Technically, an image is a function that maps a continuous domain, e.g.a box , to intensities such as (R, G, B).To store it on computer memory, an image is discretized to an array array[H][W], where each elementarray[i][j] is a pixel.

How does discretization work? How does a discrete pixel relate to the abstract notion of the underlying continuous image?These basic questions play an important role in computer graphics & computer vision algorithms.

This article discusses these low-level details, and how they affect our CNN models and deep learning libraries.If you ever wonder which resize function to use or whether you should add/subtract 0.5 or 1 to some pixel coordinates,you may find answers here.Interestingly, these details have contributed to many accuracy improvements in Detectronand Detectron2.

Formation of Discrete Image

Sampling theory tells us howa continuous 2D signal is turned into a discrete array by sampling and filtering.

1. We choose a rectangular grid of points, from which we will draw samples.In order to make the best use of the produced samples, we have to know the exact location where every sample onthis grid is chosen.

2. Values on these sampled points are not directly retrieved from the original signal, but come from afiltering step that removes high-frequency components.A bad choice of filters can lead to aliasing effects.

Sampling and filtering are both important in basic image processing operations, such as resize.Resize operation takes a discrete image, resamples it, and creates a new image.The choice of sampling grid and sampling filter will then affect how such a basic operation is implemented.

For example, the paper On Buggy Resizing Libraries and Surprising Subtleties in FID Calculationstudies the filtering issues, and shows that the resize operations in many libraries(OpenCV, PyTorch, TensorFlow) don't take into account the low-pass filtering. This then leadsto incorrect deep learning evaluation.

In this article, we ignore the issue of sampling filter, and only study the coordinates of sampling grid.We'll see that this choice is also inconsistent among libraries, and can affect the design and performance of CNN models.

Choices of Sampling Grid

Pixels are located on a sampling grid we choose.Naturally, we would like to only consider rectangular grids where pixels are spaced evenly.But there are many other factors to be concerned with:

• Offset: where is the first pixel located relative to the beginning of the signal?
• Stride: what's the distance between two neighboring pixels?
• Resolution: how many pixels are there?

(These terminologies may have a different meaning elsewhere, but this is how I define them in this article.)

For simplicity, we look at the one-dimensional case instead.We want to answer this question: for a 1D signal defined on , what is the sampling grid with stride=1?There are a few different choices:

In this figure, the green bars represent the 1D signal of length , and blue dotsrepresent the locations where point samples are taken.On top of each sample we labeled their coordinates,while on the bottom are their zero-based pixel indices.More formally, given a stride (which equals to 1 here), the offset and resolution of the grid are defined by the following table (assume is an integer):

They (or at least the first two) are all valid interpretations when we are given an array of pixels.The interpretation we choose affects how we implement operations and models,because they each have some unique weird properties.To understand them more, let's check how a 2x resize operation should be implemented under each interpretation.

2x Resize Operation

We'll now see that a simple "2x resize" operation has many possible implementations.

A unique undesired property of ① is that, stride is not the inverse of resolution.So a 2x resize is ambiguous: we have to be clear about whether we want half of stride, or twice more pixels.The new grids after resize look like these:

Resize for grid ② & ③ aren't ambiguous:

You can easily verify that the 4 different resized grids still match thecorresponding definition in our table above.

For 2D case, the 2x resize in ①(twice more pixels) and ② look liks this (image credit: here),from which you can see why ①(twice more pixels) is also called align_corners:

These 4 different versions of 2x resize have some issues:

• Extrapolation: ② and ③ both need extrapolation outside the border of the original grid to perform resize, but ① only needs interpolation.Extrapolation is sometimes undesirable.

• Asymmetry: ③ is asymmetric, and it's probably a good reason to never use it. One consequence is that resize(flip(x)) != flip(resize(x)). All others are symmetric.

• Information Loss: in ①(half of stride) and ③ , about half of the points on the new grid exist in the old grid.By not having to interpolate their values, we minimize the loss of information.However, in ①(twice more pixels) and ②, most or all of the new pixels need to be recomputed.

  For resize with other arbitrary scale factors, all versions have information loss. But 2x/0.5x resize aremost common in deep learning.

The DeepLab series of segmentation models are famous for using grid ①(half of stride) for all the 2x resize.See here for words from its author.This matches the inconvenient image shapes they use, such as 321x513.I've heard opinions that the benefits of "no information loss" and "no extrapolation" may let itoutperform ② in segmentation, but I have yet to see more evidence.

Libraries

What do libraries use? Situation is a bit messy. I'll list what I know and look forward to your help to add more.No guarantee they are all correct, since I didn't check the source code for all of them.

It seems the mess is unique in the deep learning world. How come?From what I can tell, the history looks like this:

• TensorFlow is the first place that introduces ③, in its initial open source.This was later considered as a bug and fixedin v1.14 using a new optionnamed half_pixel_centers=True that follows grid ②.

  align_corners=True(①) appeared in TensorFlow 0.7 in 2016.I guess this was probably intended for DeepLab development and not for general use.

