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* Refactored field.md

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Co-authored-by: Taichi Gardener <taichigardener@gmail.com>
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---

# Fields
Taichi fields are used to store data.
In general, fields are global data containers that can be read and written from both the Python scope and the Taichi scope.

A field has its own data type and shape and can be considered as a multi-dimensional array of elements.
An element of a field can be a **scalar**, a **vector**, a **matrix**, or a **struct**.
The sparsity of a field element is **dense** by default, but it can also be **sparse**, as detailed described in [Sparse spatial data structures](/lang/articles/advanced/sparse.md).
The term _field_ is borrowed from mathematics and physics. If you already know [scalar field](https://en.wikipedia.org/wiki/Scalar_field) (for example heat field), or vector field (for example [gravitational field](https://en.wikipedia.org/wiki/Gravitational_field)), then it is straightforward for you to understand fields in Taichi.

Fields in Taichi are the _global_ data containers that can be accessed from both the Python scope and the Taichi scope. Just like an ndarray in NumPy or a tensor in PyTorch, a field in Taichi is defined as a multi-dimensional array of elements, and elements in a field can be a scalar, a vector, a matrix, or a struct.

:::note
The term **field** is borrowed from mathematics and physics.
If you have already known [scalar field](https://en.wikipedia.org/wiki/Scalar_field) (e.g., heat field) or vector field (e.g., [gravitational field](https://en.wikipedia.org/wiki/Gravitational_field)) in mathematics and physics,
it will be straightforward to understand the fields in Taichi.
A 0D (one-dimensional) field contains *only* one element.
:::

## Scalar fields
We start introducing fields from this very basic type, the elements of scalar fields are simply scalars.
* A 0D scalar field is a single scalar.
* A 1D scalar field is an array.
* A 2D scalar field can be used to represent a 2D regular grid of values.
* A 3D scalar field can be used for volumetric data.

Scalar fields refer to the fields that store scalars and are the most basic fields. A 0D scalar field is a single scalar.


### Declaration
``` python

```python
import taichi as ti
ti.init(arch=ti.cpu)

energy = ti.field(ti.f32, shape=()) # 0-D
linear_array = ti.field(ti.i32, shape=128) # 1-D
gray_scale_image = ti.field(ti.u8, shape=(640, 480)) # 2-D
volumetric_data = ti.field(ti.f32, shape=(32, 32, 32)) # 3-D
# Declare a 0D scalar field whose data type is f32
f_0d = ti.field(ti.f32, shape=()) # 0D
# Declare a 1D scalar field whose shape is (128)
f_1d = ti.field(ti.i32, shape=128) # 1D
# Declare a 2D scalar field whose shape is (640, 480)
f_2d = ti.field(ti.u8, shape=(640, 480)) # 2D
# Declare a 3D scalar field whose data type is f32
f_3d = ti.field(ti.f32, shape=(32, 32, 32)) # 3D
```

### Access elements of scalar fields
``` python
energy[None] = 10.0
linear_array[0] = 1
gray_scale_image[1,2] = 255
volumetric_data[3,3,3] = 2.0
```
### Access elements in a scalar field

### Meta data
``` python
linear_array.shape # (128,)
volumetric_data.dtype # f32
```
The initial value of elements in a scalar filed is zero. Always use explicit indexing to access elements in a scalar field.

:::note
* Field values are initially zero.
* Fields are **always** accessed by indices. When accessing 0-D field `x`, use `x[None] = 0` instead of `x = 0`.
When accessing a 0D field `x`, use `x[None] = 0`, *not* `x = 0`.
:::

### Example
An example might help you understand scalar fields.
Assume you have a gray-scale image. At each point in the image, there would be a pixel value. The width and height of the image are similar to the `shape` of the Taichi scalar field. The pixel value (0-D scalar) is like the element of the Taichi scalar field. We could use the following code to generate a gray-scale image with random pixel values:
```python
# For a 0D field, you are required to use the index None even though it has only one element
f_0d[None] = 10.0
f_1d[0] = 1
f_2d[1, 2] = 255
f_3d[3, 3, 3] = 2.0
```

