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import collections
import math
import pathlib
import warnings
from itertools import repeat
from types import FunctionType
from typing import Any, BinaryIO, List, Optional, Tuple, Union
import numpy as np
import torch
from PIL import Image, ImageColor, ImageDraw, ImageFont
__all__ = [
"make_grid",
"save_image",
"draw_bounding_boxes",
"draw_segmentation_masks",
"draw_keypoints",
"flow_to_image",
]
@torch.no_grad()
def make_grid(
tensor: Union[torch.Tensor, List[torch.Tensor]],
nrow: int = 8,
padding: int = 2,
normalize: bool = False,
value_range: Optional[Tuple[int, int]] = None,
scale_each: bool = False,
pad_value: float = 0.0,
) -> torch.Tensor:
"""
Make a grid of images.
Args:
tensor (Tensor or list): 4D mini-batch Tensor of shape (B x C x H x W)
or a list of images all of the same size.
nrow (int, optional): Number of images displayed in each row of the grid.
The final grid size is ``(B / nrow, nrow)``. Default: ``8``.
padding (int, optional): amount of padding. Default: ``2``.
normalize (bool, optional): If True, shift the image to the range (0, 1),
by the min and max values specified by ``value_range``. Default: ``False``.
value_range (tuple, optional): tuple (min, max) where min and max are numbers,
then these numbers are used to normalize the image. By default, min and max
are computed from the tensor.
scale_each (bool, optional): If ``True``, scale each image in the batch of
images separately rather than the (min, max) over all images. Default: ``False``.
pad_value (float, optional): Value for the padded pixels. Default: ``0``.
Returns:
grid (Tensor): the tensor containing grid of images.
"""
if not torch.jit.is_scripting() and not torch.jit.is_tracing():
_log_api_usage_once(make_grid)
if not torch.is_tensor(tensor):
if isinstance(tensor, list):
for t in tensor:
if not torch.is_tensor(t):
raise TypeError(f"tensor or list of tensors expected, got a list containing {type(t)}")
else:
raise TypeError(f"tensor or list of tensors expected, got {type(tensor)}")
# if list of tensors, convert to a 4D mini-batch Tensor
if isinstance(tensor, list):
tensor = torch.stack(tensor, dim=0)
if tensor.dim() == 2: # single image H x W
tensor = tensor.unsqueeze(0)
if tensor.dim() == 3: # single image
if tensor.size(0) == 1: # if single-channel, convert to 3-channel
tensor = torch.cat((tensor, tensor, tensor), 0)
tensor = tensor.unsqueeze(0)
if tensor.dim() == 4 and tensor.size(1) == 1: # single-channel images
tensor = torch.cat((tensor, tensor, tensor), 1)
if normalize is True:
tensor = tensor.clone() # avoid modifying tensor in-place
if value_range is not None and not isinstance(value_range, tuple):
raise TypeError("value_range has to be a tuple (min, max) if specified. min and max are numbers")
def norm_ip(img, low, high):
img.clamp_(min=low, max=high)
img.sub_(low).div_(max(high - low, 1e-5))
def norm_range(t, value_range):
if value_range is not None:
norm_ip(t, value_range[0], value_range[1])
else:
norm_ip(t, float(t.min()), float(t.max()))
if scale_each is True:
for t in tensor: # loop over mini-batch dimension
norm_range(t, value_range)
else:
norm_range(tensor, value_range)
if not isinstance(tensor, torch.Tensor):
raise TypeError("tensor should be of type torch.Tensor")
if tensor.size(0) == 1:
return tensor.squeeze(0)
# make the mini-batch of images into a grid
nmaps = tensor.size(0)
xmaps = min(nrow, nmaps)
ymaps = int(math.ceil(float(nmaps) / xmaps))
height, width = int(tensor.size(2) + padding), int(tensor.size(3) + padding)
num_channels = tensor.size(1)
grid = tensor.new_full((num_channels, height * ymaps + padding, width * xmaps + padding), pad_value)
k = 0
for y in range(ymaps):
for x in range(xmaps):
if k >= nmaps:
break
# Tensor.copy_() is a valid method but seems to be missing from the stubs
# https://pytorch.org/docs/stable/tensors.html#torch.Tensor.copy_
grid.narrow(1, y * height + padding, height - padding).narrow( # type: ignore[attr-defined]
2, x * width + padding, width - padding
).copy_(tensor[k])
k = k + 1
return grid
@torch.no_grad()
def save_image(
tensor: Union[torch.Tensor, List[torch.Tensor]],
fp: Union[str, pathlib.Path, BinaryIO],
format: Optional[str] = None,
**kwargs,
) -> None:
"""
Save a given Tensor into an image file.
