transform
Module: transform
skimage.transform.downscale_local_mean (...) | Down-sample N-dimensional image by local averaging. |
skimage.transform.estimate_transform (ttype, ...) | Estimate 2D geometric transformation parameters. |
skimage.transform.frt2 (a) | Compute the 2-dimensional finite radon transform (FRT) for an n x n integer array. |
skimage.transform.hough_circle (image, radius) | Perform a circular Hough transform. |
skimage.transform.hough_ellipse (img[, ...]) | Perform an elliptical Hough transform. |
skimage.transform.hough_line (img[, theta]) | Perform a straight line Hough transform. |
skimage.transform.hough_line_peaks (hspace, ...) | Return peaks in hough transform. |
skimage.transform.ifrt2 (a) | Compute the 2-dimensional inverse finite radon transform (iFRT) for an (n+1) x n integer array. |
skimage.transform.integral_image (img) | Integral image / summed area table. |
skimage.transform.integrate (ii, start, end, ...) | Use an integral image to integrate over a given window. |
skimage.transform.iradon (radon_image[, ...]) | Inverse radon transform. |
skimage.transform.iradon_sart (radon_image[, ...]) | Inverse radon transform |
skimage.transform.matrix_transform (coords, ...) | Apply 2D matrix transform. |
skimage.transform.probabilistic_hough_line (img) | Return lines from a progressive probabilistic line Hough transform. |
skimage.transform.pyramid_expand (image[, ...]) | Upsample and then smooth image. |
skimage.transform.pyramid_gaussian (image[, ...]) | Yield images of the Gaussian pyramid formed by the input image. |
skimage.transform.pyramid_laplacian (image[, ...]) | Yield images of the laplacian pyramid formed by the input image. |
skimage.transform.pyramid_reduce (image[, ...]) | Smooth and then downsample image. |
skimage.transform.radon (image[, theta, circle]) | Calculates the radon transform of an image given specified projection angles. |
skimage.transform.rescale (image, scale[, ...]) | Scale image by a certain factor. |
skimage.transform.resize (image, output_shape) | Resize image to match a certain size. |
skimage.transform.rotate (image, angle[, ...]) | Rotate image by a certain angle around its center. |
skimage.transform.seam_carve (img, ...[, ...]) | Carve vertical or horizontal seams off an image. |
skimage.transform.swirl (image[, center, ...]) | Perform a swirl transformation. |
skimage.transform.warp (image[, inverse_map, ...]) | Warp an image according to a given coordinate transformation. |
skimage.transform.warp_coords (coord_map, shape) | Build the source coordinates for the output of a 2-D image warp. |
skimage.transform.AffineTransform ([matrix, ...]) | 2D affine transformation of the form: |
skimage.transform.PiecewiseAffineTransform () | 2D piecewise affine transformation. |
skimage.transform.PolynomialTransform ([params]) | 2D transformation of the form: |
skimage.transform.ProjectiveTransform ([matrix]) | Matrix transformation. |
skimage.transform.SimilarityTransform ([...]) | 2D similarity transformation of the form: |
downscale_local_mean
-
skimage.transform.downscale_local_mean(image, factors, cval=0, clip=True)
[source] -
Down-sample N-dimensional image by local averaging.
The image is padded with
cval
if it is not perfectly divisible by the integer factors.In contrast to the 2-D interpolation in
skimage.transform.resize
andskimage.transform.rescale
this function may be applied to N-dimensional images and calculates the local mean of elements in each block of sizefactors
in the input image.Parameters: image : ndarray
N-dimensional input image.
factors : array_like
Array containing down-sampling integer factor along each axis.
cval : float, optional
Constant padding value if image is not perfectly divisible by the integer factors.
Returns: image : ndarray
Down-sampled image with same number of dimensions as input image.
Examples
>>> a = np.arange(15).reshape(3, 5) >>> a array([[ 0, 1, 2, 3, 4], [ 5, 6, 7, 8, 9], [10, 11, 12, 13, 14]]) >>> downscale_local_mean(a, (2, 3)) array([[ 3.5, 4. ], [ 5.5, 4.5]])
estimate_transform
-
skimage.transform.estimate_transform(ttype, src, dst, **kwargs)
[source] -
Estimate 2D geometric transformation parameters.
You can determine the over-, well- and under-determined parameters with the total least-squares method.
Number of source and destination coordinates must match.
Parameters: ttype : {‘similarity’, ‘affine’, ‘piecewise-affine’, ‘projective’, ‘polynomial’}
Type of transform.
kwargs : array or int
Function parameters (src, dst, n, angle):
NAME / TTYPE FUNCTION PARAMETERS 'similarity' `src, `dst` 'affine' `src, `dst` 'piecewise-affine' `src, `dst` 'projective' `src, `dst` 'polynomial' `src, `dst`, `order` (polynomial order, default order is 2)
Also see examples below.
Returns: tform :
GeometricTransform
Transform object containing the transformation parameters and providing access to forward and inverse transformation functions.
Examples
>>> import numpy as np >>> from skimage import transform as tf
>>> # estimate transformation parameters >>> src = np.array([0, 0, 10, 10]).reshape((2, 2)) >>> dst = np.array([12, 14, 1, -20]).reshape((2, 2))
>>> tform = tf.estimate_transform('similarity', src, dst)
>>> np.allclose(tform.inverse(tform(src)), src) True
>>> # warp image using the estimated transformation >>> from skimage import data >>> image = data.camera()
>>> warp(image, inverse_map=tform.inverse)
>>> # create transformation with explicit parameters >>> tform2 = tf.SimilarityTransform(scale=1.1, rotation=1, ... translation=(10, 20))
>>> # unite transformations, applied in order from left to right >>> tform3 = tform + tform2 >>> np.allclose(tform3(src), tform2(tform(src))) True
frt2
-
skimage.transform.frt2(a)
[source] -
Compute the 2-dimensional finite radon transform (FRT) for an n x n integer array.
Parameters: a : array_like
A 2-D square n x n integer array.
Returns: FRT : 2-D ndarray
Finite Radon Transform array of (n+1) x n integer coefficients.
See also
-
ifrt2
- The two-dimensional inverse FRT.
Notes
The FRT has a unique inverse if and only if n is prime. [FRT] The idea for this algorithm is due to Vlad Negnevitski.
