---
title: "Histogram gaps and spikes in digital photography"
author: "Michael A. Covington"
mainfont: "Georgia"
sansfont: "Arial"
fig-height: 2.5
fig-width: 4.5
date: "2023 November 26"
date-format: "YYYY MMMM D"
format: pdf
editor: visual
---

```{r}
#| echo: false
#| message: false
require(ggplot2)
```

Some digital cameras scale their digital output more than once before delivering it to you. This results in gaps or spikes in the histogram, especially if the scale factor is close to 1. Here are examples showing how this happens.

Let's start with a smooth distribution of random floating-point numbers, then round each of them to the nearest integer. The histogram comes out smooth, of course.

```{r}
x = round(seq(0,200,0.2))
ggplot(data.frame(x),aes(x)) + geom_histogram(binwidth=1) + ylim(c(0,10))
```

\newpage
Now multiply each of those integers by 0.95 and round to the nearest integer again. You get spikes:

```{r}
y = round(x*0.95)
ggplot(data.frame(y),aes(y)) + geom_histogram(binwidth=1) + ylim(c(0,10))
```

The reason is that some integers occur more than once. An example:

| Original value | $\times$ 0.95 | Rounded |
|----------------|---------------|---------|
| 26             | 24.70         | 25      |
| 27             | 25.65         | 26      |
| 28             | 26.60         | 27      |
| 29             | 27.55         | 28      |
| 30             | 28.50         | 29      |
| 31             | 29.45         | 29      |
| 32             | 30.40         | 30      |

There is a spike at 29 because there are two ways to get 29. (Actually, with R's rounding rules, 28.50 rounds down, and there is a spike at 28.)

\newpage
If we scale up rather than down, we get gaps in the histogram:

```{r}
y = round(x*1.05)
ggplot(data.frame(y),aes(y)) + geom_histogram(binwidth=1) + ylim(c(0,10))
```

Now there are integers that never occur because adjacent integers in the input
round to integers that are not adjacent in the output.
Gaps are bothersome because they cause jumps (posterization) in gradients in
underexposed areas of the picture.

I am grateful to Mark Shelley, who pointed out this effect to me
in this discussion on Cloudy Nights Forums:
<https://www.cloudynights.com/topic/896181-canon-eos-r-iso-1600-colour-gradients-and-rings/>