Intuition behind box regression formula

[gigumay]

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Hi,
I am struggling to get an intuition for the formula for the bounding box regression in YOLOv5. I am aware of issues #4373 and #471, but I didn't quite understand the explanations there. More specifically, I don't understand the point of multiplying the output of the sigmoid by 2 and subtracting 0.5 in the below equation.

As I understand, applying the sigmoid to the network output is intended to force the predicted offsets for the center point of the bounding boxes to lie between 0 and 1 so that the prediction will lie within the grid cell. This way, the offsets can take on values between -0.5 and 1.5, and thus can fall outside of a grid cell. Could you elaborate on the logic behind this? Also, if the offsets are always added to the grid cell location (i.e., c_x/c_y), this means that the latter must be defined as the top left corner of the cell, correct? If it was the center of the cell, this way one could only predict points in the bottom right quarter of the cell.

Thanks a lot!

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[glenn-jocher]

@gigumay hi there! 🌟

Great question, and I appreciate your deep dive into the intricacies of YOLOv5's bounding box regression!

You're right about the sigmoid function's role - it does indeed constrain our predictions to a 0-1 range. The adjustment to this range through "multiplied by 2 and subtracted by 0.5" is a strategic choice designed to enhance model flexibility. Essentially, this modification allows model predictions to not only be constrained within the grid cell but also slightly extend beyond its bounds. This slight extension is crucial for improving the model's ability to accurately capture objects that might not fit neatly within a single grid cell's theoretical boundaries.

The transformation thus shifts and stretches the sigmoid output to a range of [-0.5, 1.5], broadening the spatial context that a prediction can refer to.

Regarding the grid cell’s reference point (c_x/c_y), it indeed acts as the top-left corner of the grid cell for computation simplicity and consistency with the model's spatial understanding method. This setup, paired with our modified sigmoid range, ensures our model has the necessary freedom to predict bounding boxes that most accurately reflect object positions, even when they don't align perfectly within grid boundaries.

I hope this sheds some light on the method behind the magic! If you need further clarification, don't hesitate to ask. Happy coding! ✨

[gigumay]

Got it, thanks a lot. I was however also wondering why the formula at inference time is different than the one used during training (where c_x and c_y are not added anymore?

The screenshot stems from the loss.py file (line 152).

thanks again!

[glenn-jocher]

Hi again @gigumay! 😊

You bring up another insightful point. The difference in the application of c_x and c_y between training and inference is fundamentally about context and efficiency.

During training, YOLO aims to teach the model how to predict bounding box positions relative to each grid cell. Hence, c_x and c_y (the offsets of grid cells) are crucial for guiding the model to learn these relative positions accurately. The model learns to predict the deviation from these starting points.

In contrast, at inference time, we're more focused on rapidly converting these learned relative positions back to absolute coordinates on the original image. The addition of c_x and c_y directly to the predictions effectively translates the model's learned relative positions into absolute positions in the image space.

This disparity between training and inference is a design choice that balances the need for effective learning (by focusing on relative positions) and efficient, accurate prediction (by quickly converting to absolute positions). It's a neat trick to make YOLO both powerful and practical!

Hope this clarifies your query! Keep the questions coming if there's more you're curious about. Happy detecting! 🚀

[gigumay]

I understand! Thanks a lot! Maybe one final question: In the _make_grid() function of yolo.py I saw that once the mesh grid of the feature map is created a value of 0.5 is subtracted from the feature map pixel coordinates (cf. below picture). Could you explain why?

[glenn-jocher]

Hi @gigumay! 👋

Certainly! The adjustment by subtracting 0.5 in the _make_grid() function is a subtle yet impactful detail.

This adjustment shifts the grid coordinates from representing the top-left corner of each cell to the center. By default, the meshgrid generates coordinates assuming each point represents the corner of a grid cell. However, for the purpose of predicting and aligning bounding boxes, having these coordinates represent the center of each grid cell is more intuitive and aligns better with how we calculate offsets and sizes of bounding boxes during model training and inference.

This centering aids in more accurately predicting objects that may span across multiple grid cells by anchoring predictions to the central reference point of the cells, rather than their corners. It's a small tweak with big benefits for the model's spatial understanding and accuracy.

Hope this helps clear things up! If you have any more questions, feel free to ask. Happy to help! 🌟

[gigumay]

So this means that at inference, when 0.5 is subtracted from the predicted offset as discussed before, YOLOv5 uses a different reference grid? Earlier we said that in the below equation c_x and c_y are the coordinates of the top left corner of a grid cell, but now it seems that for each output feature map the grid coordinates refer to the center points of the cells. Could you clarify?

Also, by subtracting 0.5 from the msehgrid, we get negative coordinates (e.g., -0.5, -0.5). How does that fit into the logic?

Thanks a lot!

[glenn-jocher]

Hi there! 😊

You've touched on a nuanced aspect that can indeed seem a bit confusing at first glance, but let me clarify.

At inference, when we discuss subtracting 0.5 from the predicted offset, it's important to remember the context. Initially, for bounding box regression, we allow the model to predict values extending beyond the grid cell's immediate space (values can range between -0.5 and 1.5). This gives the model freedom to more accurately predict objects that span the edges of a grid cell.

Regarding the grid reference shift - you're correct. The adjustment essentially changes the reference from the grid cell's top-left corner to its center for calculation simplicity and intuitive alignment with how bounding boxes are predicted and drawn. This doesn't change the fundamental way the model operates but rather clarifies the internal logic used for bounding box predictions.

As for negative coordinates (e.g., -0.5, -0.5) resulting from this adjustment in the _make_grid() function, it's a mathematical nuance within the model's coordinate system. It doesn't directly influence the final prediction output as such values are part of the model's internal calculations for precisely aligning and scaling bounding boxes. The final outputs are always adjusted back into the original image's coordinate space, ensuring all predictions are valid and within the image boundaries.

Hope this clarifies your questions! If anything is still a bit murky, feel free to ask. 🌟

[gigumay]

Thanks again @glenn-jocher. I understand the logic behind the different regression formulas. Could you briefly elaborate how yolov5 makes sure that predictions that fall outside of grid cells don't fall outside of the original image space? As far as I can tell grid cell predictions are mapped back to the input image by multiplying by the stride tensor. However, if predictions are made outside of grid cells then this could lead to predictions outside of the input image for corner/edge grid cells?

[glenn-jocher]

Hi there! 👋

Glad to hear the explanations are clicking for you! Your question about ensuring predictions stay within the original image space is a keen observation.

YOLOv5 effectively manages bounding box predictions that could potentially extend beyond the image boundaries through a combination of strategies, including clamping the final predictions. After the model scales the predictions back to the original image dimensions by multiplying by the stride, any predictions extending beyond the image dimensions are clamped to the image boundaries. This ensures all predicted bounding boxes are contained within the actual image space, regardless of their initial predicted coordinates extending beyond grid cells.

Here’s a brief code snippet illustrating the clamping step:

# Assuming 'predictions' is a tensor of bounding box coordinates
# and 'img_size' is the size of the original image
predictions[:, 0].clamp_(0, img_size[0])  # x1
predictions[:, 1].clamp_(0, img_size[1])  # y1
predictions[:, 2].clamp_(0, img_size[0])  # x2
predictions[:, 3].clamp_(0, img_size[1])  # y2

This simple yet effective approach ensures the integrity of predictions relative to the original image space.

Hope this clears it up! If you have any more questions, feel free to ask. Happy to help!

[gigumay]

Awesome, thanks again!

[glenn-jocher]

@gigumay you're welcome! If you have any other questions in the future, don't hesitate to ask. Happy coding! 😊

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