Args:
tensor (Tensor or list): Image to be saved. If given a mini-batch tensor,
saves the tensor as a grid of images by calling ``make_grid``.
fp (string or file object): A filename or a file object
format(Optional): If omitted, the format to use is determined from the filename extension.
If a file object was used instead of a filename, this parameter should always be used.
**kwargs: Other arguments are documented in ``make_grid``.
"""
if not torch.jit.is_scripting() and not torch.jit.is_tracing():
_log_api_usage_once(save_image)
grid = make_grid(tensor, **kwargs)
# Add 0.5 after unnormalizing to [0, 255] to round to the nearest integer
ndarr = grid.mul(255).add_(0.5).clamp_(0, 255).permute(1, 2, 0).to("cpu", torch.uint8).numpy()
im = Image.fromarray(ndarr)
im.save(fp, format=format)
@torch.no_grad()
def draw_bounding_boxes(
image: torch.Tensor,
boxes: torch.Tensor,
labels: Optional[List[str]] = None,
colors: Optional[Union[List[Union[str, Tuple[int, int, int]]], str, Tuple[int, int, int]]] = None,
fill: Optional[bool] = False,
width: int = 1,
font: Optional[str] = None,
font_size: Optional[int] = None,
) -> torch.Tensor:
"""
Draws bounding boxes on given image.
The values of the input image should be uint8 between 0 and 255.
If fill is True, Resulting Tensor should be saved as PNG image.
Args:
image (Tensor): Tensor of shape (C x H x W) and dtype uint8.
boxes (Tensor): Tensor of size (N, 4) containing bounding boxes in (xmin, ymin, xmax, ymax) format. Note that
the boxes are absolute coordinates with respect to the image. In other words: `0 <= xmin < xmax < W` and
`0 <= ymin < ymax < H`.
labels (List[str]): List containing the labels of bounding boxes.
colors (color or list of colors, optional): List containing the colors
of the boxes or single color for all boxes. The color can be represented as
PIL strings e.g. "red" or "#FF00FF", or as RGB tuples e.g. ``(240, 10, 157)``.
By default, random colors are generated for boxes.
fill (bool): If `True` fills the bounding box with specified color.
width (int): Width of bounding box.
font (str): A filename containing a TrueType font. If the file is not found in this filename, the loader may
also search in other directories, such as the `fonts/` directory on Windows or `/Library/Fonts/`,
`/System/Library/Fonts/` and `~/Library/Fonts/` on macOS.
font_size (int): The requested font size in points.
Returns:
img (Tensor[C, H, W]): Image Tensor of dtype uint8 with bounding boxes plotted.
"""
if not torch.jit.is_scripting() and not torch.jit.is_tracing():
_log_api_usage_once(draw_bounding_boxes)
if not isinstance(image, torch.Tensor):
raise TypeError(f"Tensor expected, got {type(image)}")
elif image.dtype != torch.uint8:
raise ValueError(f"Tensor uint8 expected, got {image.dtype}")
elif image.dim() != 3:
raise ValueError("Pass individual images, not batches")
elif image.size(0) not in {1, 3}:
raise ValueError("Only grayscale and RGB images are supported")
elif (boxes[:, 0] > boxes[:, 2]).any() or (boxes[:, 1] > boxes[:, 3]).any():
raise ValueError(
"Boxes need to be in (xmin, ymin, xmax, ymax) format. Use torchvision.ops.box_convert to convert them"
)
num_boxes = boxes.shape[0]
if num_boxes == 0:
warnings.warn("boxes doesn't contain any box. No box was drawn")
return image
if labels is None:
labels: Union[List[str], List[None]] = [None] * num_boxes # type: ignore[no-redef]
elif len(labels) != num_boxes:
raise ValueError(
f"Number of boxes ({num_boxes}) and labels ({len(labels)}) mismatch. Please specify labels for each box."
