Continuous Coordinates ¶
HoloViews is designed to work with scientific and engineering data, which is often in the form of discrete samples from an underlying continuous system. Imaging data is one clear example: measurements taken at a regular interval over a grid covering a twodimensional area. Although the measurements are discrete, they approximate a continuous distribution, and HoloViews provides extensive support for working naturally with data of this type.
2D Continuous spaces ¶
In this tutorial we will show the support provided by HoloViews for working with twodimensional regularly sampled grid data like images, and then in subsequent sections discuss how HoloViews supports onedimensional, higherdimensional, and irregularly sampled data with continuous coordinates.
import numpy as np
import holoviews as hv
hv.notebook_extension()
np.set_printoptions(precision=2, linewidth=80)
%opts HeatMap (cmap="hot")
First, let's consider:
f(x,y)

a simple function that accepts a location in a 2D plane specified in millimeters (mm)  
region

a 1mm×1mm square region of this 2D plane, centered at the origin, and</dd>  
coords

a function returning a square (s×s) grid of (x,y) coordinates regularly sampling the region in the given bounds, at the centers of each grid cell  
def f(x,y):
return x+y/3.1
region=(0.5,0.5,0.5,0.5)
def coords(bounds,samples):
l,b,r,t=bounds
hc=0.5/samples
return np.meshgrid(np.linspace(l+hc,rhc,samples),
np.linspace(b+hc,thc,samples))
Now let's build a Numpy array regularly sampling this function at a density of 5 samples per mm:
f5=f(*coords(region,5))
f5
We can visualize this array (and thus the function
f
) either using a
Raster
, which uses the array's own integerbased coordinate system (which we will call "array" coordinates), or an
Image
, which uses a continuous coordinate system, or as a
HeatMap
labelling each value explicitly:
r5 = hv.Raster(f5, label="R5")
i5 = hv.Image( f5, label="I5", bounds=region)
h5 = hv.HeatMap([(x, y, f5[4y,x]) for x in range(0,5) for y in range(0,5)], label="H5")
r5+i5+h5
Both the
Raster
and
Image
Element
types accept the same input data and show the same arrangement of colors, but a visualization of the
Raster
type reveals the underlying raw array indexing, while the
Image
type has been labelled with the coordinate system from which we know the data has been sampled. All
Image
operations work with this continuous coordinate system instead, while the corresponding operations on a
Raster
use raw array indexing.
For instance, all five of these indexing operations refer to the same element of the underlying Numpy array, i.e. the second item in the first row:
"r5[0,1]=%0.2f r5.data[0,1]=%0.2f i5[0.2,0.4]=%0.2f i5[0.24,0.37]=%0.2f i5.data[0,1]=%0.2f" % \
(r5[1,0], r5.data[0,1], i5[0.2,0.4], i5[0.24,0.37], i5.data[0,1])
You can see that the
Raster
and the underlying
.data
elements both use Numpy's raw integer indexing, while the
Image
uses floatingpoint values that are then mapped onto the appropriate array element.
This diagram should help show the relationships between the
Raster
coordinate system in the plot (which ranges from 0 at the top edge to 5 at the bottom), the underlying raw Numpy integer array indexes (labelling each dot in the
Array coordinates
figure), and the underlying
Continuous coordinates
:



Importantly, although we used a 5×5 array in this example, we could substitute a much larger array with the same continuous coordinate system if we wished, without having to change any of our continuous indexes  they will still point to the correct location in the continuous space:
f10=f(*coords(region,10))
f10
r10 = hv.Raster(f10, label="R10")
i10 = hv.Image(f10, label="I10", bounds=region)
r10+i10
The image now has higher resolution, but still visualizes the same underlying continuous function, now evaluated at 100 grid positions instead of 25:



Indexing the exact same coordinates as above now gets very different results:
"r10[0,1]=%0.2f r10.data[0,1]=%0.2f i10[0.2,0.4]=%0.2f i10[0.24,0.37]=%0.2f i10.data[0,1]=%0.2f" % \
(r10[1,0], r10.data[0,1], i10[0.2,0.4], i10[0.24,0.37], i10.data[0,1])
The arraybased indexes used by
Raster
and the Numpy array in
.data
still return the second item in the first row of the array, but this array element now corresponds to location (0.35,0.4) in the continuous function, and so the value is different. These indexes thus do
not
refer to the same location in continuous space as they did for the other array density, because raw Numpybased indexing is
not
independent of density or resolution.
Luckily, the two continuous coordinates still return very similar values to what they did before, since they always return the value of the array element corresponding to the closest location in continuous space. They now return elements just above and to the right, or just below and to the left, of the earlier location, because the array now has a higher resolution with elements centered at different locations.
Indexing in continuous coordinates always returns the value closest to the requested value, given the available resolution. Note that in the case of coordinates truly on the boundary between array elements (as for 0.2,0.4), the bounds of each array cell are taken as right exclusive and upper exclusive, and so (0.2,0.4) returns array index (3,0).
Slicing in 2D ¶
In addition to indexing (looking up a value), slicing (selecting a region) works as expected in continuous space (see the Sampling Data tutorial for more explanation). For instance, we can ask for a slice from (0.275,0.0125) to (0.025,0.2885) in continuous coordinates:
sl10=i10[0.275:0.025,0.0125:0.2885]
sl10.data
sl10
This slice has selected those array elements whose centers are contained within the specified continuous space. To do this, the continuous coordinates are first converted by HoloViews into the floatingpoint range (5.125,2.250) (2.125,5.250) of array coordinates, and all those elements whose centers are in that range are selected:



Slicing also works for
Raster
elements, but it results in an object that always reflects the contents of the underlying Numpy array (i.e., always with the upper left corner labelled 0,0):
r5[0:3,1:3] + r5[0:3,1:2]
Hopefully these examples make it clear that if you are using data that is sampled from some underlying continuous system, you should use the continuous coordinates offered by HoloViews objects like
Image
so that your programs can be independent of the resolution or sampling density of that data, and so that your axes and indexes can be expressed naturally, using the actual units of the underlying continuous space. The data will still be stored in the same Numpy array, but now you can treat it consistently like the approximation to continuous values that it is.
1D and nD Continuous coordinates ¶
All of the above examples use the common case for visualizations of a twodimensional regularly gridded continuous space, which is implemented in
holoviews.core.sheetcoords.SheetCoordinateSystem
.
Similar continuous coordinates and slicing are also supported for
Chart
elements, such as
Curve
s, but using a single index and allowing arbitrary irregular spacing, implemented in
holoviews.elements.chart.Chart
.
They also work the same for the ndimensional coordinates and slicing supported by the
container
types
HoloMap
,
NdLayout
, and
NdOverlay
, implemented in
holoviews.core.dimension.Dimensioned
and again allowing arbitrary irregular spacing.
Together, these powerful continuouscoordinate indexing and slicing operations allow you to work naturally and simply in the full n dimensional space that characterizes your data and parameter values.
Sampling ¶
The above examples focus on indexing and slicing, but as described in the
Sampling Data
tutorial there is another related operation supported for continuous spaces, called sampling. Sampling is similar to indexing and slicing, in that all of them can reduce the dimensionality of your data, but sampling is implemented in a general way that applies for any of the 1D, 2D, or nD datatypes. For instance, if we take our 10×10 array from above, we can ask for the value at a given location, which will come back as a
Table
, i.e. a dictionary with one (key,value) pair:
e10=i10.sample(x=0.275, y=0.2885)
e10
Similarly, if we ask for the value of a given
y
location in continuous space, we will get a
Curve
with the array row closest to that
y
value in the
Image
2D array returned as arrays of $x$ values and the corresponding
z
value from the image:
r10=i10.sample(y=0.2885)
r10
The same sampling syntax can be used on HoloViews objects with any number of continuouscoordinate dimensions, in each case returning a HoloViews object of the correct dimensionality. This support for working in continuous spaces makes it much more natural to work with HoloViews objects than directly with the underlying raw Numpy arrays, but the raw data always remains available when needed.
Download this notebook from GitHub (rightclick to download).