References
[FRT] A. Kingston and I. Svalbe, “Projective transforms on periodic discrete image arrays,” in P. Hawkes (Ed), Advances in Imaging and Electron Physics, 139 (2006) Examples
Generate a test image: Use a prime number for the array dimensions
>>> SIZE = 59 >>> img = np.tri(SIZE, dtype=np.int32)
Apply the Finite Radon Transform:
>>> f = frt2(img)
-
hough_circle
-
skimage.transform.hough_circle(image, radius, normalize=True, full_output=False)
[source] -
Perform a circular Hough transform.
Parameters: image : (M, N) ndarray
Input image with nonzero values representing edges.
radius : scalar or sequence of scalars
Radii at which to compute the Hough transform. Floats are converted to integers.
normalize : boolean, optional (default True)
Normalize the accumulator with the number of pixels used to draw the radius.
full_output : boolean, optional (default False)
Extend the output size by twice the largest radius in order to detect centers outside the input picture.
Returns: H : 3D ndarray (radius index, (M + 2R, N + 2R) ndarray)
Hough transform accumulator for each radius. R designates the larger radius if full_output is True. Otherwise, R = 0.
Examples
>>> from skimage.transform import hough_circle >>> from skimage.draw import circle_perimeter >>> img = np.zeros((100, 100), dtype=np.bool_) >>> rr, cc = circle_perimeter(25, 35, 23) >>> img[rr, cc] = 1 >>> try_radii = np.arange(5, 50) >>> res = hough_circle(img, try_radii) >>> ridx, r, c = np.unravel_index(np.argmax(res), res.shape) >>> r, c, try_radii[ridx] (25, 35, 23)
hough_ellipse
-
skimage.transform.hough_ellipse(img, threshold=4, accuracy=1, min_size=4, max_size=None)
[source] -
Perform an elliptical Hough transform.
Parameters: img : (M, N) ndarray
Input image with nonzero values representing edges.
threshold: int, optional (default 4)
Accumulator threshold value.
accuracy : double, optional (default 1)
Bin size on the minor axis used in the accumulator.
min_size : int, optional (default 4)
Minimal major axis length.
max_size : int, optional
Maximal minor axis length. (default None) If None, the value is set to the half of the smaller image dimension.
Returns: result : ndarray with fields [(accumulator, y0, x0, a, b, orientation)]
Where
(yc, xc)
is the center,(a, b)
the major and minor axes, respectively. Theorientation
value followsskimage.draw.ellipse_perimeter
convention.Notes
The accuracy must be chosen to produce a peak in the accumulator distribution. In other words, a flat accumulator distribution with low values may be caused by a too low bin size.
References
[R367] Xie, Yonghong, and Qiang Ji. “A new efficient ellipse detection method.” Pattern Recognition, 2002. Proceedings. 16th International Conference on. Vol. 2. IEEE, 2002 Examples
>>> from skimage.transform import hough_ellipse >>> from skimage.draw import ellipse_perimeter >>> img = np.zeros((25, 25), dtype=np.uint8) >>> rr, cc = ellipse_perimeter(10, 10, 6, 8) >>> img[cc, rr] = 1 >>> result = hough_ellipse(img, threshold=8) >>> result.tolist() [(10, 10.0, 10.0, 8.0, 6.0, 0.0)]
hough_line
-
skimage.transform.hough_line(img, theta=None)
[source] -
Perform a straight line Hough transform.
Parameters: img : (M, N) ndarray
Input image with nonzero values representing edges.
theta : 1D ndarray of double
Angles at which to compute the transform, in radians. Defaults to -pi/2 .. pi/2
Returns: H : 2-D ndarray of uint64
Hough transform accumulator.
theta : ndarray
Angles at which the transform was computed, in radians.
distances : ndarray
Distance values.
Notes
The origin is the top left corner of the original image. X and Y axis are horizontal and vertical edges respectively. The distance is the minimal algebraic distance from the origin to the detected line.
Examples
Generate a test image:
>>> img = np.zeros((100, 150), dtype=bool) >>> img[30, :] = 1 >>> img[:, 65] = 1 >>> img[35:45, 35:50] = 1 >>> for i in range(90): ... img[i, i] = 1 >>> img += np.random.random(img.shape) > 0.95
Apply the Hough transform:
>>> out, angles, d = hough_line(img)
import numpy as np import matplotlib.pyplot as plt from skimage.transform import hough_line from skimage.draw import line img = np.zeros((100, 150), dtype=bool) img[30, :] = 1 img[:, 65] = 1 img[35:45, 35:50] = 1 rr, cc = line(60, 130, 80, 10) img[rr, cc] = 1 img += np.random.random(img.shape) > 0.95 out, angles, d = hough_line(img) plt.subplot(1, 2, 1) plt.imshow(img, cmap=plt.cm.gray) plt.title('Input image') plt.subplot(1, 2, 2) plt.imshow(out, cmap=plt.cm.bone, extent=(np.rad2deg(angles[-1]), np.rad2deg(angles[0]), d[-1], d[0])) plt.title('Hough transform') plt.xlabel('Angle (degree)') plt.ylabel('Distance (pixel)') plt.subplots_adjust(wspace=0.4) plt.show()
(Source code, png, pdf)
hough_line_peaks
-
skimage.transform.hough_line_peaks(hspace, angles, dists, min_distance=9, min_angle=10, threshold=None, num_peaks=inf)
[source] -
Return peaks in hough transform.
Identifies most prominent lines separated by a certain angle and distance in a hough transform. Non-maximum suppression with different sizes is applied separately in the first (distances) and second (angles) dimension of the hough space to identify peaks.
Parameters: hspace : (N, M) array
Hough space returned by the
hough_line
function.angles : (M,) array
Angles returned by the
hough_line
function. Assumed to be continuous. (angles[-1] - angles[0] == PI
).dists : (N, ) array
Distances returned by the
hough_line
function.min_distance : int
Minimum distance separating lines (maximum filter size for first dimension of hough space).
min_angle : int
Minimum angle separating lines (maximum filter size for second dimension of hough space).
threshold : float
Minimum intensity of peaks. Default is
0.5 * max(hspace)
.num_peaks : int
Maximum number of peaks. When the number of peaks exceeds
num_peaks
, returnnum_peaks
coordinates based on peak intensity.Returns: hspace, angles, dists : tuple of array
Peak values in hough space, angles and distances.
