I recently stumbled across a fascinating question: how could I tell whether an image had changed significantly? As PHP developers, the most troublesome image problem we have to deal with is how to resize an upload with an acceptable loss of quality.

In the end I discovered what many before me have – that this problem becomes relatively simple given the application of some fundamental mathematical principles. Come along with me as we learn about them…

*You can find the code for this tutorial at https://github.com/undemanding/difference.*

### More from this author

## Bitmaps

There are two popular ways of thinking about images. The first is as a grid of individual pixels, composed of varying levels of color and contrast. Commonly, we break these colors down into their constituent red, green, and blue values. We could also think of them as hue, saturation, and lightness.

The second way of thinking about images is in terms of vectors. A line isn’t the pixels in between but rather a starting point and an ending point, with some meta data that describes a stroke in between. We’re going to focus on bitmaps because they’ll make the whole process easier.

We can break any image down into this bitmap grid, with code resembling:

```
$image = imagecreatefrompng($path);
$width = imagesx($image);
$height = imagesy($image);
$map = [];
for ($y = 0; $y < $height; $y++) {
$map[$y] = [];
for ($x = 0; $x < $width; $x++) {
$color = imagecolorat($image, $x, $y);
$map[$y][$x] = [
"r" => ($color >> 16) & 0xFF,
"g" => ($color >> 8) & 0xFF,
"b" => $color & 0xFF
];
}
}
```

Given the width and height of the image, we can use a function called `imagecolorat`

(on an image resource) to get a single integer value for the red, green, and blue at that pixel. We can then use bit shifting and masking to get the individual values of each from the single integer value.

*Each red, green, and blue value is in a range from 0 to 255. In binary, this range can be expressed as 00000000 to 11111111. A single integer value can represent 3 sets of these binary values and bit shifting is a way to get the binary values from the first, second, or third group of 8 bits.*

If we create these grids from a couple of images; how can we compare them? We got the answer 2300 years ago…

## Distance In Three Dimensions

Can you remember how to calculate the length of a line? Any line you can draw on paper can be thought of as the hypotenuse of a triangle (the long side). To measure it, we can square the horizontal and vertical sides of the right-angle triangle the hypotenuse makes, and work out their combined square root:

```
$start = [$x = 10, $y = 15];
$end = [$x = 20, $y = 30];
$width = $end[0] - $start[0];
$width *= $width;
$height = $end[1] - $start[1];
$height *= $height;
$distance = sqrt($width + $height); // ≈ 18.03
```

If the line was three-dimensional, we’d have to add a third component to the equation. There’s a general mathematical principle for this kind of distance measurement, called Euclidean distance. Older forms of it were called Pythagorean metric, because of the close relationship to the hypotenuse calculation we just did.

The formula expands to as many dimensions as we want it to, but we only need it for three:

```
$first = [$red = 100, $green = 125, $blue = 150];
$second = [$red = 125, $green = 150, $blue = 175];
$red = $second[0] - $first[0];
$red *= $red;
$green = $second[1] - $first[1];
$green *= $green;
$blue = $second[2] - $first[2];
$blue *= $blue;
$distance = sqrt($red + $green + $blue); // ≈ 43.30
```

We can apply this principle to every pixel of the bitmaps, until we have a third bitmap of just the differing values. Let’s try it…

## Simple Image Differences

We can apply these principles with very little code. Let’s make a class to load images, create their bitmaps and calculate a map of pixel differences:

```
class State
{
private $width;
private $height;
private $map = [];
public function __construct($width, $height)
{
$this->width = $width;
$this->height = $height;
}
}
```

Each map has a defined width and height. To populate the map, we need to load images into memory:

```
private static function createImage($path)
{
$image = null;
$info = getimagesize($path);
$type = $info[2];
if ($type == IMAGETYPE_JPEG) {
$image = imagecreatefromjpeg($path);
}
if ($type == IMAGETYPE_GIF) {
$image = imagecreatefromgif($path);
}
if ($type == IMAGETYPE_PNG) {
$image = imagecreatefrompng($path);
}
if (!$image) {
throw new InvalidArgumentException("image invalid");
}
return $image;
}
```

We can use the GD image library to read a number of image formats. This function tries to load a file path that could be a JPEG, a GIF, or a PNG. If none of these work, we can just raise an exception. Why is this method static? We’re going to use it in another static method:

```
public static function fromImage($path)
{
if (!file_exists($path)) {
throw new InvalidArgumentException("image not found");
}
$image = static::createImage($path);
$width = imagesx($image);
$height = imagesy($image);
$map = [];
for ($y = 0; $y < $height; $y++) {
$map[$y] = [];
for ($x = 0; $x < $width; $x++) {
$color = imagecolorat($image, $x, $y);
$map[$y][$x] = [
"r" => ($color >> 16) & 0xFF,
"g" => ($color >> 8) & 0xFF,
"b" => $color & 0xFF
];
}
}
$new = new static($width, $height);
$new->map = $map;
return $new;
}
```

This static function allows us to create new image states (or maps) from a static call to `State::fromImage("/path/to/image.png")`

. If the path doesn’t exist, we can raise another exception. Then we have the same image grid construction logic we saw previously. Finally, we create a new `State`

, with a defined width, height and the map we constructed.

