Logical and Cartesian Indexing in Julia

22 Apr, 2020
Julia, like most technical computing languages, provides a first-class array implementation with very important functions that make working with N-Dimensional arrays quite easy. One of the latest features, the dot(.) based broadcasting makes most of the “repetitive” array operations, a one-lined code.
Indexing into arrays in Julia is similar to it’s counterparts.Syntax:
x = arr[index_1, index_2, ..., index_n]
where each index_k may be a scalar integer, an array of integers, or an object that represents an array of scalar indices.
Array Indexing in Julia is of two types:
Cartesian Indexing
Logical Indexing
Cartesian Coordinates give the location of a point in 1D, 2D or 3D plane. Cartesian Indices have similar behavior. They give the value of element stored in a 1D, 2D, 3D or n-D array.For scalar indices, CartesianIndex{N}s, behave like an N-tuple of integers spanning multiple dimensions while array of scalar indices include Arrays of CartesianIndex{N}Syntax:
CartesianIndex(i, j, k...)   -> I
CartesianIndex((i, j, k...)) -> I

The above syntax creates a multidimensional index I, which can be used for indexing a multidimensional array arr. We can say that, arr[I] is equivalent to arr[i, j, k...]. It is allowed to mix integer and CartesianIndex indices.Example :
arr[Ipre, i, Ipost] 
(where Ipre and Ipost are CartesianIndex indices and i is an Int) 
can be a useful expression when writing algorithms that work along a single dimension
of an array of arbitrary dimensionality.
CartesianIndexing for 1D arrays:
# Create a 1D array of size 5arr = reshape(Vector(1:2:10), (5))  # Select index number 1arr[CartesianIndex(1)]  # Select index number 3arr[CartesianIndex(3)]
Output:
CartesianIndexing for a 2D array:
# Create a 2D array of size 3x2arr = reshape(Vector(1:2:12), (3, 2))  # Select cartesian index 2, 2 arr[CartesianIndex(2, 2)]  # Select index number 3, 1arr[CartesianIndex(3, 1)]
Output:
CartesianIndexing for 3D Array:
# Create a 3D array of size 2x1x2arr = reshape(Vector(1:2:8), (2, 1, 2))  # Select cartesian index 1, 1, 1arr[CartesianIndex(1, 1, 1)]  # Select index number 1, 1, 2arr[CartesianIndex(1, 1, 2)]
Output:
CartesianIndexing for nD array:
# Create a 5D array of size 2x2x1x2x2arr = reshape(Vector(1:2:32), (2, 2, 1, 2, 2))  # Select cartesian index 1, 2, 1, 2, 2arr[CartesianIndex(2, 2, 1, 2, 2)]
Output:
From the above examples, it becomes pretty clear that CartesianIndex simply gathers multiple integers together into one object that represents a single multidimensional index.Having discussed scalar representations, let’s talk about arrays of CartesianIndex{N}, which represent a collection of scalar indices each spanning N dimensions, such form of indexing is referred to as “pointwise” indexing.
Array based CartesianIndexing for 1D Array:
# Create a 1D array of size 5arr = reshape(Vector(2:6), 5)  # Select an array of cartesian indices 2, 3, 5arr[[CartesianIndex(2),      CartesianIndex(3), CartesianIndex(5)]]
Output:
Array based CartesianIndexing for 2D Array:
# Create a 2D array of size 5x6arr = reshape(Vector(1:30), (5, 6))  # Select an array of cartesian indices (5, 2), (3, 6), (1, 4)arr[[CartesianIndex(5, 2),      CartesianIndex(3, 6), CartesianIndex(1, 4)]]
Output:
Array based CartesianIndexing for 3D array:
# Create a 3D array of size 2x3x1arr = reshape(Vector(1:12), (2, 3, 2))  # Select an array of cartesian indices (1, 2, 1), (2, 2, 2), (1, 3, 1)arr[[CartesianIndex(1, 2, 1),     CartesianIndex(1, 3, 1), CartesianIndex(2, 2, 2)]]
Output:
Array based CartesianIndexing for nD array:
# Create a 5D array of size 2x2x1x2x2arr = reshape(Vector(1:2:32), (2, 2, 1, 2, 2))  # Select an array of cartesian indices # (1, 2, 1, 1, 1), (1, 2, 1, 2, 2), (2, 1, 1, 2, 1)arr[[CartesianIndex(1, 2, 1, 1, 1),      CartesianIndex(1, 2, 1, 2, 2),      CartesianIndex(2, 1, 1, 2, 1)]]
Output:
This is really useful, in case we want to select specific elements of an array if they are not in order. But what if we have an array of 1000 and we need to extract 100 elements, it’ll be quite tedious!This is where dot(.) broadcasting comes into play.
# Create a 3D array of size 3x3x3arr = reshape(Vector(1:27), (3, 3, 3))  # Select diagonal elements in all the # 3 planes of cartesian coordinatesarr[[CartesianIndex(1, 1, 1), CartesianIndex(2, 2, 1),      CartesianIndex(3, 3, 1), CartesianIndex(1, 1, 2),      CartesianIndex(2, 2, 2), CartesianIndex(3, 3, 2),     CartesianIndex(1, 1, 3), CartesianIndex(2, 2, 3),      CartesianIndex(3, 3, 3)]]  # Using dot broadcasting and (:)colon, to reduce codearr[CartesianIndex.(axes(arr, 1), axes(arr, 2), :)]
Output:How efficiently we have been able to reduce our code in the third step!!
What should we do with Cartesian Indexing ?We observe, CartesianIndices bear close resemblence to Cartesian coordinates and function in a similar way.Example:
We have a 3D shape in space, and it is mapped out by storing the coordinates of it's 
edges in an array and we might want to mask out a region from the shape. It becomes 
easy to deal with by using CartesianIndices, since they resemble the coordinate system. 

