Bidirectional Frame Blending

mediumPROFrame InterpolationInterpolationCV: Motion Estimation

Implement a frame interpolation technique to generate an intermediate frame between two given frames. This task is crucial in motion estimation and video processing as it enables the creation of smooth motion sequences.

The concept of frame interpolation is based on linear interpolation, where each pixel in the intermediate frame is calculated as a weighted average of the corresponding pixels in the input frames. The interpolation factor $t$ controls the blending process, ranging from 0 (first frame) to 1 (second frame).

Each pixel of the intermediate frame is computed independently, and the blended value is rounded to the nearest integer:

I_t(x, y) = (1 - t) \cdot I_0(x, y) + t \cdot I_1(x, y)

This technique is widely used in video processing and computer vision applications.

Example

Input

frame0 = [[0, 0], [0, 0]]
frame1 = [[100, 100], [100, 100]]
t = 0.5

Output

[[50, 50], [50, 50]]

Reasoning

Inputs: frame0 and frame1 are two $2 \times 2$ grayscale frames of the same size. frame0 is completely black (every intensity is 0) and frame1 is a uniform gray of intensity 100. t = 0.5 asks for the frame halfway between them.
Weights: the first frame gets weight $1 - t = 0.5$ and the second gets $t = 0.5$ . The two weights always add up to 1, so the result stays inside the range spanned by the two inputs.
One pixel: $I_{0.5}(x, y) = 0.5 \cdot 0 + 0.5 \cdot 100 = 0 + 50 = 50$ . Rounding to the nearest integer leaves it at 50.
Every pixel: both frames are uniform, so all four positions give the same value, and the output is [[50, 50], [50, 50]], a $2 \times 2$ frame of integers.
Notice: $t$ slides the result between the two frames. At $t = 0$ you get frame0 back unchanged, at $t = 1$ you get frame1, and $t = 0.25$ here would give 25 everywhere.

Constraints

frame0 and frame1 are 2D grayscale images of same size
t is interpolation factor (0 to 1)
Return blended frame with values rounded to nearest integer

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Bidirectional Frame Blending: Background & Strategy

Background Knowledge

Frame Interpolation Fundamentals

Frame interpolation is the process of generating intermediate frames between two consecutive frames in a video sequence. The basic linear blending approach you've described treats each pixel independently, assuming no motion occurs between frames. However, real-world video contains motion, which causes the simple linear interpolation to produce ghosting artifacts—blurred, semi-transparent duplicates of moving objects. To address this, modern frame interpolation methods estimate optical flow, which represents the apparent motion of pixels between frames, allowing us to warp pixels to their new positions before blending.

Bidirectional Flow and Motion Compensation

The "bidirectional" aspect refers to estimating motion in both directions: from frame $I_0$ to $I_1$ (forward flow) and from $I_1$ to $I_0$ (backward flow). Rather than simply averaging pixel values at fixed coordinates, bidirectional methods warp both input frames toward the intermediate time step using their respective flows, then blend the warped results. This approach handles occlusions (areas visible in one frame but not the other) and large motion more gracefully than naive blending. The intermediate frame synthesis then becomes a problem of intelligently combining these warped frames while managing artifacts like ghosting and blur.

Context-Aware Blending

Beyond simple averaging, advanced methods use contextual information—such as confidence maps, occlusion masks, or learned features—to determine how much each warped frame should contribute to the final result. Some approaches use neural networks to learn adaptive blending weights that vary spatially across the image, allowing the algorithm to favor one warped frame over another in regions where one provides higher-quality information.

Algorithm/Approach

The general pattern for bidirectional frame blending follows this structure:

Optical Flow Estimation: Compute bidirectional optical flow between $I_0$ and $I_1$
Flow Interpolation: Estimate flows from an intermediate frame (at time $t$ ) to both input frames
Frame Warping: Warp both input frames toward the intermediate time using the interpolated flows
Adaptive Blending: Combine the warped frames using learned or heuristic blending weights
Artifact Mitigation: Apply post-processing or masking to reduce ghosting and blur

Step-by-Step Strategy

Step 1: Understand the Motion Model

Recognize that optical flow provides a 2D motion vector $(u, v)$ for each pixel
Understand that forward warping (pushing pixels) and backward warping (pulling pixels) have different properties
Backward warping is typically preferred because it avoids holes in the output

Step 2: Implement Basic Warping

Start by implementing a backward warping function that samples from an input frame using non-integer coordinates
Use bilinear interpolation to handle sub-pixel accuracy
Handle boundary conditions (pixels that warp outside the frame)

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solution.py

Python 3.13

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