Examples
>>> from skimage.transform import hough_line, hough_line_peaks >>> from skimage.draw import line >>> img = np.zeros((15, 15), dtype=np.bool_) >>> rr, cc = line(0, 0, 14, 14) >>> img[rr, cc] = 1 >>> rr, cc = line(0, 14, 14, 0) >>> img[cc, rr] = 1 >>> hspace, angles, dists = hough_line(img) >>> hspace, angles, dists = hough_line_peaks(hspace, angles, dists) >>> len(angles) 2
ifrt2
-
skimage.transform.ifrt2(a)
[source] -
Compute the 2-dimensional inverse finite radon transform (iFRT) for an (n+1) x n integer array.
Parameters: a : array_like
A 2-D (n+1) row x n column integer array.
Returns: iFRT : 2-D n x n ndarray
Inverse Finite Radon Transform array of n x n integer coefficients.
See also
-
frt2
- The two-dimensional FRT
Notes
The FRT has a unique inverse if and only if n is prime. See [R368] for an overview. The idea for this algorithm is due to Vlad Negnevitski.
References
[R368] (1, 2) A. Kingston and I. Svalbe, “Projective transforms on periodic discrete image arrays,” in P. Hawkes (Ed), Advances in Imaging and Electron Physics, 139 (2006) Examples
>>> SIZE = 59 >>> img = np.tri(SIZE, dtype=np.int32)
Apply the Finite Radon Transform:
>>> f = frt2(img)
Apply the Inverse Finite Radon Transform to recover the input
>>> fi = ifrt2(f)
Check that it’s identical to the original
>>> assert len(np.nonzero(img-fi)[0]) == 0
-
integral_image
-
skimage.transform.integral_image(img)
[source] -
Integral image / summed area table.
The integral image contains the sum of all elements above and to the left of it, i.e.:
Parameters: img : ndarray
Input image.
Returns: S : ndarray
Integral image/summed area table of same shape as input image.
References
[R369] F.C. Crow, “Summed-area tables for texture mapping,” ACM SIGGRAPH Computer Graphics, vol. 18, 1984, pp. 207-212.
integrate
-
skimage.transform.integrate(ii, start, end, *args)
[source] -
Use an integral image to integrate over a given window.
Parameters: ii : ndarray
Integral image.
start : List of tuples, each tuple of length equal to dimension of
ii
Coordinates of top left corner of window(s). Each tuple in the list contains the starting row, col, ... index i.e
[(row_win1, col_win1, ...), (row_win2, col_win2,...), ...]
.end : List of tuples, each tuple of length equal to dimension of
ii
Coordinates of bottom right corner of window(s). Each tuple in the list containing the end row, col, ... index i.e
[(row_win1, col_win1, ...), (row_win2, col_win2, ...), ...]
.args: optional
For backward compatibility with versions prior to 0.12. The earlier function signature was
integrate(ii, r0, c0, r1, c1)
, wherer0
,c0
are int(lists) specifying start coordinates of window(s) to be integrated andr1
,c1
the end coordinates.Returns: S : scalar or ndarray
Integral (sum) over the given window(s).
Examples
>>> arr = np.ones((5, 6), dtype=np.float) >>> ii = integral_image(arr) >>> integrate(ii, (1, 0), (1, 2)) # sum from (1, 0) to (1, 2) array([ 3.]) >>> integrate(ii, [(3, 3)], [(4, 5)]) # sum from (3, 3) to (4, 5) array([ 6.]) >>> # sum from (1, 0) to (1, 2) and from (3, 3) to (4, 5) >>> integrate(ii, [(1, 0), (3, 3)], [(1, 2), (4, 5)]) array([ 3., 6.])
iradon
-
skimage.transform.iradon(radon_image, theta=None, output_size=None, filter='ramp', interpolation='linear', circle=False)
[source] -
Inverse radon transform.
Reconstruct an image from the radon transform, using the filtered back projection algorithm.
Parameters: radon_image : array_like, dtype=float
Image containing radon transform (sinogram). Each column of the image corresponds to a projection along a different angle. The tomography rotation axis should lie at the pixel index
radon_image.shape[0] // 2
along the 0th dimension ofradon_image
.theta : array_like, dtype=float, optional
Reconstruction angles (in degrees). Default: m angles evenly spaced between 0 and 180 (if the shape of
radon_image
is (N, M)).output_size : int
Number of rows and columns in the reconstruction.
filter : str, optional (default ramp)
Filter used in frequency domain filtering. Ramp filter used by default. Filters available: ramp, shepp-logan, cosine, hamming, hann. Assign None to use no filter.
interpolation : str, optional (default ‘linear’)
Interpolation method used in reconstruction. Methods available: ‘linear’, ‘nearest’, and ‘cubic’ (‘cubic’ is slow).
circle : boolean, optional
Assume the reconstructed image is zero outside the inscribed circle. Also changes the default output_size to match the behaviour of
radon
called withcircle=True
.Returns: reconstructed : ndarray
Reconstructed image. The rotation axis will be located in the pixel with indices
(reconstructed.shape[0] // 2, reconstructed.shape[1] // 2)
.Notes
It applies the Fourier slice theorem to reconstruct an image by multiplying the frequency domain of the filter with the FFT of the projection data. This algorithm is called filtered back projection.
iradon_sart
-
skimage.transform.iradon_sart(radon_image, theta=None, image=None, projection_shifts=None, clip=None, relaxation=0.15)
[source] -
Inverse radon transform
Reconstruct an image from the radon transform, using a single iteration of the Simultaneous Algebraic Reconstruction Technique (SART) algorithm.
Parameters: radon_image : 2D array, dtype=float
Image containing radon transform (sinogram). Each column of the image corresponds to a projection along a different angle. The tomography rotation axis should lie at the pixel index
radon_image.shape[0] // 2
along the 0th dimension ofradon_image
.theta : 1D array, dtype=float, optional
Reconstruction angles (in degrees). Default: m angles evenly spaced between 0 and 180 (if the shape of
radon_image
is (N, M)).image : 2D array, dtype=float, optional
Image containing an initial reconstruction estimate. Shape of this array should be
(radon_image.shape[0], radon_image.shape[0])
. The default is an array of zeros.projection_shifts : 1D array, dtype=float
Shift the projections contained in
radon_image
(the sinogram) by this many pixels before reconstructing the image. The i’th value defines the shift of the i’th column ofradon_image
.clip : length-2 sequence of floats
Force all values in the reconstructed tomogram to lie in the range
[clip[0], clip[1]]
relaxation : float
Relaxation parameter for the update step. A higher value can improve the convergence rate, but one runs the risk of instabilities. Values close to or higher than 1 are not recommended.
