Histropy: A Computer Program for Quantifications of Histograms of 2D Gray-Scale Images

1. Introduction

The term histogram was coined around 1892 by statistician Karl Pearson to describe the visual representation of the distribution of data that quantified the frequency of data that fell into "bins" of a certain range. Histograms themselves were "first conceived as a visual aid to statistical approximations" [2]. The visual analysis that a histogram facilitates, combined with the quantitative information that can be extracted from it, gives histograms a wide range of applications, ranging from analyzing the distributions of test scores in a classroom to probabilistically characterizing the behavior of river discharge.

One of the common applications of histograms is in digital image processing. Pixel intensity histograms, which represent the distribution of the intensities of all pixels of an image, provide a measure of the image that can be useful for identifying similar images, compressing the image, and more. The here briefly described computer program "Histropy" generates a pixel intensity histogram for images that are either in grayscale, Red-Green-Blue color (RGB), or a uniform hue with a color-tone-range and allows for the numerical analysis of user-selected bin-ranges within the histogram. (One of its name-giving features is the calculation of the so-called Monkey-Model [3] Shannon entropy of 2D images.)

The computer program’s prospective use within our research group is to quantitatively distinguish between symmetries and pseudosymmetries in a series of noisy as well as noise-filtered 2D-periodic images, i.e. crystal patterns. That noise-filtering of experimentally obtained and synthetic crystal patterns will involve crystallographic image processing ([4] after objective, i.e. information-theory-based [5], crystallographic symmetry classifications [6]. See appendix for an example of a highly pseudosymmetric crystal pattern. As part of such quantitative distinctions between symmetries and pseudosymmetries, both the so-called "Monkey-Model" version [3] of the Shannon entropy and the standard root-mean-square (RMS) contrast in original or converted gray-scale crystal patterns need to be calculated.

For gray-scale patterns such as the one shown in the background of Figure A1a, where there are a few pronounced peaks in the corresponding histogram (Figure A1b), it will be informative to make quantifying calculations for selected ranges of the pixel intensity values. (This may be considered as constituting a very basic form of pattern segmentation.) Competing computer programs, e.g. the histogram routines that are part of the well-known electron crystallography software CRISP [4], do not typically offer the functionality desired for our studies. (Figure A2 illustrates the effect of crystallographic image processing in two non-disjoint plane symmetry groups on a noisy version of the crystal pattern that is shown in the background of Figure A1b).

Since the source code of our program is freely available on GitHub [1], other researchers are invited to download and modify it to support their own studies that do not need to be based on information in two-dimensional crystal patterns or gray-scale images. Other applications of histograms of 2D images may arise over time as well when parts of our code get reused.

2. Description of the Computer Program

As implied by the title of the paper, gray-scale images are the standard input option. Note that Histropy can only handle images of 16-bit depth when they are gray-scale so 16-bit color images must be converted before opening in Histropy. This conversion can be done using the open access program GIMP [7] by going to the "Image" dropdown menu and then selecting "Grayscale" for the "Mode". When the user selects an 8-bit color image to analyze through Histropy, the program converts the image to grayscale using the following equation to compute the intensity value for each individual pixel based on the pixel’s RGB value:

$$\mathrm{Grayscale}\mathrm{value}=0.299\ast R+0.587\ast G+0.114\ast B,$$

(1)

where R stands for red, G for green, and B for blue. The coefficients reflect the fact that humans are most sensitive to green light and least sensitive to blue. Histropy’s histogram-workspace plot consists of the histogram of the individual pixel intensities of a user-selected 2D image (or crystal pattern), a toolbar at the bottom of the screen that allows the user to navigate the histogram, four selection spaces to the right of the histogram that allow the user to quantify the histogram and image’s information, and a display of the selected image being plotted to the right of the selection spaces, Figure 1. Note that the Histropy window must be viewed in full-screen mode for the layout seen on the next page. When processing 8-bit images, Histropy runs all actions in one to five seconds. Processing 16-bit images is about fifteen times slower on average with most operations taking around thirty seconds. These processing times will increase as the user overlays more images. These timings were found using a MacBook Pro, which has an Apple M3 processor with a 4.05 GHz max CPU clock rate.

