x, y)
表示初始图像
a is the diffusivity coefficient
t=0
t=1
t=2
t=4
t=8
t=16
Isotropic diffusion (Koenderink, Witkin)
• All “equivalent:”
– Gaussian filtering of the image – Heat flow
Similarly the
2 I y 2
Discretizing the
2 I x 2
I 2 y
2
I (i 1, j ) I (i 1, j ) 2I (i, j )
n n n
Putting it all together :
I I ( x, y, t ) t 2 I 2 x
2 2 2
• 即上述过程可以看成在3个相互垂直方向上的一维扩散 之和，并且在3个相互垂直方向上的传导系数相同，所 以是“各项同性扩散”
对于一个图像来
I ( x, y, t ) 说，上述演化方程为
2 2
Linear Heat Flow Equation
I I I ( x, y, t ) a 2 a 2 t x y 其中， ( x, y, t ) 为演化中的图像，而 I
F div( f )dv
V
• 上述式子中的V表示由曲面S围成的体积。可见F的物理 意义是通过曲面S的单位时间流出的热量总和，即热量 的损失率
• 已知，总热量为：
I ( x, y, z, t )dv
V
则热量的损失率为：
I ( x, y, z, t )dv div( f )dv t V V
I (x, y,t) I (x, y,(n 1)t) I (x, y,nt) t t

I
(n 1)
(x, y) I (x, y) t
( n)
•
Space discretization we compute the solution I(x,y,t) only at particular points(i,j). Actually, the digital image itself have been discretized in column and row direction, and the spacial discretization parameter in column and row is unit .
• 为了达到去噪并且同时保护图像的边界，一个很 自然的想法是令扩散过程中的“扩散系数”依赖 于图像的局部特征。 • 具体说，在图像比较平坦的区域，即，此处对 应的梯度模值很小，传导系数能自动增大，这样 就可使得平坦区域中噪音被平滑掉；而在图像的 边缘附近，即，此处对应的梯度模值很大，传导 系数能自动减小，那么图像的边缘几乎不受影响。
• Perona and Malik suggested an anisotropic diffusion（各向异性扩散） technique for smoothing images in 1990.<Scale space and edge detection using anisotropic diffusion>
Solve the image diffusion equation time t.
I ( x, y, t ) at any point (x,y) at
2 I 2 I I ( x, y, t ) a 2 a 2 t x y
with the initial condition:
• 于是：
I ( x, y, z, t ) div( f ) div(aI ) t
I I I I ( x, y, z, t ) div(aI ) (a ) () (a ) t x x y y z z
即：
I I I I ( x, y, z, t ) a 2 a 2 a 2 a * I ( x, y, z, t ) t x y z
( x, y) I ( x, y) a* n n n t a *{I (i 1, j ) I (i 1, j ) 2I (i, j )}
(n)
{I n (i, j 1) I n (i, j 1) 2I n (i, j )}
Putting it all together :
• The concept of diffusion emerged from physical sciences. Diffusion is the movement of particles from an area of high concentration to an area of low concentration. • The evolution of the temperature in a room. • The evolution of the concentration of chemical components in a solution. • Image diffusion is equivalent to Image smoothing.
所以，可以将介质中温度在宏观上的扩散，看成是有梯 度产生的作用力 I ( x, y, z, t ) 来推动的，其中负号，表 示作用力指向这个时刻、这个位置的温度值减小的方向。
•
如果介质是各向同性的 (Isotropic)，那么在上述作用力下， 将产生流密度，即单位时间内通 过与temperature gradient矢量垂 直的单位面积的热量
4,BackGround Anisotropic Diffusion
Often image and surface denoising is done with a low-pass filter (isotropic filtering) which reduces noise but also blurs sharp features and details, such as image edges.
Discretizing the
2 I x 2
I n n |W I (i, j 1) I (i, j ) x
I n n |E I (i, j ) I (i, j 1) x
2 I x 2
So we have discretization of
2
I n n n n I (i, j 1) I (i, j ) ( I (i, j ) I (i, j 1)) 2 x n n n I (i, j 1) I (i, j 1) 2 I (i, j )
•
Suppose function I ( x, y, z, t ) be temperature
variance with space and time.
We can describe the asymmetry of temperature distribution With temperature gradient I ( x, y, z, t )
I ( x, y, t ) |t 0 I 0 ( x, y)
For each pixel（i,j）, looking for the diffusion result after some time.
The first step is to discretize I(x,y,t): • Time discretization we compute the solution I(x,y,t) only at particular times ts, multiples of the time step, denoted by Δt: ts = n*Δt
I
(n+1) (n)
(i, j )
n
I (i, j ) t *{4a * I (i, j ) I (i, j 1) I (i, j 1) I (i 1, j ) I (i 1, j )}
n n n n
• If Δt and Δx are too large, the I(x,y,n* Δt )is a very poor approximation of the true solution. • If they are too small, the gain on the approximation is marginal, whereas the computationnal cost and time increase and becomes much too large.
t
– Minimize the L2 norm

2
Linear Heat Flow Equation:
I ( x, y, t ) 2 I ( x, y, t ) 2 I ( x, y, t ) I ( x, y, t ) 2 t x y 2
Isotropic diffusion: I ( x, y, t ) I ( x, y,0) G(t ) scale A Gaussian filter with zero mean and variance of t
3, Numerical Implementation
Solve the image diffusion equation at any point (x,y) at time t.
2 I 2 I I ( x, y, t ) a 2 a 2 t x y
n-1 m-1 m m+1 n n+1
2，Heat diffusion equation.