Returns: reconstructed : ndarray
Reconstructed image. The rotation axis will be located in the pixel with indices
(reconstructed.shape[0] // 2, reconstructed.shape[1] // 2)
.Notes
Algebraic Reconstruction Techniques are based on formulating the tomography reconstruction problem as a set of linear equations. Along each ray, the projected value is the sum of all the values of the cross section along the ray. A typical feature of SART (and a few other variants of algebraic techniques) is that it samples the cross section at equidistant points along the ray, using linear interpolation between the pixel values of the cross section. The resulting set of linear equations are then solved using a slightly modified Kaczmarz method.
When using SART, a single iteration is usually sufficient to obtain a good reconstruction. Further iterations will tend to enhance high-frequency information, but will also often increase the noise.
References
[R370] AC Kak, M Slaney, “Principles of Computerized Tomographic Imaging”, IEEE Press 1988. [R371] AH Andersen, AC Kak, “Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm”, Ultrasonic Imaging 6 pp 81–94 (1984) [R372] S Kaczmarz, “Angenäherte auflösung von systemen linearer gleichungen”, Bulletin International de l’Academie Polonaise des Sciences et des Lettres 35 pp 355–357 (1937) [R373] Kohler, T. “A projection access scheme for iterative reconstruction based on the golden section.” Nuclear Science Symposium Conference Record, 2004 IEEE. Vol. 6. IEEE, 2004. [R374] Kaczmarz’ method, Wikipedia, http://en.wikipedia.org/wiki/Kaczmarz_method
matrix_transform
-
skimage.transform.matrix_transform(coords, matrix)
[source] -
Apply 2D matrix transform.
Parameters: coords : (N, 2) array
x, y coordinates to transform
matrix : (3, 3) array
Homogeneous transformation matrix.
Returns: coords : (N, 2) array
Transformed coordinates.
probabilistic_hough_line
-
skimage.transform.probabilistic_hough_line(img, threshold=10, line_length=50, line_gap=10, theta=None)
[source] -
Return lines from a progressive probabilistic line Hough transform.
Parameters: img : (M, N) ndarray
Input image with nonzero values representing edges.
threshold : int, optional (default 10)
Threshold
line_length : int, optional (default 50)
Minimum accepted length of detected lines. Increase the parameter to extract longer lines.
line_gap : int, optional, (default 10)
Maximum gap between pixels to still form a line. Increase the parameter to merge broken lines more aggresively.
theta : 1D ndarray, dtype=double, optional, default (-pi/2 .. pi/2)
Angles at which to compute the transform, in radians.
Returns: lines : list
List of lines identified, lines in format ((x0, y0), (x1, y0)), indicating line start and end.
References
[R375] C. Galamhos, J. Matas and J. Kittler, “Progressive probabilistic Hough transform for line detection”, in IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 1999.
pyramid_expand
-
skimage.transform.pyramid_expand(image, upscale=2, sigma=None, order=1, mode='reflect', cval=0)
[source] -
Upsample and then smooth image.
Parameters: image : array
Input image.
upscale : float, optional
Upscale factor.
sigma : float, optional
Sigma for Gaussian filter. Default is
2 * upscale / 6.0
which corresponds to a filter mask twice the size of the scale factor that covers more than 99% of the Gaussian distribution.order : int, optional
Order of splines used in interpolation of upsampling. See
skimage.transform.warp
for detail.mode : {‘reflect’, ‘constant’, ‘edge’, ‘symmetric’, ‘wrap’}, optional
The mode parameter determines how the array borders are handled, where cval is the value when mode is equal to ‘constant’.
cval : float, optional
Value to fill past edges of input if mode is ‘constant’.
Returns: out : array
Upsampled and smoothed float image.
References
[R376] http://web.mit.edu/persci/people/adelson/pub_pdfs/pyramid83.pdf
pyramid_gaussian
-
skimage.transform.pyramid_gaussian(image, max_layer=-1, downscale=2, sigma=None, order=1, mode='reflect', cval=0)
[source] -
Yield images of the Gaussian pyramid formed by the input image.
Recursively applies the
pyramid_reduce
function to the image, and yields the downscaled images.Note that the first image of the pyramid will be the original, unscaled image. The total number of images is
max_layer + 1
. In case all layers are computed, the last image is either a one-pixel image or the image where the reduction does not change its shape.Parameters: image : array
Input image.
max_layer : int
Number of layers for the pyramid. 0th layer is the original image. Default is -1 which builds all possible layers.
downscale : float, optional
Downscale factor.
sigma : float, optional
Sigma for Gaussian filter. Default is
2 * downscale / 6.0
which corresponds to a filter mask twice the size of the scale factor that covers more than 99% of the Gaussian distribution.order : int, optional
Order of splines used in interpolation of downsampling. See
skimage.transform.warp
for detail.mode : {‘reflect’, ‘constant’, ‘edge’, ‘symmetric’, ‘wrap’}, optional
The mode parameter determines how the array borders are handled, where cval is the value when mode is equal to ‘constant’.
cval : float, optional
Value to fill past edges of input if mode is ‘constant’.
Returns: pyramid : generator
Generator yielding pyramid layers as float images.
References
[R377] http://web.mit.edu/persci/people/adelson/pub_pdfs/pyramid83.pdf
pyramid_laplacian
-
skimage.transform.pyramid_laplacian(image, max_layer=-1, downscale=2, sigma=None, order=1, mode='reflect', cval=0)
[source] -
Yield images of the laplacian pyramid formed by the input image.
Each layer contains the difference between the downsampled and the downsampled, smoothed image:
layer = resize(prev_layer) - smooth(resize(prev_layer))
Note that the first image of the pyramid will be the difference between the original, unscaled image and its smoothed version. The total number of images is
max_layer + 1
. In case all layers are computed, the last image is either a one-pixel image or the image where the reduction does not change its shape.Parameters: image : array
Input image.
max_layer : int
Number of layers for the pyramid. 0th layer is the original image. Default is -1 which builds all possible layers.
downscale : float, optional
Downscale factor.
sigma : float, optional
Sigma for Gaussian filter. Default is
2 * downscale / 6.0
which corresponds to a filter mask twice the size of the scale factor that covers more than 99% of the Gaussian distribution.order : int, optional
Order of splines used in interpolation of downsampling. See
skimage.transform.warp
for detail.mode : {‘reflect’, ‘constant’, ‘edge’, ‘symmetric’, ‘wrap’}, optional
The mode parameter determines how the array borders are handled, where cval is the value when mode is equal to ‘constant’.
cval : float, optional
Value to fill past edges of input if mode is ‘constant’.
