skimage2.feature.hessian_matrix#

skimage2.feature.hessian_matrix(image, sigma=1, mode='constant', cval=0, order='rc', use_gaussian_derivatives=None)[source]#

Compute the Hessian matrix.

In 2D, the Hessian matrix is defined as:

H = [Hrr Hrc]
    [Hrc Hcc]

which is computed by convolving the image with the second derivatives of the Gaussian kernel in the respective r- and c-directions.

The implementation here also supports n-dimensional data.

Parameters:

imagendarray: Input image.
sigmafloat: Standard deviation used for the Gaussian kernel, which is used as weighting function for the auto-correlation matrix.
mode{‘constant’, ‘reflect’, ‘wrap’, ‘nearest’, ‘mirror’}, optional: How to handle values outside the image borders.
cvalfloat, optional: Used in conjunction with mode ‘constant’, the value outside the image boundaries.
order{‘rc’, ‘xy’}, optional: For 2D images, this parameter allows for the use of reverse or forward order of the image axes in gradient computation. ‘rc’ indicates the use of the first axis initially (Hrr, Hrc, Hcc), whilst ‘xy’ indicates the usage of the last axis initially (Hxx, Hxy, Hyy). Images with higher dimension must always use ‘rc’ order.
use_gaussian_derivativesbool, optional: Indicates whether the Hessian is computed by convolving with Gaussian derivatives, or by a simple finite-difference operation.

Returns:

H_elemslist of ndarray: Upper-diagonal elements of the hessian matrix for each pixel in the input image. In 2D, this will be a three element list containing [Hrr, Hrc, Hcc]. In nD, the list will contain (n**2 + n) / 2 arrays.

Notes

The distributive property of derivatives and convolutions allows us to restate the derivative of an image, I, smoothed with a Gaussian kernel, G, as the convolution of the image with the derivative of G.

\[\frac{\partial }{\partial x_i}(I * G) = I * \left( \frac{\partial }{\partial x_i} G \right)\]

When use_gaussian_derivatives is True, this property is used to compute the second order derivatives that make up the Hessian matrix.

When use_gaussian_derivatives is False, simple finite differences on a Gaussian-smoothed image are used instead.

Examples

>>> from _skimage2.feature import hessian_matrix
>>> square = np.zeros((5, 5))
>>> square[2, 2] = 4
>>> Hrr, Hrc, Hcc = hessian_matrix(square, sigma=0.1, order='rc',
...                                use_gaussian_derivatives=False)
>>> Hrc
array([[ 0.,  0.,  0.,  0.,  0.],
       [ 0.,  1.,  0., -1.,  0.],
       [ 0.,  0.,  0.,  0.,  0.],
       [ 0., -1.,  0.,  1.,  0.],
       [ 0.,  0.,  0.,  0.,  0.]])