Now, let’s make a way to compare multiple images:

```
public function withDifference(State $state, callable $method)
{
$map = [];
for ($y = 0; $y < $this->height; $y++) {
$map[$y] = [];
for ($x = 0; $x < $this->width; $x++) {
$map[$y][$x] = $method(
$this->map[$y][$x],
$state->map[$y][$x]
);
}
}
return $this->cloneWith("map", $map);
}
private function cloneWith($property, $value)
{
$clone = clone $this;
$clone->$property = $value;
return $clone;
}
```

We go through each pixel, in each image. We pass them to a difference function and assign the resulting value to a new map. Finally, we create a clone of the current `State`

, so it keeps the width and height, but sets a new state. This ensures we don’t modify existing `State`

instances, but rather have access to a new instance with its own map.

What does that difference function look like?

```
class EuclideanDistance
{
public function __invoke(array $p, array $q)
{
$r = $p["r"] - $q["r"];
$r *= $r;
$g = $p["g"] - $q["g"];
$g *= $g;
$b = $p["b"] - $q["b"];
$b *= $b;
return sqrt($r + $g + $b);
}
}
```

It turns out we can use classes as functions, if we give them an `__invoke`

method. In this class we can put the Euclidian distance logic we saw previously. We can put all of these pieces together like this:

```
$state1 = State::fromImage("/path/to/image1.png");
$state2 = State::fromImage("/path/to/image2.png");
$state3 = $state1->withDifference(
$state2,
new EuclideanDistance()
);
```

This method works great for images that are almost identical. When we try to work out the differences in very similar photos or even lossy versions of almost identical images, we’re presented with many slight differences.

## Standard Deviation

To get around this problem, we need to remove the noise and focus only on the biggest problem. We can do that by working out how spread out the differences are. This measure of how spread out numbers are is called Standard Deviation.

*Image from Wikipedia*

Most of the small differences between images are within the standard deviation (or the dark blue band between -1σ and 1σ). If we eliminate all the small differences within the standard deviation, then we should be left with the big differences. To work out which pixels are within the standard deviation, we need to work out the average pixel value:

```
public function average()
{
$average = 0;
for ($y = 0; $y < $this->height; $y++) {
for ($x = 0; $x < $this->width; $x++) {
if (!is_numeric($this->map[$y][$x])) {
throw new LogicException("pixel is not numeric");
}
$average += $this->map[$y][$x];
}
}
$average /= ($this->width * $this->height);
return $average;
}
```

Averages are easy! Just add together everything and divide by the number of things you added together. An average of two things is their total value divided by two. An average of 400 x 300 things is their total value divided by 120,000.

*Notice how we’re expecting numeric values during the calculation? That means we first need to generate the purely numeric state ( $state3 in the above example), or a state generated by the EuclideanDistance difference function.*

```
public function standardDeviation()
{
$standardDeviation = 0;
$average = $this->average();
for ($y = 0; $y < $this->height; $y++) {
for ($x = 0; $x < $this->width; $x++) {
if (!is_numeric($this->map[$y][$x])) {
throw new LogicException(
"pixel is not numeric"
);
}
$delta = $this->map[$y][$x] - $average;
$standardDeviation += ($delta * $delta);
}
}
$standardDeviation /= (($this->width * $this->height) - 1);
$standardDeviation = sqrt($standardDeviation);
return $standardDeviation;
}
```

This calculation is a little similar to the distance calculation we did before. The basic idea is that we work out the average of all the pixels. We then work out how far each is from that average, and then work out the average of all of these distances.

We can apply this back to the `State`

:

```
public function withReducedStandardDeviation()
{
$map = array_slice($this->map, 0);
$deviation = $this->standardDeviation();
for ($y = 0; $y < $this->height; $y++) {
for ($x = 0; $x < $this->width; $x++) {
if (abs($map[$y][$x]) < $deviation) {
$map[$y][$x] = 0;
}
}
}
return $this->cloneWith("map", $map);
}
```

This way, if differences are inside the standard deviation band, we exclude them from a new `State`

. The result is a cleaner picture of changes.

```
public function boundary()
{
$ax = $this->width;
$bx = 0;
$ay = $this->width;
$by = 0;
for ($y = 0; $y < $this->height; $y++) {
for ($x = 0; $x < $this->width; $x++) {
if ($this->map[$y][$x] > 0) {
if ($x > $bx) {
$bx = $x;
}
if ($x < $ax) {
$ax = $x;
}
if ($y > $by) {
$by = $y;
}
if ($y < $ay) {
$ay = $y;
}
}
}
}
if ($ax > $bx) {
throw new LogicException("ax is greater than bx");
}
if ($ay > $by) {
throw new LogicException("ay is greater than by");
}
$ax = ($ax / $this->width) * $this->width;
$bx = ((($bx + 1) / $this->width) * $this->width) - $ax;
$ay = ($ay / $this->height) * $this->height;
$by = ((($by + 1) / $this->height) * $this->height) - $ay;
return [
"left" => $ax,
"top" => $ay,
"width" => $bx,
"height" => $by
];
}
```

The final piece of the puzzle is a function to work out where the boundaries of the changes are. We can use this function as a way to draw a box around the changes from one image to another. The box starts on the edges of the grid and slowly moves inward until it reaches changes in the grid.

## Conclusion

I started this experiment to find the differences between two images. You see, I wanted to take screenshots of an interface, during automated testing, and tell if something significant had changed.

Euclidian distance, applied to bitmaps, told me about every changed pixel. Then I wanted to allow for slight changes (like small text or colour changes), so I applied Standard Deviation noise removal, so that only significant changes would come through. Finally, I could work out exactly how many pixels were different as a percentage of the total number of pixels on the screen. I could tell if one screenshot was within 10% or 20% of a test fixture image.

Perhaps you have need of this for something wildly different. Perhaps you have ideas of how this can be improved. Let us know in the comments!