Returns: pyramid : generator
Generator yielding pyramid layers as float images.
References
[R378] http://web.mit.edu/persci/people/adelson/pub_pdfs/pyramid83.pdf [R379] http://sepwww.stanford.edu/~morgan/texturematch/paper_html/node3.html
pyramid_reduce
-
skimage.transform.pyramid_reduce(image, downscale=2, sigma=None, order=1, mode='reflect', cval=0)
[source] -
Smooth and then downsample image.
Parameters: image : array
Input image.
downscale : float, optional
Downscale factor.
sigma : float, optional
Sigma for Gaussian filter. Default is
2 * downscale / 6.0
which corresponds to a filter mask twice the size of the scale factor that covers more than 99% of the Gaussian distribution.order : int, optional
Order of splines used in interpolation of downsampling. See
skimage.transform.warp
for detail.mode : {‘reflect’, ‘constant’, ‘edge’, ‘symmetric’, ‘wrap’}, optional
The mode parameter determines how the array borders are handled, where cval is the value when mode is equal to ‘constant’.
cval : float, optional
Value to fill past edges of input if mode is ‘constant’.
Returns: out : array
Smoothed and downsampled float image.
References
[R380] http://web.mit.edu/persci/people/adelson/pub_pdfs/pyramid83.pdf
radon
-
skimage.transform.radon(image, theta=None, circle=False)
[source] -
Calculates the radon transform of an image given specified projection angles.
Parameters: image : array_like, dtype=float
Input image. The rotation axis will be located in the pixel with indices
(image.shape[0] // 2, image.shape[1] // 2)
.theta : array_like, dtype=float, optional (default np.arange(180))
Projection angles (in degrees).
circle : boolean, optional
Assume image is zero outside the inscribed circle, making the width of each projection (the first dimension of the sinogram) equal to
min(image.shape)
.Returns: radon_image : ndarray
Radon transform (sinogram). The tomography rotation axis will lie at the pixel index
radon_image.shape[0] // 2
along the 0th dimension ofradon_image
.
rescale
-
skimage.transform.rescale(image, scale, order=1, mode='constant', cval=0, clip=True, preserve_range=False)
[source] -
Scale image by a certain factor.
Performs interpolation to upscale or down-scale images. For down-sampling N-dimensional images with integer factors by applying the arithmetic sum or mean, see
skimage.measure.local_sum
andskimage.transform.downscale_local_mean
, respectively.Parameters: image : ndarray
Input image.
scale : {float, tuple of floats}
Scale factors. Separate scale factors can be defined as
(row_scale, col_scale)
.Returns: scaled : ndarray
Scaled version of the input.
Other Parameters: order : int, optional
The order of the spline interpolation, default is 1. The order has to be in the range 0-5. See
skimage.transform.warp
for detail.mode : {‘constant’, ‘edge’, ‘symmetric’, ‘reflect’, ‘wrap’}, optional
Points outside the boundaries of the input are filled according to the given mode. Modes match the behaviour of
numpy.pad
.cval : float, optional
Used in conjunction with mode ‘constant’, the value outside the image boundaries.
clip : bool, optional
Whether to clip the output to the range of values of the input image. This is enabled by default, since higher order interpolation may produce values outside the given input range.
preserve_range : bool, optional
Whether to keep the original range of values. Otherwise, the input image is converted according to the conventions of
img_as_float
.Examples
>>> from skimage import data >>> from skimage.transform import rescale >>> image = data.camera() >>> rescale(image, 0.1).shape (51, 51) >>> rescale(image, 0.5).shape (256, 256)
resize
-
skimage.transform.resize(image, output_shape, order=1, mode='constant', cval=0, clip=True, preserve_range=False)
[source] -
Resize image to match a certain size.
Performs interpolation to up-size or down-size images. For down-sampling N-dimensional images by applying the arithmetic sum or mean, see
skimage.measure.local_sum
andskimage.transform.downscale_local_mean
, respectively.Parameters: image : ndarray
Input image.
output_shape : tuple or ndarray
Size of the generated output image
(rows, cols[, dim])
. Ifdim
is not provided, the number of channels is preserved. In case the number of input channels does not equal the number of output channels a 3-dimensional interpolation is applied.Returns: resized : ndarray
Resized version of the input.
Other Parameters: order : int, optional
The order of the spline interpolation, default is 1. The order has to be in the range 0-5. See
skimage.transform.warp
for detail.mode : {‘constant’, ‘edge’, ‘symmetric’, ‘reflect’, ‘wrap’}, optional
Points outside the boundaries of the input are filled according to the given mode. Modes match the behaviour of
numpy.pad
.cval : float, optional
Used in conjunction with mode ‘constant’, the value outside the image boundaries.
clip : bool, optional
Whether to clip the output to the range of values of the input image. This is enabled by default, since higher order interpolation may produce values outside the given input range.
preserve_range : bool, optional
Whether to keep the original range of values. Otherwise, the input image is converted according to the conventions of
img_as_float
.Notes
Modes ‘reflect’ and ‘symmetric’ are similar, but differ in whether the edge pixels are duplicated during the reflection. As an example, if an array has values [0, 1, 2] and was padded to the right by four values using symmetric, the result would be [0, 1, 2, 2, 1, 0, 0], while for reflect it would be [0, 1, 2, 1, 0, 1, 2].
Examples
>>> from skimage import data >>> from skimage.transform import resize >>> image = data.camera() >>> resize(image, (100, 100)).shape (100, 100)
rotate
-
skimage.transform.rotate(image, angle, resize=False, center=None, order=1, mode='constant', cval=0, clip=True, preserve_range=False)
[source] -
Rotate image by a certain angle around its center.
Parameters: image : ndarray
Input image.
angle : float
Rotation angle in degrees in counter-clockwise direction.
resize : bool, optional
Determine whether the shape of the output image will be automatically calculated, so the complete rotated image exactly fits. Default is False.
center : iterable of length 2
The rotation center. If
center=None
, the image is rotated around its center, i.e.center=(rows / 2 - 0.5, cols / 2 - 0.5)
.Returns: rotated : ndarray
Rotated version of the input.
