Reference Guide

Table of Contents

polarTransform.convertToCartesianImage(image)	Convert polar image to cartesian image.
polarTransform.convertToPolarImage(image[, …])	Convert cartesian image to polar image.
polarTransform.getCartesianPointsImage(…)	Convert list of polar points from image to cartesian image points based on transform metadata
polarTransform.getPolarPointsImage(points, …)	Convert list of cartesian points from image to polar image points based on transform metadata
polarTransform.ImageTransform(center, …)	Class to store settings when converting between cartesian and polar domain

polarTransform Module

polarTransform.convertToCartesianImage(image, center=None, initialRadius=None, finalRadius=None, initialAngle=None, finalAngle=None, imageSize=None, hasColor=False, order=3, border='constant', borderVal=0.0, useMultiThreading=False, settings=None)

Convert polar image to cartesian image.

Using a polar image, this function creates a cartesian image. This function is versatile because it can automatically calculate an appropriate cartesian image size and center given the polar image. In addition, parameters for converting to the polar domain are necessary for the conversion back to the cartesian domain.

Parameters:

image : N-dimensional numpy.ndarray

Polar image to convert to cartesian domain

Image should be structured in C-order, i.e. the axes should be ordered (…, z, theta, r, [ch]). The channel axes should only be present if hasColor is True. This format is arbitrary but is selected to stay consistent with the traditional C-order representation in the Cartesian domain.

In the mathematical domain, Cartesian coordinates are traditionally represented as (x, y, z) and as (r, theta, z) in the polar domain. When storing Cartesian data in C-order, the axes are usually flipped and the data is saved as (z, y, x). Thus, the polar domain coordinates are also flipped to stay consistent, hence the format (z, theta, r).

Note

For multi-dimensional images above 2D, the cartesian transformation is applied individually across each 2D slice. The last two dimensions should be the r & theta dimensions, unless hasColor is True in which case the 2nd and 3rd to last dimensions should be. The multidimensional shape will be preserved for the resulting cartesian image (besides the polar dimensions).

center : str or (2,) list, tuple or numpy.ndarray of int, optional

Specifies the center in the cartesian image to use as the origin in polar domain. The center in the cartesian domain will be (0, 0) in the polar domain.

If center is not set, then it will default to middle-middle. If the image size is None, the center is calculated after the image size is determined.

For relative positioning within the image, center can be one of the string values in the table below. The quadrant column contains the visible quadrants for the given center. initialAngle and finalAngle must contain at least one of the quadrants, otherwise an error will be thrown because the resulting cartesian image is blank. An example cartesian image is given below with annotations to what the center will be given a center string.

Valid center strings

Value	Quadrant	Location in image
top-left	IV	1
top-middle	III, IV	2
top-right	III	3
middle-left	I, IV	4
middle-middle	I, II, III, IV	5
middle-right	II, III	6
bottom-left	I	7
bottom-middle	I, II	8
bottom-right	II	9

initialRadius : int, optional

Starting radius in pixels from the center of the cartesian image in the polar image

The polar image begins at this radius, i.e. the first row of the polar image corresponds to this starting radius.

If initialRadius is not set, then it will default to 0.

finalRadius : int, optional

Final radius in pixels from the center of the cartesian image in the polar image

The polar image ends at this radius, i.e. the last row of the polar image corresponds to this ending radius.

Note

The polar image does not include this radius. It includes all radii starting from initial to final radii excluding the final radius. Rather, it will stop one step size before the final radius. Assuming the radial resolution (see radiusSize) is small enough, this should not matter.

If finalRadius is not set, then it will default to the maximum radius which is the size of the radial (1st) dimension of the polar image.

initialAngle : float, optional

Starting angle in radians in the polar image

The polar image begins at this angle, i.e. the first column of the polar image corresponds to this starting angle.

Radian angle is with respect to the x-axis and rotates counter-clockwise. The angle should be in the range of 0 to \(2\pi\).

If initialAngle is not set, then it will default to 0.0.

finalAngle : float, optional

Final angle in radians in the polar image

The polar image ends at this angle, i.e. the last column of the polar image corresponds to this ending angle.

