Normalized Integer

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

A Normalized Integer is an integer which is used to store a decimal floating point number. When formats use such an integer, OpenGL will automatically convert them to/from floating point values as needed. This allows normalized integers to be treated equivalently with floating-point values, acting as a form of compression.

For example, if a 2D Texture's Image Format uses normalized integers, it will still be treated as a floating-point texture. The sampler type the shader uses will be sampler2D, just like for a floating-point texture. If you use this image in the framebuffer and write to it from the Fragment Shader, the output variables will be floating-point vectors, not integer ones.

The idea behind a normalized integer is that, in various commands, an integer is specified with a certain amount of bits. These bits together form minimum and maximum integer values, giving you an integer range (which can be signed or unsigned). This range can be used along with the specified integer to map the integer to another range, like a floating-point one.

The downside to integer normalization is that they can only represent floating-point values on the range [0.0, 1.0] or [-1.0, 1.0], depending on whether they are unsigned or signed integers. This is sufficient in many cases for colors, but it can also be used for some vertex inputs like texture coordinates and normals.

Use cases

Normalized integers are useful in many parts of OpenGL. The most common uses for them are:

• Colors, whether stored in Texture Image Formats or Vertex Formats. Non-HDR color data has a maximum intensity. As such, a normalized integer per channel is a reasonable representation of colors. The most typical format is 8-bits per channel and unsigned (as negative colors are not frequently useful).
• Texture coordinates. Most Texture Sampling functions accept normalized texture coordinates, with 0 representing the bottom/left/front of the image and 1 representing the top/right/rear. Indeed, Repeat Filtering relies on normalized coordinates, as this affects what happens when a texture coordinate is outside of the normalized range.
• Normals. These can commonly use signed, normalized integers. They can be stored in vertex arrays or textures. In vertex arrays, they often use the GL_INT_2_10_10_10_REV format.

Storage and bitdepths

Every normalized integer has some bitdepth. These are usually 8 or 16, but some normalized integers use unusual numbers like 2, 10, or even 32. Regardless of the bitdepth, the way they are converted is identical. Only the specific numbers change.

In all of the following equations, the bitdepth will be represented by B.

Unsigned

For unsigned, normalized integers, the conversion is fairly simple. For a given integer of bitdepth B, the maximum representable unsigned integer is ${\displaystyle MAX=2^{B}-1}$.

Unsigned, normalized integers map into the floating-point range [0, 1.0]. It does this by mapping the entire integer range to that. So it maps [0, MAX] to [0, 1.0] linearly, using the following simple equation:

${\displaystyle float={\tfrac {int}{MAX}}}$

The conversion back to integers uses the inverse equation. Implementations are allowed to round the converted integer any way it likes. So a floating point 0.25 converted to an 8-bit unsigned integer may be 63 or 64, even though 0.25 * 255 = 63.75, and is closer to 64.

Signed

For signed, normalized integers, the conversion is slightly more complicated. Signed integers in OpenGL are represented as Two's complement numbers. Therefore, for a given integer of bitdepth B, the maximum representable signed integer is ${\displaystyle MAX=2^{B-1}-1}$, while the minimum signed integer is ${\displaystyle MIN=-2^{B-1}}$. Notice that the absolute value of MIN is larger than MAX.

In all cases, signed, normalized integers map to the floating-point range [-1.0, 1.0]. How exactly this mapping happens is version-specific.

In OpenGL 4.2 and above, the conversion always happens by mapping the signed integer range [-MAX, MAX] to the float range [-1, 1]. Notice that the range distribution is equal on both sides of zero. More importantly, this range allows signed, normalized integers to store a floating-point 0 exactly. However, the mapping for the signed integer value of MIN itself is also stated to resolve exactly to -1.0, so you can't get more negative values. Therefore, the mapping function is:

${\displaystyle float=max({\tfrac {int}{MAX}},-1.0)}$

The max function returns the largest value of its arguments, which ensures that if int is MIN, then the return value remains exactly -1.0.

As with unsigned integers, the conversion to integers happens via the inverse function. The behavior of rounding is not specified for signed integers either, so the conversion from 0.25 could yield, in 8-bit signed integers, either 31 or 32, even though the starting number before rounding is 31.75.

Alternate mapping

In OpenGL versions less than 4.2, there are instances where an alternate conversion is used. These instances are:

The alternate mapping directly maps the signed integer range [MIN, MAX] to [-1.0, 1.0]. The equation for this is simple:

${\displaystyle float={\tfrac {2int+1}{2^{B}-1}}}$

While this allows the full signed integer range to be expressed, it also does not allow a signed integer to exactly express zero.