Equation 2 defines the gradient orientation and value, G x is the gradient value of x orientation, G y is the gradient value of y orientation, G x,y) represents the gradient value, and α x,y) represents the orientation of the pixel.
If the function (g ( X ) = langle Q,X^{T}X rangle) with (X inmathbb{R}^{m times n}) and (Q inmathbb{R}^{n times n}), then the gradient of (g ( X )) is (nabla g ( X ) = 2XQ).
With the proximity function (3), they proposed an optimization model min { p ( x ) | x ∈ X } (4). to approach the (2) and exerted the projection gradient method to solve it with x k + 1 = P C { x k − s ∇ p ( x k ) }, where ∇p denotes the gradient of p ( x ) and can be shown as follows (see [4]): ∇ p ( x ) = ∑ i = 1 t a i ( x − P C i ( x ) ) + ∑ j = 1 r b j ( A x − P Q j ( A x ) ).
where H x, H y, and H v are the gradient histograms of x, y, and v respectively, ⊗ is the convolution operator, and c·R(H x ) is a regularization term.
The gradient of f (x, y) can calculated as ∇ f ( x, y ) = 1 1 n 2 c y x + c y - ln ( 1 + c y x ) - 1 1 n 2 ( c x x + c y ).
To see this, we compute the gradient of Fσ(x), denoted as g ' = g 1 ', g 2 ', …, g N ' T, whose element is given by g i ' = x i / σ 2 e − x i 2 / 2 σ 2. (12..
The sensitivity can be represented by the gradient of f(x) with respect to R, so according to (27) the sensitivity of the primitive objective function can be formulated as: S_{Y} = left.
If the gradient of u(x) is nth power locally integrable on Euclidean n-space, then the integral average over a ball B of the exponential of a constant multiple of |u(x)−uB|n/(n−1), uB=average of u over B, tends to 1 as the radius of B shrinks to zero for quasi almost all center points.
For solving (1), the conjugate gradient method generates a sequence ({x_{k}} ): (x_{k+1}=x_{k}+alpha_{k}d_{k}), (d_{0}= -g_{0}), and (d_{k}=-g_{k}+beta_{k}d_{k-1}), where the stepsize (alpha_{k}>0) is obtained by the line search, (d_{k}) is the search direction, (g_{k}=nabla f{(x_{k})}) is the gradient of (f(x)) at the point (x_{k}), and (beta_{k}) is known as the conjugate gradient parameter.
The Clarke's generalized directional derivative of h at x ∈ X in the direction v ∈ ℝ n, denoted by h0 x, v), is defined as h 0 ( x, v ) = l i m s u p y → x t ↓ 0 h ( y + t v ) - h ( y ) t. Definition 2.6[16]The Clarke's generalized gradient of h at x ∈ X, denoted by ∂h(x) is defined as ∂ h ( x ) = { ξ ∈ ℝ n : h 0 ( x, v ) ≥ ⟨ ξ, v ⟩ f o r a l l v ∈ ℝ n }.