Note

The polar image does not include this angle. It includes all angles starting from initial to final angle excluding the final angle. Rather, it stops one step size before the final angle. Assuming the angular resolution (see angleSize) is small enough, this should not matter.

Radian angle is with respect to the x-axis and rotates counter-clockwise. The angle should be in the range of 0 to \(2\pi\).

If finalAngle is not set, then it will default to \(2\pi\).

imageSize : (2,) list, tuple or numpy.ndarray of int, optional

Desired size of cartesian image where 1st dimension is number of rows and 2nd dimension is number of columns

If imageSize is not set, then it defaults to the size required to fit the entire polar image on a cartesian image.

hasColor : bool, optional

Whether or not the polar image contains color channels

This means that the image is structured as (…, y, x, ch) or (…, theta, r, ch) for Cartesian or polar images, respectively. If color channels are present, the last dimension (channel axes) will be shifted to the front, converted and then shifted back to its original location.

Default is False

Note

If an alpha band (4th channel of image is present), then it will be converted. Typically, this is unwanted, so the recommended solution is to transform the first 3 channels and set the 4th channel to fully on.

order : int (0-5), optional

The order of the spline interpolation, default is 3. The order has to be in the range 0-5.

The following orders have special names:

• 0 - nearest neighbor
• 1 - bilinear
• 3 - bicubic

border : {‘constant’, ‘nearest’, ‘wrap’, ‘reflect’}, optional

Polar points outside the cartesian image boundaries are filled according to the given mode.

Default is ‘constant’

The following table describes the mode and expected output when seeking past the boundaries. The input column is the 1D input array whilst the extended columns on either side of the input array correspond to the expected values for the given mode if one extends past the boundaries.

Valid border modes and expected output

Mode	Ext.	Input	Ext.
mirror	4 3 2	1 2 3 4 5 6 7 8	7 6 5
reflect	3 2 1	1 2 3 4 5 6 7 8	8 7 6
nearest	1 1 1	1 2 3 4 5 6 7 8	8 8 8
constant	0 0 0	1 2 3 4 5 6 7 8	0 0 0
wrap	6 7 8	1 2 3 4 5 6 7 8	1 2 3

Refer to scipy.ndimage.map_coordinates() for more details on this argument.

borderVal : same datatype as image, optional

Value used for polar points outside the cartesian image boundaries if border = ‘constant’.

Default is 0.0

useMultiThreading : bool, optional

Whether to use multithreading when applying transformation for 3D images. This considerably speeds up the execution time for large images but adds overhead for smaller 3D images.

Default is False

settings : ImageTransform, optional

Contains metadata for conversion between polar and cartesian image.

Settings contains many of the arguments in convertToPolarImage() and convertToCartesianImage() and provides an easy way of passing these parameters along without having to specify them all again.

Warning

Cleaner and more succint to use ImageTransform.convertToCartesianImage()

If settings is not specified, then the other arguments are used in this function and the defaults will be calculated if necessary. If settings is given, then the values from settings will be used.

Returns:

cartesianImage : N-dimensional numpy.ndarray

Cartesian image

Resulting image is structured in C-order, i.e. the axes are ordered as (…, z, y, x, [ch]). This format is the traditional method of storing image data in Python.

Resulting image shape will be the same as the input image except for the polar dimensions are replaced with the Cartesian dimensions.

settings : ImageTransform

Contains metadata for conversion between polar and cartesian image.

Settings contains many of the arguments in convertToPolarImage() and convertToCartesianImage() and provides an easy way of passing these parameters along without having to specify them all again.

polarTransform.convertToPolarImage(image, center=None, initialRadius=None, finalRadius=None, initialAngle=None, finalAngle=None, radiusSize=None, angleSize=None, hasColor=False, order=3, border='constant', borderVal=0.0, useMultiThreading=False, settings=None)

Convert cartesian image to polar image.

Using a cartesian image, this function creates a polar domain image where the first dimension is radius and second dimension is the angle. This function is versatile because it allows different starting and stopping radii and angles to extract the polar region you are interested in.

