Alpha Matte Compositing

mediumPROImage MattingMattingCV: Computational Photography

Implement alpha compositing for foreground/background blending, a key technique in image matting. This process involves combining a foreground image with a background image using a matte, which is a grayscale image defining the opacity of the foreground.

The matte, represented by $\alpha$ , is used to control the blending of the foreground and background colors. The compositing equation is based on the principle of linear blending, where the resulting color is a weighted sum of the foreground and background colors.

Define the foreground, background, and matte images.
Apply the compositing equation to each pixel.

C = \alpha F + (1 - \alpha) B

Return the composited image with the same $(H, W, 3)$ shape, stored as 8-bit unsigned integers (uint8). Fractional values are truncated, not rounded: $127.5$ becomes $127$ .

This technique is widely used in digital image editing.

Example

Input

foreground = person image
background = beach scene
alpha = person matte (1 inside, 0 outside)

Output

Person composited onto beach

Reasoning

Inputs: foreground and background are $H \times W \times 3$ RGB images with values 0 to 255; alpha is an $H \times W$ matte, one opacity per pixel: $1$ on the person, $0$ on the beach, fractional along hair and soft edges.
Every pixel follows $C = \alpha F + (1-\alpha) B$ , with that pixel's single $\alpha$ weighting all three colour channels.
Take one edge pixel: red foreground $F = (255, 0, 0)$ , green background $B = (0, 255, 0)$ , $\alpha = 0.5$ . Then $C = (127.5,\ 127.5,\ 0)$ , and stored as 8-bit integers with the fraction dropped it becomes $(127, 127, 0)$ , a dark yellow halfway between.
With $\alpha = 1$ that pixel is exactly $F$ , [255, 0, 0]; with $\alpha = 0$ it is exactly $B$ .
Output: an $H \times W \times 3$ image, the person composited onto beach: person pixels where the matte is 1, beach where it is 0, smooth blends in between. Notice $\alpha + (1-\alpha) = 1$ , so a blend never leaves the 0 to 255 range.

Constraints

foreground: Foreground image (H, W, 3)
background: Background image (H, W, 3)
alpha: Alpha matte (H, W) in [0, 1]
Return: Composited image

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Background Knowledge

Image matting is about separating an image into a foreground color $F$ , a background color $B$ , and a per-pixel opacity (the alpha matte) $\alpha \in [0,1]$ . The observed pixel color $C$ is assumed to be a linear mixture of foreground and background: $C = \alpha F + (1 - \alpha) B$ This is done per color channel (e.g., R, G, B independently). When you have F, B, and \alpha, you can composite the foreground onto any new background by applying this equation.

There are two common representations of images with alpha: straight alpha and premultiplied alpha. In straight alpha, F stores the true foreground colors (not multiplied by \alpha); in premultiplied alpha, the stored foreground colors are already multiplied by alpha: F_{\text{stored}} = \alpha F. In that case, the compositing equation becomes: $C = F_{\text{stored}} + (1 - \alpha) B$ Understanding which convention the problem uses is important for implementing the equation correctly and avoiding double-multiplying by $\alpha$ .

Algorithm / General Approach

At a high level, the algorithm is:

Treat each pixel independently.
For each pixel, and for each color channel:
Apply the compositing formula using the given $\alpha$ , foreground, and background.
Handle straight vs premultiplied alpha correctly:
Straight alpha: use $C = \alpha F + (1 - \alpha) B$ .
Premultiplied alpha: assume input foreground is already $\alpha F$ and use $C = F + (1 - \alpha) B$ .
Implement this in a vectorized way over the whole image (e.g., NumPy, PyTorch) instead of looping in Python, for efficiency.

Step-by-Step Strategy

Assume:

Foreground image: F with shape (H, W, 3) or (H, W, C)
Background image: B with same shape
Alpha matte: A with shape (H, W) or (H, W, 1) and values in [0, 1]

Straight alpha case:

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Example

Constraints

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