In computing/electronics, the basis is a logic that is deterministic in nature. It is:
true, false
one, zero
on, off

It is basically a selection of elements at the indices where the values of our logical indexing array are true.A logical indexing array is called a “mask” since it masks out the values that are false. A mask is of type bool(boolean).
Relation with CartesianIndexing:
Indexing by a N-dimensional boolean array is equivalent to indexing by the vector 
of CartesianIndex{N}s where its values are true. 

Example : Implementation of a logical mask
# Create a 2D array of size 5x5arr = reshape(Vector(1:25), (5, 5))  # Apply a logical mask to the arrayarr[[true, false, true, false, true], :]
Output:So we see how only the rows whose index matches with the index of trues in our mask[true, false, true, false, true] are selected.
Any condition that yields a boolean value can be used as a mask.Example for 1D array:
# Create a 1D array of size 10arr = reshape(Vector(4:2:22), 10)  # Create a power of 2 logic mask # and map it with the size of arrmask = map(ispow2, arr)  # Apply it to the arrayarr[mask]
Output:
Example for 2D array:
# Create a 2D array of size 10x10arr = reshape(Vector(1:100), (10, 10))  # Create a prime number logic mask and # map it with the size of arr # Julia library for working with prime numbers using Primes mask = map(isprime, arr)  # Apply it to the arrayarr[mask]
Output:
Example for 3D array:
# Create a 3D array of size 3x2x3arr = reshape(Vector(2:19), (3, 2, 3))  # Create a prime number logic mask # and map it with the size of arr# Julia library for working with prime numbers using Primesmask = map(isprime, arr)  # Apply it to the arrayarr[mask]
Output:
One of the most used cases of logical indexing is when we want to filter elements out of an array on the basis of multiple conditions.Example:
# Create a 3D array of size 3x2x3arr = reshape(Vector(2:19), (3, 2, 3))  # Create a function that checks for # both prime number or power of two# Julia library for working with prime numbers using Primes   function prime_and_2pow(arr)    if isprime(arr) || ispow2(arr)        return true    else         return false    endend  # Create a logic mask and map it# with the size of arrmask = map(prime_and_2pow, arr)      # Apply it to the arrayarr[mask]
Output:
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