Other Parameters: order : int, optional
The order of the spline interpolation, default is 1. The order has to be in the range 0-5. See
skimage.transform.warp
for detail.mode : {‘constant’, ‘edge’, ‘symmetric’, ‘reflect’, ‘wrap’}, optional
Points outside the boundaries of the input are filled according to the given mode. Modes match the behaviour of
numpy.pad
.cval : float, optional
Used in conjunction with mode ‘constant’, the value outside the image boundaries.
clip : bool, optional
Whether to clip the output to the range of values of the input image. This is enabled by default, since higher order interpolation may produce values outside the given input range.
preserve_range : bool, optional
Whether to keep the original range of values. Otherwise, the input image is converted according to the conventions of
img_as_float
.Examples
>>> from skimage import data >>> from skimage.transform import rotate >>> image = data.camera() >>> rotate(image, 2).shape (512, 512) >>> rotate(image, 2, resize=True).shape (530, 530) >>> rotate(image, 90, resize=True).shape (512, 512)
seam_carve
-
skimage.transform.seam_carve(img, energy_map, mode, num, border=1, force_copy=True)
[source] -
Carve vertical or horizontal seams off an image.
Carves out vertical/horizontal seams from an image while using the given energy map to decide the importance of each pixel.
Parameters: image : (M, N) or (M, N, 3) ndarray
Input image whose seams are to be removed.
energy_map : (M, N) ndarray
The array to decide the importance of each pixel. The higher the value corresponding to a pixel, the more the algorithm will try to keep it in the image.
mode : str {‘horizontal’, ‘vertical’}
Indicates whether seams are to be removed vertically or horizontally. Removing seams horizontally will decrease the height whereas removing vertically will decrease the width.
num : int
Number of seams are to be removed.
border : int, optional
The number of pixels in the right, left and bottom end of the image to be excluded from being considered for a seam. This is important as certain filters just ignore image boundaries and set them to
0
. By default border is set to1
.force_copy : bool, optional
If set, the
image
andenergy_map
are copied before being used by the method which modifies it in place. Set this toFalse
if the original image and the energy map are no longer needed after this opetration.Returns: out : ndarray
The cropped image with the seams removed.
References
[R381] Shai Avidan and Ariel Shamir “Seam Carving for Content-Aware Image Resizing” http://www.cs.jhu.edu/~misha/ReadingSeminar/Papers/Avidan07.pdf
swirl
-
skimage.transform.swirl(image, center=None, strength=1, radius=100, rotation=0, output_shape=None, order=1, mode='constant', cval=0, clip=True, preserve_range=False)
[source] -
Perform a swirl transformation.
Parameters: image : ndarray
Input image.
center : (row, column) tuple or (2,) ndarray, optional
Center coordinate of transformation.
strength : float, optional
The amount of swirling applied.
radius : float, optional
The extent of the swirl in pixels. The effect dies out rapidly beyond
radius
.rotation : float, optional
Additional rotation applied to the image.
Returns: swirled : ndarray
Swirled version of the input.
Other Parameters: output_shape : tuple (rows, cols), optional
Shape of the output image generated. By default the shape of the input image is preserved.
order : int, optional
The order of the spline interpolation, default is 1. The order has to be in the range 0-5. See
skimage.transform.warp
for detail.mode : {‘constant’, ‘edge’, ‘symmetric’, ‘reflect’, ‘wrap’}, optional
Points outside the boundaries of the input are filled according to the given mode. Modes match the behaviour of
numpy.pad
.cval : float, optional
Used in conjunction with mode ‘constant’, the value outside the image boundaries.
clip : bool, optional
Whether to clip the output to the range of values of the input image. This is enabled by default, since higher order interpolation may produce values outside the given input range.
preserve_range : bool, optional
Whether to keep the original range of values. Otherwise, the input image is converted according to the conventions of
img_as_float
.
warp
-
skimage.transform.warp(image, inverse_map=None, map_args={}, output_shape=None, order=1, mode='constant', cval=0.0, clip=True, preserve_range=False)
[source] -
Warp an image according to a given coordinate transformation.
Parameters: image : ndarray
Input image.
inverse_map : transformation object, callable
cr = f(cr, **kwargs)
, or ndarrayInverse coordinate map, which transforms coordinates in the output images into their corresponding coordinates in the input image.
There are a number of different options to define this map, depending on the dimensionality of the input image. A 2-D image can have 2 dimensions for gray-scale images, or 3 dimensions with color information.
- For 2-D images, you can directly pass a transformation object, e.g.
skimage.transform.SimilarityTransform
, or its inverse. - For 2-D images, you can pass a
(3, 3)
homogeneous transformation matrix, e.g.skimage.transform.SimilarityTransform.params
. - For 2-D images, a function that transforms a
(M, 2)
array of(col, row)
coordinates in the output image to their corresponding coordinates in the input image. Extra parameters to the function can be specified throughmap_args
. - For N-D images, you can directly pass an array of coordinates. The first dimension specifies the coordinates in the input image, while the subsequent dimensions determine the position in the output image. E.g. in case of 2-D images, you need to pass an array of shape
(2, rows, cols)
, whererows
andcols
determine the shape of the output image, and the first dimension contains the(row, col)
coordinate in the input image. Seescipy.ndimage.map_coordinates
for further documentation.
Note, that a
(3, 3)
matrix is interpreted as a homogeneous transformation matrix, so you cannot interpolate values from a 3-D input, if the output is of shape(3,)
.See example section for usage.
map_args : dict, optional
Keyword arguments passed to
inverse_map
.output_shape : tuple (rows, cols), optional
Shape of the output image generated. By default the shape of the input image is preserved. Note that, even for multi-band images, only rows and columns need to be specified.
order : int, optional
- The order of interpolation. The order has to be in the range 0-5:
-
- 0: Nearest-neighbor
- 1: Bi-linear (default)
- 2: Bi-quadratic
- 3: Bi-cubic
- 4: Bi-quartic
- 5: Bi-quintic
mode : {‘constant’, ‘edge’, ‘symmetric’, ‘reflect’, ‘wrap’}, optional
Points outside the boundaries of the input are filled according to the given mode. Modes match the behaviour of
numpy.pad
.cval : float, optional
Used in conjunction with mode ‘constant’, the value outside the image boundaries.
clip : bool, optional
Whether to clip the output to the range of values of the input image. This is enabled by default, since higher order interpolation may produce values outside the given input range.
preserve_range : bool, optional
Whether to keep the original range of values. Otherwise, the input image is converted according to the conventions of
img_as_float
.Returns: warped : double ndarray
The warped input image.
Notes
- The input image is converted to a
double
image. - In case of a
SimilarityTransform
,AffineTransform
andProjectiveTransform
andorder
in [0, 3] this function uses the underlying transformation matrix to warp the image with a much faster routine.
