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Similar calculations are carried out to determine pixel positions along a line with negative slope. Thus, if the absolute value of the slope is less than 1, we set dx=1 if x s t a r t < x e n d {\displaystyle x_{\rm {start}}<x_{\rm {end}}} i.e. the starting extreme point is at the left.
This result is obtained by setting the gradient of the function equal to zero, noticing that the resulting equation is a rational function of . For small N {\displaystyle N} the polynomials can be determined exactly and Sturm's theorem can be used to determine the number of real roots , while the roots can be bounded in the region of | x i ...
is the derivative with respect to y (gradient in the y direction). The derivative of an image can be approximated by finite differences . If central difference is used, to calculate ∂ f ∂ y {\displaystyle \textstyle {\frac {\partial f}{\partial y}}} we can apply a 1-dimensional filter to the image A {\displaystyle \mathbf {A} } by convolution :
Normalized y-gradient from Sobel–Feldman operator The images below illustrate the change in the direction of the gradient on a grayscale circle. When the sign of G x {\displaystyle \mathbf {G_{x}} } and G y {\displaystyle \mathbf {G_{y}} } are the same the gradient's angle is positive, and negative when different.
Here is an example gradient method that uses a line search in step 5: Set iteration counter k = 0 {\displaystyle k=0} and make an initial guess x 0 {\displaystyle \mathbf {x} _{0}} for the minimum.
The x-coordinate is defined here as increasing in the "left"-direction, and the y-coordinate is defined as increasing in the "up"-direction. At each point in the image, the resulting gradient approximations can be combined to give the gradient magnitude, using:
Coordinate descent is an optimization algorithm that successively minimizes along coordinate directions to find the minimum of a function.At each iteration, the algorithm determines a coordinate or coordinate block via a coordinate selection rule, then exactly or inexactly minimizes over the corresponding coordinate hyperplane while fixing all other coordinates or coordinate blocks.
The geometric interpretation of Newton's method is that at each iteration, it amounts to the fitting of a parabola to the graph of () at the trial value , having the same slope and curvature as the graph at that point, and then proceeding to the maximum or minimum of that parabola (in higher dimensions, this may also be a saddle point), see below.