Note

Traditionally images are loaded such that the origin is in the upper-left hand corner. In these cases the initialAngle and finalAngle will rotate clockwise from the x-axis. For simplicitly, it is recommended to flip the image along first dimension before passing to this function.

Parameters:

image : N-dimensional numpy.ndarray

Cartesian image to convert to polar domain

Image should be structured in C-order, i.e. the axes should be ordered as (…, z, y, x, [ch]). This format is the traditional method of storing image data in Python.

Note

For multi-dimensional images above 2D, the polar transformation is applied individually across each 2D slice. The last two dimensions should be the x & y dimensions, unless hasColor is True in which case the 2nd and 3rd to last dimensions should be. The multidimensional shape will be preserved for the resulting polar image (besides the Cartesian dimensions).

center : (2,) list, tuple or numpy.ndarray of int, optional

Specifies the center in the cartesian image to use as the origin in polar domain. The center in the cartesian domain will be (0, 0) in the polar domain.

The center is structured as (x, y) where the first item is the x-coordinate and second item is the y-coordinate.

If center is not set, then it will default to round(image.shape[::-1] / 2).

initialRadius : int, optional

Starting radius in pixels from the center of the cartesian image that will appear in the polar image

The polar image will begin at this radius, i.e. the first row of the polar image will correspond to this starting radius.

If initialRadius is not set, then it will default to 0.

finalRadius : int, optional

Final radius in pixels from the center of the cartesian image that will appear in the polar image

The polar image will end at this radius, i.e. the last row of the polar image will correspond to this ending radius.

Note

The polar image will not include this radius. It will include all radii starting from initial to final radii excluding the final radius. Rather, it will stop one step size before the final radius. Assuming the radial resolution (see radiusSize) is small enough, this should not matter.

If finalRadius is not set, then it will default to the maximum radius of the cartesian image. Using the furthest corner from the center, the finalRadius can be calculated as:

\[finalRadius = \sqrt{((X_{max} - X_{center})^2 + (Y_{max} - Y_{center})^2)}\]

initialAngle : float, optional

Starting angle in radians that will appear in the polar image

The polar image will begin at this angle, i.e. the first column of the polar image will correspond to this starting angle.

Radian angle is with respect to the x-axis and rotates counter-clockwise. The angle should be in the range of 0 to \(2\pi\).

If initialAngle is not set, then it will default to 0.0.

finalAngle : float, optional

Final angle in radians that will appear in the polar image

The polar image will end at this angle, i.e. the last column of the polar image will correspond to this ending angle.

Note

The polar image will not include this angle. It will include all angle starting from initial to final angle excluding the final angle. Rather, it will stop one step size before the final angle. Assuming the angular resolution (see angleSize) is small enough, this should not matter.

Radian angle is with respect to the x-axis and rotates counter-clockwise. The angle should be in the range of 0 to \(2\pi\).

If finalAngle is not set, then it will default to \(2\pi\).

radiusSize : int, optional

Size of polar image for radial (1st) dimension

This in effect determines the resolution of the radial dimension of the polar image based on the initialRadius and finalRadius. Resolution can be calculated using equation below in radial px per cartesian px:

\[radialResolution = \frac{radiusSize}{finalRadius - initialRadius}\]

If radiusSize is not set, then it will default to the minimum size necessary to ensure that image information is not lost in the transformation. The minimum resolution necessary can be found by finding the smallest change in radius from two connected pixels in the cartesian image. Through experimentation, there is a surprisingly close relationship between the maximum difference from width or height of the cartesian image to the center times two.

The radiusSize is calculated based on this relationship and is proportional to the initialRadius and finalRadius given.

angleSize : int, optional

Size of polar image for angular (2nd) dimension

This in effect determines the resolution of the angular dimension of the polar image based on the initialAngle and finalAngle. Resolution can be calculated using equation below in angular px per cartesian px:

\[angularResolution = \frac{angleSize}{finalAngle - initialAngle}\]

If angleSize is not set, then it will default to the minimum size necessary to ensure that image information is not lost in the transformation. The minimum resolution necessary can be found by finding the smallest change in angle from two connected pixels in the cartesian image.