Examples
>>> from skimage.transform import warp >>> from skimage import data >>> image = data.camera()
The following image warps are all equal but differ substantially in execution time. The image is shifted to the bottom.
Use a geometric transform to warp an image (fast):
>>> from skimage.transform import SimilarityTransform >>> tform = SimilarityTransform(translation=(0, -10)) >>> warped = warp(image, tform)
Use a callable (slow):
>>> def shift_down(xy): ... xy[:, 1] -= 10 ... return xy >>> warped = warp(image, shift_down)
Use a transformation matrix to warp an image (fast):
>>> matrix = np.array([[1, 0, 0], [0, 1, -10], [0, 0, 1]]) >>> warped = warp(image, matrix) >>> from skimage.transform import ProjectiveTransform >>> warped = warp(image, ProjectiveTransform(matrix=matrix))
You can also use the inverse of a geometric transformation (fast):
>>> warped = warp(image, tform.inverse)
For N-D images you can pass a coordinate array, that specifies the coordinates in the input image for every element in the output image. E.g. if you want to rescale a 3-D cube, you can do:
>>> cube_shape = np.array([30, 30, 30]) >>> cube = np.random.rand(*cube_shape)
Setup the coordinate array, that defines the scaling:
>>> scale = 0.1 >>> output_shape = (scale * cube_shape).astype(int) >>> coords0, coords1, coords2 = np.mgrid[:output_shape[0], ... :output_shape[1], :output_shape[2]] >>> coords = np.array([coords0, coords1, coords2])
Assume that the cube contains spatial data, where the first array element center is at coordinate (0.5, 0.5, 0.5) in real space, i.e. we have to account for this extra offset when scaling the image:
>>> coords = (coords + 0.5) / scale - 0.5 >>> warped = warp(cube, coords)
- For 2-D images, you can directly pass a transformation object, e.g.
warp_coords
-
skimage.transform.warp_coords(coord_map, shape, dtype=<type 'numpy.float64'>)
[source] -
Build the source coordinates for the output of a 2-D image warp.
Parameters: coord_map : callable like GeometricTransform.inverse
Return input coordinates for given output coordinates. Coordinates are in the shape (P, 2), where P is the number of coordinates and each element is a
(row, col)
pair.shape : tuple
Shape of output image
(rows, cols[, bands])
.dtype : np.dtype or string
dtype for return value (sane choices: float32 or float64).
Returns: coords : (ndim, rows, cols[, bands]) array of dtype
dtype
Coordinates for
scipy.ndimage.map_coordinates
, that will yield an image of shape (orows, ocols, bands) by drawing from source points according to thecoord_transform_fn
.Notes
This is a lower-level routine that produces the source coordinates for 2-D images used by
warp()
.It is provided separately from
warp
to give additional flexibility to users who would like, for example, to re-use a particular coordinate mapping, to use specific dtypes at various points along the the image-warping process, or to implement different post-processing logic thanwarp
performs after the call tondi.map_coordinates
.Examples
Produce a coordinate map that shifts an image up and to the right:
>>> from skimage import data >>> from scipy.ndimage import map_coordinates >>> >>> def shift_up10_left20(xy): ... return xy - np.array([-20, 10])[None, :] >>> >>> image = data.astronaut().astype(np.float32) >>> coords = warp_coords(shift_up10_left20, image.shape) >>> warped_image = map_coordinates(image, coords)
AffineTransform
-
class skimage.transform.AffineTransform(matrix=None, scale=None, rotation=None, shear=None, translation=None)
[source] -
Bases:
skimage.transform._geometric.ProjectiveTransform
2D affine transformation of the form:
..:math:
- X = a0*x + a1*y + a2 =
- = sx*x*cos(rotation) - sy*y*sin(rotation + shear) + a2
- Y = b0*x + b1*y + b2 =
- = sx*x*sin(rotation) + sy*y*cos(rotation + shear) + b2
where
sx
andsy
are zoom factors in the x and y directions, and the homogeneous transformation matrix is:[[a0 a1 a2] [b0 b1 b2] [0 0 1]]
Parameters: matrix : (3, 3) array, optional
Homogeneous transformation matrix.
scale : (sx, sy) as array, list or tuple, optional
Scale factors.
rotation : float, optional
Rotation angle in counter-clockwise direction as radians.
shear : float, optional
Shear angle in counter-clockwise direction as radians.
translation : (tx, ty) as array, list or tuple, optional
Translation parameters.
Attributes
params ((3, 3) array) Homogeneous transformation matrix. -
__init__(matrix=None, scale=None, rotation=None, shear=None, translation=None)
[source]
-
rotation
-
scale
-
shear
-
translation
PiecewiseAffineTransform
-
class skimage.transform.PiecewiseAffineTransform
[source] -
Bases:
skimage.transform._geometric.GeometricTransform
2D piecewise affine transformation.
Control points are used to define the mapping. The transform is based on a Delaunay triangulation of the points to form a mesh. Each triangle is used to find a local affine transform.
Attributes
affines (list of AffineTransform objects) Affine transformations for each triangle in the mesh. inverse_affines (list of AffineTransform objects) Inverse affine transformations for each triangle in the mesh. -
__init__()
[source]
-
estimate(src, dst)
[source] -
Set the control points with which to perform the piecewise mapping.
Number of source and destination coordinates must match.
Parameters: src : (N, 2) array
Source coordinates.
dst : (N, 2) array
Destination coordinates.
Returns: success : bool
True, if model estimation succeeds.
-
inverse(coords)
[source] -
Apply inverse transformation.
Coordinates outside of the mesh will be set to
- 1
.Parameters: coords : (N, 2) array
Source coordinates.
Returns: coords : (N, 2) array
Transformed coordinates.
-
PolynomialTransform
-
class skimage.transform.PolynomialTransform(params=None)
[source] -
Bases:
skimage.transform._geometric.GeometricTransform
2D transformation of the form:
..:math:
X = sum[j=0:order]( sum[i=0:j]( a_ji * x**(j - i) * y**i )) Y = sum[j=0:order]( sum[i=0:j]( b_ji * x**(j - i) * y**i ))Parameters: params : (2, N) array, optional
Polynomial coefficients where
N * 2 = (order + 1) * (order + 2)
. So, a_ji is defined inparams[0, :]
and b_ji inparams[1, :]
.Attributes
params ((2, N) array) Polynomial coefficients where N * 2 = (order + 1) * (order + 2)
. So, a_ji is defined inparams[0, :]
and b_ji inparams[1, :]
.-
__init__(params=None)
[source]
-
estimate(src, dst, order=2)
[source] -
Set the transformation matrix with the explicit transformation parameters.
