2D histograms
Multidimensional histograms are the extension of the concept of Image Histogram to Multi Channel images. In the simplest case, a 2D histogram for a Two Channel image, the horizontal axis represents the pixels with values in the first channel only; the vertical axis those existing only in the second channel. The different bins in the quadrant delimited by these axes represent the amount of VoXels occuring in the image with a specific (x,y) combination of intensities in the two channels. The amount of pixels in a given (x,y) bin is represented by a color scale only, not with the height of any bar as in 1D Image Histograms. Higher values are represented with colors closer to red.
Adding another dimension, histograms of 3-channel images are represented by cubes of bins. In Huygens Professional, the Slicer in the image viewer can be used to inspect the values slice by slice.
To demonstrate the usefulness of image histograms, consider the two-channel image in this figure:
Its 2D histogram follows:
There are no many pixels in the horizontal axis: the image does not contain many VoXels with signal only in the first channel. Wherever signal appears in the first channel, there is also signal in the second channel. The contrary is not true: there are many pixels in the vertical axis, indicating the existence of voxels with intensity recorded only in the second channel.
As the original image is represented with first channel in red and second channel in green, we can see there what the histogram shows: we only see green (pure second channel) and yellow (1st-2nd channel combinations) regions, but not any red region where only first channel intensities would exist.
The following remarks can be made in this particular example:
- The histogram shows a single red pixel at the origin, indicating a high value for bin 0. This is suspicious because it means that this is probably a Clipped Image, with clipping at the background level. Normally the noise in the background is Gauss-distributed. The bins around the origin should have been filled to a larger extent.
- Many high values (green pixels) are located at the top. Quite probably the values in the second channel are clipped.
Another extreme situation is shown in the following figures. Two identical one-channel images are joined together to create a two-channel image (in a red-green representation, all the image is yellow). All pixels in the final image belong equally to both channels, and thus the 2D histogram is a straight line at values y=x.
The Colocalization Analyzer also makes use of 2D histograms. For two channels with a high degree of overlapping, the histogram pixels trend to concrentrate along this y=x line, the example above being the extreme case of total overlapping.
You can find more examples of 2D histograms used on colocalization in the following reference: Analysis of protein co-localization using wide-field fluorescence microscopy and image-restoration for co-visualisation of CFP and YFP conjugated signalling proteins inside living cells. J. Weitzman, R. Lizundia, B. Blumen, M. Marchand, S. Shorte. http://www.pasteur.fr/recherche/unites/Pfid/html/un_coloc/?en
See also Cooccurence Theory.
