mathematics

Sum of absolute differences

Sum of absolute differences is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Sum of absolute differences rather than just read about it. In short: In digital image processing, the sum of absolute differences (SAD) is a measure of the similarity between image blocks. It is calculated by taking the absolute difference between each pixel in the original block and the corresponding pixel in the block being used for comparison.

Key takeaways

Sum of absolute differences belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Sum of absolute differences to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Sum of absolute differences from memory before moving on to harder problems.

Reference excerpt

In digital image processing, the sum of absolute differences (SAD) is a measure of the similarity between image blocks. It is calculated by taking the absolute difference between each pixel in the original block and the corresponding pixel in the block being used for comparison. These differences are summed to create a simple metric of block similarity, the L1 norm of the difference image or Manhattan distance between two image blocks. The sum of absolute differences may be used for a variety of purposes, such as object recognition, the generation of disparity maps for stereo images, and motion estimation for video compression.

Example This example uses the sum of absolute differences to identify which part of a search image is most similar to a template image. In this example, the template image is 3 by 3 pixels in size, while the search image is 3 by 5 pixels in size. Each pixel is represented by a single integer from 0 to 9.

Template Search image 2 5 5 2 7 5 8 6 4 0 7 1 7 4 2 7 7 5 9 8 4 6 8 5

There are exactly three unique locations within the search image where the template may fit: the left side of the image, the center of the image, and the right side of the image. To calculate the SAD values, the absolute value of the difference between each corresponding pair of pixels is used: the difference between 2 and 2 is 0, 4 and 1 is 3, 7 and 8 is 1, and so forth. Calculating the values of the absolute differences for each pixel, for the three possible template locations, gives the following:

Left Center Right 0 2 0 5 0 3 3 3 1 3 7 3 3 4 5 0 2 0 1 1 3 3 1 1 1 3 4

For each of these three image patches, the 9 absolute differences are added together, giving SAD values of 20, 25, and 17, respectively. From these SAD values, it could be asserted that the right side of the search image is the most similar to the template image, because it has the lowest sum of absolute differences as compared to the other two locations.

Comparison to other metrics

Object recognition The sum of absolute differences provides a simple way to automate the searching for objects inside an image, but may be unreliable due to the effects of contextual factors such as changes in lighting, color, viewing direction, size, or shape. The SAD may be used in conjunction with other object recognition methods, such as edge detection, to improve the reliability of results.

Video compression SAD is an extremely fast metric due to its simplicity; it is effectively the simplest possible metric that takes into account every pixel in a block. Therefore, it is very effective for a wide motion search of many different blocks. SAD is also easily parallelizable since it analyzes each pixel separately, making it easily implementable with such instructions as ARM NEON or x86 SSE2. For example, SSE has packed sum of absolute differences instruction (PSADBW) specifically for this purpose. Once candidate blocks are found, the final refinement of the motion estimation process is often done with other slower but more accurate metrics, which better take into account human perception. These include the sum of absolute transformed differences (SATD), the sum of squared differences (SSD), and rate–distortion optimization.

See also Computer stereo vision Hadamard transform Motion compensation Motion estimation Object recognition (computer vision) Rate–distortion optimization

References E. G. Richardson, Iain (2003). H.264 and MPEG-4 Video Compression: Video Coding for Next-generation Multimedia. Chichester: John Wiley & Sons Ltd.

Worked examples

Example 1 — a first encounter with Sum of absolute differences

Start with the simplest possible case. Write down what Sum of absolute differences claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Sum of absolute differences before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Sum of absolute differences ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Sum of absolute differences

In research: Sum of absolute differences appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Sum of absolute differences in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Sum of absolute differences is common in secondary-school and first-year university syllabi. It links to neighbouring topics Loss functions, Signal processing metrics, Video compression, so understanding it makes those chapters shorter.
In everyday life: Look for Sum of absolute differences outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Sum of absolute differences in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Sum of absolute differences means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Sum of absolute differences out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Sum of absolute differences in simple terms?

Why does Sum of absolute differences matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Sum of absolute differences?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Sum of absolute differences.