visualizing colormaps for complex functions

Hi,

Is there a way to generate colormaps for complex-valued functions using matplotlib? The type of plots I’m looking for are like the plots in:
http://commons.wikimedia.org/wiki/User:Jan_Homann/Mathematics

Thanks in advance,

Guy

···

http://www.guyrutenberg.com

Hi Guy,

I am also interested in the answer to this. The cplot function in the mpmath module does exactly this using matplotlib, but very inefficiently, as it computes the colour of each pixel in the image in hls colour-space and generates the corresponding rgb value directly. I suspect this is how it has to be done, as colormaps in matplotlib are 1D sequences and the black-white (lightness) value is really another dimension. However mpmath's method can be improved by doing the mapping using array operations instead of computing it for each pixel.

I've attached a function I wrote to reproduce the Sage cplot command in my own work. It's a bit old and can be improved. It takes the Arg and Abs of a complex array as the first two arguments - you can easily change this to compute these inside the function if you prefer. The line
np.vectorize(hls_to_rgb) can be replaced - recent versions of matplotlib have a vectorized function called hsv_to_rgb() inside colors.py - so you replace the return line with the commented-out version if you first import hsv_to_rgb from colors.

I hope this helps.

I'm also curious: the plots you point to also show plots of the function "extrema", which are the phase singularities - does mathematica have a function that gives you these, or did you write your own function to find them?

regards,
Gary

Guy Rutenberg wrote:

cplot_like.py (2.13 KB)

···

Hi,

Is there a way to generate colormaps for complex-valued functions using matplotlib? The type of plots I'm looking for are like the plots in:
http://commons.wikimedia.org/wiki/User:Jan_Homann/Mathematics

Thanks in advance,

Guy

Hi Gary,

Thanks for responding. It looks like the mpmath does what I’m looking for. Your code looks interesting, as it lowers the number of dependencies needed. I hope I’ll find the time to really incorporate this features to something that can come as part to matplotlib.

Regarding the plots I’ve pointed to, I didn’t make them, so I don’t know how the author did it (expect that he used Mathematica).

Thanks,

Guy

···

On Sat, Apr 3, 2010 at 4:29 AM, Gary Ruben <gruben@…636…> wrote:

Hi Guy,

I am also interested in the answer to this. The cplot function in the mpmath module does exactly this using matplotlib, but very inefficiently, as it computes the colour of each pixel in the image in hls colour-space and generates the corresponding rgb value directly. I suspect this is how it has to be done, as colormaps in matplotlib are 1D sequences and the black-white (lightness) value is really another dimension. However mpmath’s method can be improved by doing the mapping using array operations instead of computing it for each pixel.

I’ve attached a function I wrote to reproduce the Sage cplot command in my own work. It’s a bit old and can be improved. It takes the Arg and Abs of a complex array as the first two arguments - you can easily change this to compute these inside the function if you prefer. The line

np.vectorize(hls_to_rgb) can be replaced - recent versions of matplotlib have a vectorized function called hsv_to_rgb() inside colors.py - so you replace the return line with the commented-out version if you first import hsv_to_rgb from colors.

I hope this helps.

I’m also curious: the plots you point to also show plots of the function “extrema”, which are the phase singularities - does mathematica have a function that gives you these, or did you write your own function to find them?

regards,

Gary

Guy Rutenberg wrote:

Hi,

Is there a way to generate colormaps for complex-valued functions using matplotlib? The type of plots I’m looking for are like the plots in:

http://commons.wikimedia.org/wiki/User:Jan_Homann/Mathematics

Thanks in advance,

Guy

def cplot_like(ph, intens=None, int_exponent=1.0, s=1.0, l_bias=1.0, drape=0, is_like_mpmath=False):

'''

Implements the mpmath cplot-like default_color_function

The combined image is generated in hls colourspace then transformed to rgb

*phase*

    A filename or 2D n x m array containing phase data in the range -pi->pi

*intens*

    If None, set to 1.0

    A filename or 2D n x m array containing intensity or amplitude data in the range 0->max

*int_exponent*

    Default 1.0 applies the intens mask directly to the hls lightness-channel

    0.6 works well when drape==0

*s*

    saturation. Defaults to 1.0. mpmath uses 0.8.

*l_bias*

    biases the mean lightness value away from 0.5. mpmath uses 1.0.

    Examples are: l_bias=2 -> mean=0.33 (ie darker), l_bias=0.5 -> mean=0.66 (lighter)

*drape*

    If >1, drapes a structured maximum filter of size drape x drape over the intensity data

*is_like_mpmath*

    If True, sets int_exponent = 0.3, s = 0.8

'''

from colorsys import hls_to_rgb



if type(ph) is str:

    cph = plt.imread(ph)/256.*2*pi-pi              # -pi->pi

    if len(cph.shape) == 3: cph = cph[...,0]      # if ph is RGB or RGBA, extract the R-plane

else:

    cph = ph.copy()



if intens is None:

    cintens = np.ones_like(cph)

elif type(intens) is str:

    cintens = plt.imread(intens)/255.                     # 0->1

    if len(cintens.shape) == 3: cintens = cintens[...,0]   # if intens is RGB or RGBA, extract the R-plane

else:

    cintens = intens.copy()

cintens /= cintens.max()                         # autoscale intensity data to 0->1



if drape > 1:

    # envelope the intensity

    cintens = maximum_filter(cintens, size=drape)



h = ((cph + pi) / (2*pi)) % 1.0



if is_like_mpmath:

    # apply mpmath values

    int_exponent = 0.3

    s = 0.8



l = 1.0 - l_bias/(l_bias+cintens**int_exponent)

v_hls_to_rgb = np.vectorize(hls_to_rgb)



#~ return hsv_to_rgb(dstack((h,np.ones_like(h),l)))

return dstack(v_hls_to_rgb(h,l,s))