Multidimensional discrete convolution

Key takeaways

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

Reference excerpt

In signal processing, multidimensional discrete convolution refers to the mathematical operation between two functions f and g on an n-dimensional lattice that produces a third function, also of n-dimensions. Multidimensional discrete convolution is the discrete analog of the multidimensional convolution of functions on Euclidean space. It is also a special case of convolution on groups when the group is the group of n-tuples of integers.

Definition

Problem statement and basics Similar to the one-dimensional case, an asterisk is used to represent the convolution operation. The number of dimensions in the given operation is reflected in the number of asterisks. For example, an M-dimensional convolution would be written with M asterisks. The following represents a M-dimensional convolution of discrete signals:

y ( n 1 , n 2 , . . . , n M ) = x ( n 1 , n 2 , . . . , n M ) ∗ ⋯ M ∗ h ( n 1 , n 2 , . . . , n M ) {\displaystyle y(n_{1},n_{2},...,n_{M})=x(n_{1},n_{2},...,n_{M})*{\overset {M}{\cdots }}*h(n_{1},n_{2},...,n_{M})}

For discrete-valued signals, this convolution can be directly computed via the following:

∑ k 1 = − ∞ ∞ ∑ k 2 = − ∞ ∞ . . . ∑ k M = − ∞ ∞ h ( k 1 , k 2 , . . . , k M ) x ( n 1 − k 1 , n 2 − k 2 , . . . , n M − k M ) {\displaystyle \sum _{k_{1}=-\infty }^{\infty }\sum _{k_{2}=-\infty }^{\infty }...\sum _{k_{M}=-\infty }^{\infty }h(k_{1},k_{2},...,k_{M})x(n_{1}-k_{1},n_{2}-k_{2},...,n_{M}-k_{M})}

The resulting output region of support of a discrete multidimensional convolution will be determined based on the size and regions of support of the two input signals. Listed are several properties of the two-dimensional convolution operator. Note that these can also be extended for signals of N {\displaystyle N} -dimensions. Commutative Property:

x ∗ ∗ h = h ∗ ∗ x {\displaystyle x**h=h**x}

Associate Property:

( x ∗ ∗ h ) ∗ ∗ g = x ∗ ∗ ( h ∗ ∗ g ) {\displaystyle (x**h)**g=x**(h**g)}

Distributive Property:

x ∗ ∗ ( h + g ) = ( x ∗ ∗ h ) + ( x ∗ ∗ g ) {\displaystyle x**(h+g)=(x**h)+(x**g)}

These properties are seen in use in the figure below. Given some input x ( n 1 , n 2 ) {\displaystyle x(n_{1},n_{2})} that goes into a filter with impulse response h ( n 1 , n 2 ) {\displaystyle h(n_{1},n_{2})} and then another filter with impulse response g ( n 1 , n 2 ) {\displaystyle g(n_{1},n_{2})} , the output is given by y ( n 1 , n 2 ) {\displaystyle y(n_{1},n_{2})} . Assume that the output of the first filter is given by w ( n 1 , n 2 ) {\displaystyle w(n_{1},n_{2})} , this means that:

w = x ∗ ∗ h {\displaystyle w=x**h}

… excerpt ends here. Continue reading the full article.

Illustrations

Multidimensional discrete convolution: Impulse Responses of Typical Multidimensional Low Pass Filters — Impulse Responses of Typical Multidimensional Low Pass Filters

Multidimensional discrete convolution: Typical Impulse Responses for Edge Detection — Typical Impulse Responses for Edge Detection

Worked examples

Example 1 — a first encounter with Multidimensional discrete convolution

Start with the simplest possible case. Write down what Multidimensional discrete convolution claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, 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 Multidimensional discrete convolution 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 Multidimensional discrete convolution 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 Multidimensional discrete convolution

In research: Multidimensional discrete convolution appears in science 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 Multidimensional discrete convolution 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: Multidimensional discrete convolution is common in secondary-school and first-year university syllabi. It links to neighbouring topics Multidimensional signal processing, so understanding it makes those chapters shorter.
In everyday life: Look for Multidimensional discrete convolution 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.

How to study Multidimensional discrete convolution in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Multidimensional discrete convolution 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 Multidimensional discrete convolution out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Multidimensional discrete convolution in simple terms?

In signal processing, multidimensional discrete convolution refers to the mathematical operation between two functions f and g on an n-dimensional lattice that produces a third function, also of n-dimensions. Multidimensional discrete convolution is the discrete analog of the multidimensional convo…

Why does Multidimensional discrete convolution matter?

Because it connects several science 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 Multidimensional discrete convolution?

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 Multidimensional discrete convolution.