  In TensorFlow v2, grid ② becomes the only version of resize, but it was too late.During all these years, the uncommon version (①) and the wrong version (③) have propagated to people'smodels and other libraries.

• PyTorch's interpolate comes originally from upsample operation. Nearest upsample was buggy when it's firstadded in LuaTorch in 2014.Bilinear upsample was first added in LuaTorch in 2016 andused grid ①. Grid ② was added in 2018 to PyTorch under an align_corners=False option,and became the default since then.

• Due to this mess, resize operator in ONNX has to support5 versions of coordinate transform!Kudos to ONNX maintainers.

Literature

Many computer graphics textbooks and papers talk about this topic and choose ②, for example:

• Sec. 3.2, "Images, Pixels and Geometry" of Fundamentals of Computer Graphics
• Sec. 7.1.7, "Understanding Pixels" of Physically Based Rendering
• "What Are the Coordinates of a Pixel?" from Graphics Gems. Explained again in Real-time Rendering.
• A well-known tech memo, A Pixel is Not a Little Square

(Note that some of them uses ② but defines the continuous signal in the range .We'll discuss this more.)

Given all the graphics literature, computer vision and deep learning libraries promoting grid ②, we use ② as the convention.

Choices of Origin

We pick ② as the convention for grid locations, but this is not the end of the story!We now know the grid locations relative to the beginning of the signal are 0.5, 1.5, , but what are their absolute coordinates?In other words, where is the origin (0, 0) ?

This is just a choice of convention and has no substantial effect on any algorithms.Two of the graphics literature I listed above put the origin on the first pixel.This has the benefit that all pixel locations have integer coordinates, but then it's weird that the signal lies oninterval .This convention is sometimes referred to as "integer centers".

Another convention, "integer corners", or "half-integer centers", puts the origin at the beginning of the signal, so the first pixel is centered at (0.5, 0.5).The two conventions are demonstrated in this figure:

We choose "integer corners", and then willhave the following relationship between continuous coordinates and discrete pixel indices:

The choice doesn't matter for resize because absolute coordinates are not part of its API.However, for functions that accept or return absolute coordinates, we should be aware of their convention. For example:

• cv2.findContours returns integer polygons represented by indices. So we always add 0.5 pixel to itsresults to obtain coordinates that match our convention.
• cv2.warpAffine uses "integer centers" and this is complained about in this issue.In fact most OpenCV functions use the "integer centers" convention.
• pycocotools.mask.frPyObjects renders polygons as masks. It accepts polygons in "integer corners"convention.Same for PIL.ImageDraw.polygon, but its results are 0.5 pixel "fatter" due to howits implemented. This has affected cityscapes annotations.
• RoIAlign in torchvision takes a box in absolute coordinates that match our "integer corners" convention.
• scipy.ndimage.map_coordinates takes coordinates in "integer centers" convention.

If a dataset is annotated with coordinates, we also need to know its choice of coordinate system. Thisinformation is often not provided by dataset owner, so we make guesses. For example, in COCO itappears that polygon annotations match our convention, butkeypoint annotations do not and should be incremented by 0.5.

Now that we have a convention for the coordinate system, it's a good practice in computer vision systems toalways use coordinates rather than indices to represent geometries, such as boxes and polygons. This is because indices are integers, andcan easily lose precision during geometric operations.Using indices for bounding boxes has caused some issues in Detectron.

Improvements in Detectron & Detectron2

Models in Detectron / Detectron2all involve localization of objects in images, so the convention of pixels and coordinates matters a lot.Various improvements and bugfixes in the two libraries are related to pixels.

Box Regression Transform

In detection models, bounding box regression typically predicts "deltas" between the ground truth (GT) box and a reference box (e.g. anchor).In training, GT box is encoded to deltas as training target. In inference, the predicted deltas are decoded to become output boxes.

Boxes in Detectron often use integer indices, instead of coordinates. So the width of abox is given by instead of . Its box transform code looked like this for a long time (showing only one dimension for brevity):

ref_x0, ref_x1: int                     # the reference box
ref_w = ref_x1 - ref_x0 + 1
ref_center = ref_x0 + 0.5 * ref_w

def encode(x0, x1):                     # given reference box and gt box
  w = x1 - x0 + 1
  center = x0 + 0.5 * w
  dx = (center - ref_center) / ref_w    # delta between centers
  dw = log(w / ref_w)                   # delta between widths
  return dx, dw

def decode(dx, dw):                     # given reference box and deltas
  center = dx * ref_w + ref_center      # undo the encoding
  w = exp(dw) * ref_w
  x0, x1 = center - 0.5 * w, center + 0.5 * w
  return x0, x1

As innocent as the code seems, the two functions are not inverse of each other: decode(encode(x0, x1)) != (x0, x1).x1 is incorrectly decoded: it should be center + 0.5 * w - 1 instead.