For a cartesian image with either dimension greater than 500px, the angleSize is set to be two times larger than the largest dimension proportional to initialAngle and finalAngle. Otherwise, for a cartesian image with both dimensions less than 500px, the angleSize is set to be four times larger the largest dimension proportional to initialAngle and finalAngle.

Note

The above logic estimates the necessary angleSize to reduce image information loss. No algorithm currently exists for determining the required angleSize.

hasColor : bool, optional

Whether or not the cartesian image contains color channels

This means that the image is structured as (…, y, x, ch) or (…, theta, r, ch) for Cartesian or polar images, respectively. If color channels are present, the last dimension (channel axes) will be shifted to the front, converted and then shifted back to its original location.

Default is False

Note

If an alpha band (4th channel of image is present), then it will be converted. Typically, this is unwanted, so the recommended solution is to transform the first 3 channels and set the 4th channel to fully on.

order : int (0-5), optional

The order of the spline interpolation, default is 3. The order has to be in the range 0-5.

The following orders have special names:

• 0 - nearest neighbor
• 1 - bilinear
• 3 - bicubic

border : {‘constant’, ‘nearest’, ‘wrap’, ‘reflect’}, optional

Polar points outside the cartesian image boundaries are filled according to the given mode.

Default is ‘constant’

The following table describes the mode and expected output when seeking past the boundaries. The input column is the 1D input array whilst the extended columns on either side of the input array correspond to the expected values for the given mode if one extends past the boundaries.

Valid border modes and expected output

Mode	Ext.	Input	Ext.
mirror	4 3 2	1 2 3 4 5 6 7 8	7 6 5
reflect	3 2 1	1 2 3 4 5 6 7 8	8 7 6
nearest	1 1 1	1 2 3 4 5 6 7 8	8 8 8
constant	0 0 0	1 2 3 4 5 6 7 8	0 0 0
wrap	6 7 8	1 2 3 4 5 6 7 8	1 2 3

Refer to scipy.ndimage.map_coordinates() for more details on this argument.

borderVal : same datatype as image, optional

Value used for polar points outside the cartesian image boundaries if border = ‘constant’.

Default is 0.0

useMultiThreading : bool, optional

Whether to use multithreading when applying transformation for 3D images. This considerably speeds up the execution time for large images but adds overhead for smaller 3D images.

Default is False

settings : ImageTransform, optional

Contains metadata for conversion between polar and cartesian image.

Settings contains many of the arguments in convertToPolarImage() and convertToCartesianImage() and provides an easy way of passing these parameters along without having to specify them all again.

Warning

Cleaner and more succint to use ImageTransform.convertToPolarImage()

If settings is not specified, then the other arguments are used in this function and the defaults will be calculated if necessary. If settings is given, then the values from settings will be used.

Returns:

polarImage : N-dimensional numpy.ndarray

Polar image

Resulting image is structured in C-order, i.e. the axes are be ordered as (…, z, theta, r, [ch]) depending on if the input image was 3D. This format is arbitrary but is selected to stay consistent with the traditional C-order representation in the Cartesian domain.

In the mathematical domain, Cartesian coordinates are traditionally represented as (x, y, z) and as (r, theta, z) in the polar domain. When storing Cartesian data in C-order, the axes are usually flipped and the data is saved as (z, y, x). Thus, the polar domain coordinates are also flipped to stay consistent, hence the format (z, theta, r).

Resulting image shape will be the same as the input image except for the Cartesian dimensions are replaced with the polar dimensions.

settings : ImageTransform

Contains metadata for conversion between polar and cartesian image.

Settings contains many of the arguments in convertToPolarImage() and convertToCartesianImage() and provides an easy way of passing these parameters along without having to specify them all again.

class polarTransform.ImageTransform(center, initialRadius, finalRadius, initialAngle, finalAngle, cartesianImageSize, polarImageSize, hasColor)

Class to store settings when converting between cartesian and polar domain

convertToCartesianImage(image, order=3, border='constant', borderVal=0.0, useMultiThreading=False)

Convert polar image to cartesian image.

Using a polar image, this function creates a cartesian image. This function is versatile because it can automatically calculate an appropiate cartesian image size and center given the polar image. In addition, parameters for converting to the polar domain are necessary for the conversion back to the cartesian domain.