You can determine the over-, well- and under-determined parameters with the total least-squares method.
Number of source and destination coordinates must match.
The transformation is defined as:
X = sum[j=0:order]( sum[i=0:j]( a_ji * x**(j - i) * y**i )) Y = sum[j=0:order]( sum[i=0:j]( b_ji * x**(j - i) * y**i ))
These equations can be transformed to the following form:
0 = sum[j=0:order]( sum[i=0:j]( a_ji * x**(j - i) * y**i )) - X 0 = sum[j=0:order]( sum[i=0:j]( b_ji * x**(j - i) * y**i )) - Y
which exist for each set of corresponding points, so we have a set of N * 2 equations. The coefficients appear linearly so we can write A x = 0, where:
A = [[1 x y x**2 x*y y**2 ... 0 ... 0 -X] [0 ... 0 1 x y x**2 x*y y**2 -Y] ... ... ] x.T = [a00 a10 a11 a20 a21 a22 ... ann b00 b10 b11 b20 b21 b22 ... bnn c3]
In case of total least-squares the solution of this homogeneous system of equations is the right singular vector of A which corresponds to the smallest singular value normed by the coefficient c3.
Parameters: src : (N, 2) array
Source coordinates.
dst : (N, 2) array
Destination coordinates.
order : int, optional
Polynomial order (number of coefficients is order + 1).
Returns: success : bool
True, if model estimation succeeds.
-
inverse(coords)
[source]
-
ProjectiveTransform
-
class skimage.transform.ProjectiveTransform(matrix=None)
[source] -
Bases:
skimage.transform._geometric.GeometricTransform
Matrix transformation.
Apply a projective transformation (homography) on coordinates.
For each homogeneous coordinate , its target position is calculated by multiplying with the given matrix, , to give :
[[a0 a1 a2] [b0 b1 b2] [c0 c1 1 ]].
E.g., to rotate by theta degrees clockwise, the matrix should be:
[[cos(theta) -sin(theta) 0] [sin(theta) cos(theta) 0] [0 0 1]]
or, to translate x by 10 and y by 20:
[[1 0 10] [0 1 20] [0 0 1 ]].
Parameters: matrix : (3, 3) array, optional
Homogeneous transformation matrix.
Attributes
params ((3, 3) array) Homogeneous transformation matrix. -
__init__(matrix=None)
[source]
-
estimate(src, dst)
[source] -
Set the transformation matrix with the explicit transformation parameters.
You can determine the over-, well- and under-determined parameters with the total least-squares method.
Number of source and destination coordinates must match.
The transformation is defined as:
X = (a0*x + a1*y + a2) / (c0*x + c1*y + 1) Y = (b0*x + b1*y + b2) / (c0*x + c1*y + 1)
These equations can be transformed to the following form:
0 = a0*x + a1*y + a2 - c0*x*X - c1*y*X - X 0 = b0*x + b1*y + b2 - c0*x*Y - c1*y*Y - Y
which exist for each set of corresponding points, so we have a set of N * 2 equations. The coefficients appear linearly so we can write A x = 0, where:
A = [[x y 1 0 0 0 -x*X -y*X -X] [0 0 0 x y 1 -x*Y -y*Y -Y] ... ... ] x.T = [a0 a1 a2 b0 b1 b2 c0 c1 c3]
In case of total least-squares the solution of this homogeneous system of equations is the right singular vector of A which corresponds to the smallest singular value normed by the coefficient c3.
In case of the affine transformation the coefficients c0 and c1 are 0. Thus the system of equations is:
A = [[x y 1 0 0 0 -X] [0 0 0 x y 1 -Y] ... ... ] x.T = [a0 a1 a2 b0 b1 b2 c3]
Parameters: src : (N, 2) array
Source coordinates.
dst : (N, 2) array
Destination coordinates.
Returns: success : bool
True, if model estimation succeeds.
-
inverse(coords)
[source] -
Apply inverse transformation.
Parameters: coords : (N, 2) array
Source coordinates.
Returns: coords : (N, 2) array
Transformed coordinates.
-
SimilarityTransform
-
class skimage.transform.SimilarityTransform(matrix=None, scale=None, rotation=None, translation=None)
[source] -
Bases:
skimage.transform._geometric.ProjectiveTransform
2D similarity transformation of the form:
..:math:
- X = a0 * x - b0 * y + a1 =
- = m * x * cos(rotation) - m * y * sin(rotation) + a1
- Y = b0 * x + a0 * y + b1 =
- = m * x * sin(rotation) + m * y * cos(rotation) + b1
where
m
is a zoom factor and the homogeneous transformation matrix is:[[a0 b0 a1] [b0 a0 b1] [0 0 1]]
Parameters: matrix : (3, 3) array, optional
Homogeneous transformation matrix.
scale : float, optional
Scale factor.
rotation : float, optional
Rotation angle in counter-clockwise direction as radians.
translation : (tx, ty) as array, list or tuple, optional
x, y translation parameters.
Attributes
params ((3, 3) array) Homogeneous transformation matrix. -
__init__(matrix=None, scale=None, rotation=None, translation=None)
[source]
-
estimate(src, dst)
[source] -
Set the transformation matrix with the explicit parameters.
You can determine the over-, well- and under-determined parameters with the total least-squares method.
Number of source and destination coordinates must match.
The transformation is defined as:
X = a0 * x - b0 * y + a1 Y = b0 * x + a0 * y + b1
These equations can be transformed to the following form:
0 = a0 * x - b0 * y + a1 - X 0 = b0 * x + a0 * y + b1 - Y
which exist for each set of corresponding points, so we have a set of N * 2 equations. The coefficients appear linearly so we can write A x = 0, where:
A = [[x 1 -y 0 -X] [y 0 x 1 -Y] ... ... ] x.T = [a0 a1 b0 b1 c3]
In case of total least-squares the solution of this homogeneous system of equations is the right singular vector of A which corresponds to the smallest singular value normed by the coefficient c3.
Parameters: src : (N, 2) array
Source coordinates.
dst : (N, 2) array
Destination coordinates.
Returns: success : bool
True, if model estimation succeeds.
-
rotation
-
scale
-
translation
© 2011 the scikit-image team
Licensed under the BSD 3-clause License.
http://scikit-image.org/docs/0.12.x/api/skimage.transform.html