``` python {5}
import taichi as ti
As mentioned above, you can use a 2D scalar field to represent a 2D grid of values. The following code snippet creates and displays a 640&times;480 image with randomly-generated gray scales:

```python
import taichi as ti
ti.init(arch=ti.cpu)

width, height = 640,480
# Create a 640x480 scalar field, each of its elements representing a pixel value (f32)
gray_scale_image = ti.field(dtype=ti.f32, shape=(width, height))

@ti.kernel
def fill_image():
# Fill the image with random gray
for i,j in gray_scale_image:
gray_scale_image[i,j] = ti.random()

fill_image()

# Create a GUI of same size as the gray-scale image
gui = ti.GUI('gray-scale image with random values', (width, height))
while gui.running:
gui.set_image(gray_scale_image)
gui.show()
```

:::tip
In earlier versions of Taichi, you could not allocate new fields after executing the first kernel. Since Taichi v0.8.0, you can use a new class `FieldsBuilder` for dynamic field allocation and destruction. For more details, please see [Field (advanced)](/lang/articles/advanced/layout.md).
With Taichi versions earlier than v0.8.0, you cannot allocate new fields after executing a kernel. Starting from v0.8.0, you can use the `FieldsBuilder` class to dynamically allocate or destruct fields. See the [Field (advanced)](/lang/articles/advanced/layout) for more information.
:::

:::caution WARNING
Taichi does not support slicing on a Taichi field. For example, with the 2D scalar field `f_2d`, you can do `f_2d[1, 2]`, but *not* `f_2d[1]`.
:::

### Metadata

Metadata provides the basic information of a scalar field. You can retrieve the data type and shape of a scalar field via its `shape` and `dtype` property:

```python
f_1d.shape # (128)
f_3d.dtype # f32
```

## Vector fields
We are all living in a gravitational field, which is a vector field. At each position in 3D space, there is a gravity force vector. The gravitational field could be represented by:

As the name suggests, vector fields are the fields whose elements are vectors.

- You can use a vector field to represent an RGB image. Then, each of its elements is an (r, g, b) triple.
- You can use a vector field to represent a volumetric field. Then, each of its elements can be the velocity of the corresponding particle.

### Declaration

The following code snippet declares a 3D field of 2D vectors:

```python
gravitational_field = ti.Vector.field(n=3, dtype=ti.f32, shape=(x, y, z))
# Declare a 1x2x3 vector field, whose vector dimension is n=2
f = ti.Vector(ti.f32, n=2).field(shape=(1,2,3))
```
`x, y, z` are the sizes of each dimension of the 3D space respectively. `n` is the number of elements of the gravity force vector.

### Access elements of vector fields
There are **two** indexing operators `[]` when you access a member of a vector field: the first is for field indexing, and the second is for vector indexing.
- The gravity force vector could be accessed by `gravitational_field[i, j, k]` (`0 <= i < x, 0 <= j < y, 0 <= k < z`).
- The `p`-th member of the gravity force vector could be accessed by `gravitational_field[i, j, k][p]` (`0 <= p < n`).
- The 0-D vector field `x = ti.Vector.field(n=3, dtype=ti.f32, shape=())` should be accessed by `x[None][p]` (`0 <= p < n`).
The following code snippet declares a `300x300x300` vector field `volumetric_field`, whose vector dimension is 3:

### Example
This example helps you understand how to access vector fields:
``` python
import taichi as ti
ti.init(arch=ti.cpu)
```python
box_size = (300, 300, 300) # A 300x300x300 grid in a 3D space
# Declare a 300x300x300 vector field, whose vector dimension is n=3
volumetric_field = ti.Vector.field(n=3, dtype=ti.f32, shape=box_size)
```

### Access elements in a vector field

Accessing a vector field is similar to accessing a multi-dimensional array: You use an index operator `[]` to access an element in the field. The only difference is that, to access a specific component of an element (vector in this case), you need an *extra* index operator `[]`:

- To access the velocity vector at a specific position of the volumetric field above:

`volumetric_field[i,j,k]`

- To access the `l`-th component of the velocity vector:

`volumetric_field[i, j, k][l]`

The following code snippet generates and prints a random vector field:

```python
# n: vector dimension; w: width; h: height
n,w,h = 3,128,64
vec_field = ti.Vector.field(n, dtype=ti.f32, shape=(w,h))

Expand All @@ -114,56 +144,100 @@ fill_vector()
print(vec_field[w-1,h-1][n-1])
```

:::note
To access the `p`-th component of the 0D vector field `x = ti.Vector.field(n=3, dtype=ti.f32, shape=())`:

`x[None][p]` (0 &le; p < n).
:::

## Matrix fields
Field elements can also be matrices. In continuum mechanics, each
infinitesimal point in a material exists a strain and a stress tensor. The strain and stress tensor is a 3 by 3 matrix in the 3D space. To represent this tensor field we could use:

As the name suggests, matrix fields are the fields whose elements are matrices. In continuum mechanics, at each infinitesimal point in a 3D material exists a strain and stress tensor. The strain and stress tensor is a 3 x 2 matrix. Then, you can use a matrix field to represent such a tensor field.

### Declaration

The following code snippet declares a tensor field:

```python
strain_tensor_field = ti.Matrix.field(n=3, m=3, dtype=ti.f32, shape=(x, y, z))
# Declare a 300x400x500 matrix field, each of its elements being a 3x2 matrix
tensor_field = ti.Matrix.field(n=3, m=2, dtype=ti.f32, shape=(300, 400, 500))
```
`x, y, z` are the sizes of each dimension of the 3D material respectively. `n, m` are the dimensions of the strain tensor.

In a general case, suppose you have a `128 x 64` field called `A`, and each element is
a `3 x 2` matrix, you can define it with `A = ti.Matrix.field(3, 2, dtype=ti.f32, shape=(128, 64))`.

### Access elements of matrix fields
There are **two** indexing operators `[]` when you access a member of a matrix from a matrix field:
the first is for field indexing, and the second is for matrix indexing.
- If you want to get the element `i, j` of the matrix field, please use `mat = A[i, j]`. `mat` is simply a `3 x 2` matrix.
- To get the member on the first row and second column of that element `mat`, use `mat[0, 1]` or `A[i, j][0, 1]`.
- The 0-D matrix field `x = ti.Matrix.field(n=3, m=4, dtype=ti.f32, shape=())` should be accessed by `x[None][p, q]` (`0 <= p < n, 0 <= q < m`).
- `ti.Vector` is simply an alias of `ti.Matrix`.

### Example
This example helps you understand element and member in matrix fields:
``` python
matrix_field = ti.Matrix.field(n = 2, m = 3, dtype = ti.f32, shape = (2, 2))
Element = matrix_field[0, 0]
Member = matrix_field[0, 1][1,1]
```
![image](https://raw.githubusercontent.com/taichi-dev/public_files/master/taichi/doc/matrix_field.jpg)

### Matrix size
For performance reasons, matrix operations will be unrolled during the compile stage.
Therefore we suggest using only small matrices. For example, `2x1`, `3x3`, `4x4`
matrices are fine, yet `32x6` is probably too big as a matrix size.

If you have a dimension that is too large (e.g. `64`), it's better to
declare a field of size `64`. E.g., instead of declaring
`ti.Matrix.field(64, 32, dtype=ti.f32, shape=(3, 2))`, declare
`ti.Matrix.field(3, 2, dtype=ti.f32, shape=(64, 32))`. Try to put large
dimensions to fields instead of matrices.

:::caution
Due to the unrolling mechanism, operating on large matrices (e.g.
`32x128`) can lead to a very long compilation time and low performance.

### Access elements in a matrix field

Accessing a matrix field is similar to accessing a vector field: You use an index operator `[]` for field indexing and a second `[]` for matrix indexing.

- To retrieve the `i, j` element of the matrix field `tensor_field`:

`mat = tensor_field[i, j]`

- To retrieve the member on the first row and second column of the element `mat`:

`mat[0, 1]` or `tensor_field[i, j][0, 1]`

:::note

To access the 0D matrix field `x = ti.Matrix.field(n=3, m=4, dtype=ti.f32, shape=())`:

`x[None][p, q]` (0 &le; p < n, 0 &le; q < m)

:::

### Considerations: Matrix size

Matrix operations are unrolled during compile time. Take a look at the following example:

```python
import taichi as ti
ti.init()

a = ti.Matrix.field(n=2, m=3, dtype=ti.f32, shape=(2, 2))
@ti.kernel
def test():
for i in ti.grouped(a):
# a[i] is a 2x3 matrix
a[i] = [[1, 1, 1], [1, 1, 1]]
# The assignment is unrolled to the following during compile time:
# a[i][0, 0] = 1
# a[i][0, 1] = 1
# a[i][0, 2] = 1
# a[i][1, 0] = 1
# a[i][1, 1] = 1
# a[i][1, 2] = 1
```

Operating on large matrices (for example `32x128`) can lead to long compilation time and poor performance. For performance reasons, it is recommended that you keep your matrices small:

- `2x1`, `3x3`, and `4x4` matrices work fine.
- `32x6` is a bit too large.

**Workaround:**

When declaring the matrix field, leave large dimensions to the fields, rather than to the matrices. If you have a `3x2` field of `64x32` matrices:

- Not recommended:
`ti.Matrix.field(64, 32, dtype=ti.f32, shape=(3, 2))`
- Recommended:
`ti.Matrix.field(3, 2, dtype=ti.f32, shape=(64, 32))`

## Struct fields
Field elements can be user-defined structs.
Struct fields are created by providing the name and data type of each member variable in a dictionary format.
Member variables of struct fields might be scalars, vectors, matrices, or other struct fields.
For example, a 1-D field of particles with position, velocity, acceleration, and mass can be declared as:

Struct fields are fields that store user-defined structs. Members of a struct element can be:

- Scalars
- Vectors
- Matrices
- Other struct fields.

### Declaration

The following code snippet declares a 1D field of particle information (position, velocity, acceleration, and mass) using `ti.Struct.field()`. Note that:

- Member variables `pos`, `vel`, `acc`, and `mass` are provided in the dictionary format.
- [Compound types](type.md#compound-types), such as `ti.types.vector`, `ti.types.matrix`, and `ti.types.struct`, can be used to declare vectors, matrices, or structs as struct members.

```python
# Declare a 1D struct field using the ti.Struct.field() method
particle_field = ti.Struct.field({
"pos": ti.types.vector(3, ti.f32),
"vel": ti.types.vector(3, ti.f32),
Expand All @@ -172,25 +246,30 @@ particle_field = ti.Struct.field({
}, shape=(n,))
```

[Compound types](type.md#compound-types) (`ti.types.vector`, `ti.types.matrix`, and `ti.types.struct`) are used to declare vectors, matrices, or structs as field members. Apart from using `ti.Struct.field`, the above particle field can also be declared by using the field of compound types:
Alternatively, besides *directly* using `ti.Struct.field()`, you can first declare a compound type `particle` and then create a field of it:

```python
# Declare a compound type vec3f to represent position, velocity, and acceleration.
vec3f = ti.types.vector(3, ti.f32)
# Declare a struct composed of three vectors and one f32 floating-point number
particle = ti.types.struct(
pos=vec3f, vel=vec3f, acc=vec3f, mass=ti.f32,
)
# Declare a 1D field of the struct particle using field()
particle_field = particle.field(shape=(n,))
```

Members of a struct field can be accessed either locally (i.e., member of a struct field element) or globally (i.e., member field of a struct field):
### Access elements in a struct field

You can access members of elements in a struct field either one by one or universally:

```python
# set the position of the first particle to origin
particle_field[0] # local ti.Struct
particle_field[0].pos = ti.Vector([0.0, 0.0, 0.0])
# Set the position of the first particle in the field to origin [0.0, 0.0, 0.0]
particle_field[0].pos = ti.Vector([0.0, 0.0, 0.0]) # pos is a 3D vector

# set the first member of the second position to 1.0
particle_field[1].pos[0] = 1.0
# Set the second particle's pos[0] in the field to 1.0
particle_field[1].pos[0] = 1.0 # pos[0] is the first member of pos

# make the mass of all particles be 1
particle_field.mass # global ti.Vector.field
# Universally set the mass of all particles to 1.0
particle_field.mass.fill(1.0)
```

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