Parameters:

image : N-dimensional numpy.ndarray

Polar image to convert to cartesian domain

Image should be structured in C-order, i.e. the axes should be ordered (…, z, theta, r, [ch]). The channel axes should only be present if hasColor is True. This format is arbitrary but is selected to stay consistent with the traditional C-order representation in the Cartesian domain.

In the mathematical domain, Cartesian coordinates are traditionally represented as (x, y, z) and as (r, theta, z) in the polar domain. When storing Cartesian data in C-order, the axes are usually flipped and the data is saved as (z, y, x). Thus, the polar domain coordinates are also flipped to stay consistent, hence the format (z, theta, r).

Note

For multi-dimensional images above 2D, the cartesian transformation is applied individually across each 2D slice. The last two dimensions should be the r & theta dimensions, unless hasColor is True in which case the 2nd and 3rd to last dimensions should be. The multidimensional shape will be preserved for the resulting cartesian image (besides the polar dimensions).

order : int (0-5), optional

The order of the spline interpolation, default is 3. The order has to be in the range 0-5.

The following orders have special names:

• 0 - nearest neighbor
• 1 - bilinear
• 3 - bicubic

border : {‘constant’, ‘nearest’, ‘wrap’, ‘reflect’}, optional

Polar points outside the cartesian image boundaries are filled according to the given mode.

Default is ‘constant’

The following table describes the mode and expected output when seeking past the boundaries. The input column is the 1D input array whilst the extended columns on either side of the input array correspond to the expected values for the given mode if one extends past the boundaries.

Valid border modes and expected output

Mode	Ext.	Input	Ext.
mirror	4 3 2	1 2 3 4 5 6 7 8	7 6 5
reflect	3 2 1	1 2 3 4 5 6 7 8	8 7 6
nearest	1 1 1	1 2 3 4 5 6 7 8	8 8 8
constant	0 0 0	1 2 3 4 5 6 7 8	0 0 0
wrap	6 7 8	1 2 3 4 5 6 7 8	1 2 3

Refer to scipy.ndimage.map_coordinates() for more details on this argument.

borderVal : same datatype as image, optional

Value used for polar points outside the cartesian image boundaries if border = ‘constant’.

Default is 0.0

useMultiThreading : bool, optional

Whether to use multithreading when applying transformation for 3D images. This considerably speeds up the execution time for large images but adds overhead for smaller 3D images.

Default is False

Returns:

cartesianImage : N-dimensional numpy.ndarray

Cartesian image

Resulting image is structured in C-order, i.e. the axes are ordered as (…, z, y, x, [ch]). This format is the traditional method of storing image data in Python.

Resulting image shape will be the same as the input image except for the polar dimensions are replaced with the Cartesian dimensions.

See also

convertToCartesianImage()

convertToPolarImage(image, order=3, border='constant', borderVal=0.0, useMultiThreading=False)

Convert cartesian image to polar image.

Using a cartesian image, this function creates a polar domain image where the first dimension is radius and second dimension is the angle. This function is versatile because it allows different starting and stopping radii and angles to extract the polar region you are interested in.

Note

Traditionally images are loaded such that the origin is in the upper-left hand corner. In these cases the initialAngle and finalAngle will rotate clockwise from the x-axis. For simplicitly, it is recommended to flip the image along first dimension before passing to this function.

Parameters:

image : N-dimensional numpy.ndarray

Cartesian image to convert to polar domain

Image should be structured in C-order, i.e. the axes should be ordered as (…, z, y, x, [ch]). This format is the traditional method of storing image data in Python.

Note

For multi-dimensional images above 2D, the polar transformation is applied individually across each 2D slice. The last two dimensions should be the x & y dimensions, unless hasColor is True in which case the 2nd and 3rd to last dimensions should be. The multidimensional shape will be preserved for the resulting polar image (besides the Cartesian dimensions).

order : int (0-5), optional

The order of the spline interpolation, default is 3. The order has to be in the range 0-5.