This bug appeared in the py-faster-rcnn project around 2015, and is still theretoday.It was carried into Detectron and negatively affected results in the Mask R-CNN paper.Then it's fixed in late 2017after I found it, and contributed to an improvement of 0.4~0.7 box AP.Detectron went open source in 2018 with this fix.In Detectron2, we adopt the rule to always use floating-point coordinates for boxes, so the issue nolonger exists.

Flip Augmentation

How to horizontally flip a geometry? Although pixel indices should be flipped by ,we should follow the rule to always use coordinates, and coordinates should be flipped by under "integer corner" system.

Detectron isn't so rigorous on this and it uses for coordinates. IIRC, fixing the issue leads to ~0.5 mask AP improvement.

The augmentation library "imgaug" also made this fix.

Delay Conversion to Mask Representation

COCO's instance segmentation data is annotated with polygons that have sub-pixel precision.Converting polygons to binary masks loses the precision due to quantization,and the lost might become more severe during augmentations.Therefore it's preferrable to keep the polygon representation and delay the conversion as much as possible.

In both Detectron and Detectron2, polygon representation are kept during flipping, scaling, and RoI cropping.Masks are not created until the second stage's box predictions are made, where the boxes are used to crop the groundtruth polygonsand generate the mask training target.

On the contrary, in TensorFlow's detection code hereand herepolygons are turned to binary masks immediately at dataset creation time.

Anchor Generation

The code to generate anchors in Detectron is quite long,because it tries to generate integer-valued anchor boxes.By adopting coordinates for all boxes in Detectron2, integer boxes are not needed.This simplifies all the logic to just a few lines of code.

This does not affect accuracy, because the exact values of anchors are not that important as long asthe same is used in training & testing.

RoIAlign

The RoIAlign operation crops a region from an image and resize it to certain shape.It's easy to make mistakes becausetwo images and two coordinate systems are involved.Let's derive how to perform RoIAlign.

Given an image and a region (the green box), we want to resample a K K output imagethat corresponds to the region.W.l.o.g. we assume the input image has stride=1.Since we know the resolution and the absolute length of output,the output stride derived from the definition of grid ② is .Because grid offset is 0.5stride, the location of output pixel [i,j] isLet's call it .To compute resampled values at location , an easy way is to do bilinear interpolation with its 4nearest pixels (this corresponds to RoIAlign with sampling_ratio=1).We show the 4 neighboring input pixels of output[0,0] in the figure.The indices of 4 nearest pixels of are obtained after subtracting 0.5 to align their coordinate system:

The original implementation of RoIAlign in Detectron doesn't subtract 0.5 in the end, so it's actually not very aligned.It turns out this detail does not affect accuracy of R-CNNs, because RoIAlign is applied on CNN features, and CNNis believed to be able to fit slightly misaligned features.

However, we have new use cases of RoIAlign in other places, e.g. to crop mask head training targets from the ground truth mask, soI fixed it in the detectron2 / torchvision RoIAlign with an aligned=True option.Its unittestdemonstrates how the old version is misaligned.

Btw, once we figured out the coordinate transform formula, it's easy to implement RoIAlign using grid_sample.This shows that RoIAlign is nothing more than a fused bilinear sampling + averaging.Using grid_sample is about 10%-50% slower than the RoIAlign CUDA kernel.

Paste Mask

Mask R-CNN is trained to predict masks of fixed resolution (e.g. 28x28) restrained inside given boxes(we call it "RoIMask").But in the end, we often want to obtain full-image masks.A "paste mask" operation is needed to paste the small RoIMask into the given region in the image.

This operation is an inverse of RoIAlign, so it should be implemented similar to our derivation above.In Detectron, this was implementedwith some magic rounding & resize that are not exactly the inverse of RoIAlign.Fixing it in detectron2 increases the mask AP by 0.1~0.4.

Point-based Algorithms

Obviously, the paste mask operation can introduce aliasing in the results due to the low resolution RoIMask.This is the motivation behind our work ofPointRend.

PointRend is a segmentation method that focuses on point-wise features, where a "point" isnot necessarily a pixel, but any real-valued coordinates.Pointly-Supervised Instance Segmentation, also from our team, uses point-wise annotations to train segmentation models.Both projects involve heavy use of point sampling and coordinate transforms.Having a clear and consistent convention of pixels and coordinates was important to their success.

Summary

Due to some sloppy code in the early days of deep learning libraries,today we're facing multiple versions of resize functions.Together with the two different coordinate system conventions, they easily cause hidden bugs in computer vision code.

This article revisits these historical technical debts and shows how these fun details matter in modeling and training.I hope they will help you make proper choices.