2.1. Selection Space 1: Scale

The first selection space, “Scale,” shown in Figure 2, allows the user to click-change between a linear and log base 10 scale, which affects the y-axis (display of number of pixels in a standard bin of unity) scale on the histogram. It also contains an input field for a y-axis limit, which defaults to the maximum y-value in the histogram. This value represents how many pixels have the most common pixel intensity value or the height of the most prominent peak in the histogram.

2.2. Selection Space 2: Intensity Range

The text fields in the second selection space, “Intensity Range,” displayed in Figure 3, set the range for calculations performed in the third selection space. These text fields default to the minimum and maximum pixel intensity in the image. They can be set by directly typing into the fields themselves or by clicking directly on the histogram, which will automatically set the range in the selection space text fields to the x-value corresponding to where the user’s mouse is pointing. When the user’s mouse is hovering over the histogram, the text in the bottom right corner of the screen will display the coordinates that the mouse is over, Figure 4. These coordinates can be used to accurately select the x-values for the range. The range selected is visually represented by the vertical bars and a translucent blue rectangle shown on the histogram, Figure 1. This range selection allows the user to perform a simple form of segmentation by separating out the pixel bins that contribute to a certain histogram peak.

2.3. Selection Space 3: Calculations

The third selection space, “Calculations,” shown in Figure 5, is a display of the following calculations over a user-selected range (which defaults to the entire range of the histogram):

• The number of pixels on the given range
• The percent of total pixels that are found in the given range
• A form of the Shannon entropy on the given range
• The mean on the given range
• The RMS contrast on the range
• The total pixel intensity on the range (a sum)

Note that these calculations are not impacted by y-axis limitings done in the “Scale” selection space. The one-dimensional Shannon entropy is calculated using the following equation:

$$\sum _{i=1}^N{\displaystyle \frac{p_i}{N}}·{log}_2\left({\displaystyle \frac{p_i}{N}}\right),$$

(2)

where N is the number of pixels on the range and $p_i$ is the intensity of the $i^{th}$ pixel. The mean of the range is the sum of the pixel intensities over a range divided by the number of pixels on the range:

$$\overline{p}=\left({\displaystyle \frac{1}{N}}\right)\sum _{i=1}^N{p}_i.$$

(3)

The total intensity is calculated as $N·\overline{p}$. Finally, the RMS Contrast is calculated with the equation

$${\displaystyle \frac{1}{d}}\sqrt{\sum _{i=1}^N{\displaystyle \frac{{\left(p_i-\overline{p}\right)}^2}{N}}},$$

(4)

where d is $2^{\mathrm{bit}\mathrm{depth}}-1$ and ${\displaystyle \frac{1}{d}}$ normalizes the output between 0 and 1.

2.4. Selection Space 4: Histogram Overlay

The final selection space, “Histogram Overlays,” Figure 6, allows the user to add images whose pixel intensity data will be overlaid onto the histogram of the first image. The second image that a user adds will have its quantifying calculation data displayed in the color corresponding to its plot in the “Histogram Overlays” selection space, Figure 7. Currently, Histropy is only capable of displaying these quantifications for one image in addition to the user’s first image. As another image is read into the program using the fourth selection space, the selection space will switch to displaying the file names of the added images in colors corresponding to how they appear on the histogram, Figure 7. The first four images that the user overlays will appear on the right underneath the original image with their title colors corresponding to their appearance on the histogram, Figure 8. Up to 22 images

can be overlaid but it is recommended to stick to 10 overlaid images or less in order to maximize program performance. If the user wishes to remove the overlays, one has to click on the “Clear Overlays” button in the fourth selection space, which will reset the histogram, the selection space, and the image displays to their original states.

2.5. Navigating the Histogram

The buttons in the bottom left corner of the Histropy workspace, Figure 1, which appear underneath the histogram, allow the user to zoom in on specific parts of the histogram using the magnifying glass and move around the viewing window along the x- and y-axes with the axes button. When the user wishes to go back to a previous view, they can do so using the back arrow or they can move forward to the next view with the forward arrow. All viewing changes can be reset with the home button and the user can save the full plot image as a PNG with the save button.

3. Concluding Remarks

Histropy is a versatile program that facilitates the visual analysis and quantification of images. The functionality and a potential use of the computer program Histropy has been briefly demonstrated in a few examples based on versions of the crystal pattern in the appendix. An expanded description (handbook) of this program is freely available, together with the source code, in the supporting material for this paper. The authors hope that other researchers will find this paper interesting and eventually make good use of this program or parts of its source code in their own work.