The following orders have special names:

• 0 - nearest neighbor
• 1 - bilinear
• 3 - bicubic

border : {‘constant’, ‘nearest’, ‘wrap’, ‘reflect’}, optional

Polar points outside the cartesian image boundaries are filled according to the given mode.

Default is ‘constant’

The following table describes the mode and expected output when seeking past the boundaries. The input column is the 1D input array whilst the extended columns on either side of the input array correspond to the expected values for the given mode if one extends past the boundaries.

Valid border modes and expected output

Mode	Ext.	Input	Ext.
mirror	4 3 2	1 2 3 4 5 6 7 8	7 6 5
reflect	3 2 1	1 2 3 4 5 6 7 8	8 7 6
nearest	1 1 1	1 2 3 4 5 6 7 8	8 8 8
constant	0 0 0	1 2 3 4 5 6 7 8	0 0 0
wrap	6 7 8	1 2 3 4 5 6 7 8	1 2 3

Refer to scipy.ndimage.map_coordinates() for more details on this argument.

borderVal : same datatype as image, optional

Value used for polar points outside the cartesian image boundaries if border = ‘constant’.

Default is 0.0

Returns:

polarImage : N-dimensional numpy.ndarray

Polar image

Resulting image is structured in C-order, i.e. the axes are be ordered as (…, z, theta, r, [ch]) depending on if the input image was 3D. This format is arbitrary but is selected to stay consistent with the traditional C-order representation in the Cartesian domain.

In the mathematical domain, Cartesian coordinates are traditionally represented as (x, y, z) and as (r, theta, z) in the polar domain. When storing Cartesian data in C-order, the axes are usually flipped and the data is saved as (z, y, x). Thus, the polar domain coordinates are also flipped to stay consistent, hence the format (z, theta, r).

Resulting image shape will be the same as the input image except for the Cartesian dimensions are replaced with the polar dimensions.

getCartesianPointsImage(points)

Convert list of polar points from image to cartesian image points based on transform metadata

Note

This does not convert from polar to cartesian points, but rather converts pixels from polar image to pixels from cartesian image using ImageTransform.

The returned points are not rounded to the nearest point. User must do that by hand if desired.

Parameters:

points : (N, 2) or (2,) numpy.ndarray

List of polar points to convert to cartesian domain

First column is r and second column is theta

Returns:

cartesianPoints : (N, 2) or (2,) numpy.ndarray

Corresponding cartesian points from polar points using ImageTransform

See also

getCartesianPointsImage(), getCartesianPoints(), getCartesianPoints2()

getPolarPointsImage(points)

Convert list of cartesian points from image to polar image points based on transform metadata

Note

This does not convert from cartesian to polar points, but rather converts pixels from cartesian image to pixels from polar image using ImageTransform.

The returned points are not rounded to the nearest point. User must do that by hand if desired.

Parameters:

points : (N, 2) or (2,) numpy.ndarray

List of cartesian points to convert to polar domain

First column is x and second column is y

Returns:

polarPoints : (N, 2) or (2,) numpy.ndarray

Corresponding polar points from cartesian points using ImageTransform

See also

getPolarPointsImage(), getPolarPoints(), getPolarPoints2()

polarTransform.getCartesianPointsImage(points, settings)

Convert list of polar points from image to cartesian image points based on transform metadata

Warning

Cleaner and more succinct to use ImageTransform.getCartesianPointsImage()

Note

This does not convert from polar to cartesian points, but rather converts pixels from polar image to pixels from cartesian image using ImageTransform.

The returned points are not rounded to the nearest point. User must do that by hand if desired.

Parameters:

points : (N, 2) or (2,) numpy.ndarray

List of polar points to convert to cartesian domain

First column is r and second column is theta

settings : ImageTransform

Contains metadata for conversion from polar to cartesian domain

Settings contains many of the arguments in convertToPolarImage() and convertToCartesianImage() and provides an easy way of passing these parameters along without having to specify them all again.

Returns:

cartesianPoints : (N, 2) or (2,) numpy.ndarray

Corresponding cartesian points from polar points using settings

See also

ImageTransform.getCartesianPointsImage(), getCartesianPoints(), getCartesianPoints2()