)
colors = _parse_colors(colors, num_objects=num_boxes)
if font is None:
if font_size is not None:
warnings.warn("Argument 'font_size' will be ignored since 'font' is not set.")
txt_font = ImageFont.load_default()
else:
txt_font = ImageFont.truetype(font=font, size=font_size or 10)
# Handle Grayscale images
if image.size(0) == 1:
image = torch.tile(image, (3, 1, 1))
ndarr = image.permute(1, 2, 0).cpu().numpy()
img_to_draw = Image.fromarray(ndarr)
img_boxes = boxes.to(torch.int64).tolist()
if fill:
draw = ImageDraw.Draw(img_to_draw, "RGBA")
else:
draw = ImageDraw.Draw(img_to_draw)
for bbox, color, label in zip(img_boxes, colors, labels): # type: ignore[arg-type]
if fill:
fill_color = color + (100,)
draw.rectangle(bbox, width=width, outline=color, fill=fill_color)
else:
draw.rectangle(bbox, width=width, outline=color)
if label is not None:
margin = width + 1
draw.text((bbox[0] + margin, bbox[1] + margin), label, fill=color, font=txt_font)
return torch.from_numpy(np.array(img_to_draw)).permute(2, 0, 1).to(dtype=torch.uint8)
@torch.no_grad()
def draw_segmentation_masks(
image: torch.Tensor,
masks: torch.Tensor,
alpha: float = 0.8,
colors: Optional[Union[List[Union[str, Tuple[int, int, int]]], str, Tuple[int, int, int]]] = None,
) -> torch.Tensor:
"""
Draws segmentation masks on given RGB image.
The image values should be uint8 in [0, 255] or float in [0, 1].
Args:
image (Tensor): Tensor of shape (3, H, W) and dtype uint8 or float.
masks (Tensor): Tensor of shape (num_masks, H, W) or (H, W) and dtype bool.
alpha (float): Float number between 0 and 1 denoting the transparency of the masks.
0 means full transparency, 1 means no transparency.
colors (color or list of colors, optional): List containing the colors
of the masks or single color for all masks. The color can be represented as
PIL strings e.g. "red" or "#FF00FF", or as RGB tuples e.g. ``(240, 10, 157)``.
By default, random colors are generated for each mask.
Returns:
img (Tensor[C, H, W]): Image Tensor, with segmentation masks drawn on top.
"""
if not torch.jit.is_scripting() and not torch.jit.is_tracing():
_log_api_usage_once(draw_segmentation_masks)
if not isinstance(image, torch.Tensor):
raise TypeError(f"The image must be a tensor, got {type(image)}")
elif not (image.dtype == torch.uint8 or image.is_floating_point()):
raise ValueError(f"The image dtype must be uint8 or float, got {image.dtype}")
elif image.dim() != 3:
raise ValueError("Pass individual images, not batches")
elif image.size()[0] != 3:
raise ValueError("Pass an RGB image. Other Image formats are not supported")
if masks.ndim == 2:
masks = masks[None, :, :]
if masks.ndim != 3:
raise ValueError("masks must be of shape (H, W) or (batch_size, H, W)")
if masks.dtype != torch.bool:
raise ValueError(f"The masks must be of dtype bool. Got {masks.dtype}")
if masks.shape[-2:] != image.shape[-2:]:
raise ValueError("The image and the masks must have the same height and width")
num_masks = masks.size()[0]
overlapping_masks = masks.sum(dim=0) > 1
if num_masks == 0:
warnings.warn("masks doesn't contain any mask. No mask was drawn")
return image
original_dtype = image.dtype
colors = [
torch.tensor(color, dtype=original_dtype, device=image.device)
for color in _parse_colors(colors, num_objects=num_masks, dtype=original_dtype)
]
img_to_draw = image.detach().clone()
# TODO: There might be a way to vectorize this
for mask, color in zip(masks, colors):
img_to_draw[:, mask] = color[:, None]
img_to_draw[:, overlapping_masks] = 0
out = image * (1 - alpha) + img_to_draw * alpha
# Note: at this point, out is a float tensor in [0, 1] or [0, 255] depending on original_dtype
return out.to(original_dtype)
@torch.no_grad()
def draw_keypoints(
image: torch.Tensor,
keypoints: torch.Tensor,
connectivity: Optional[List[Tuple[int, int]]] = None,
colors: Optional[Union[str, Tuple[int, int, int]]] = None,
radius: int = 2,
width: int = 3,
visibility: Optional[torch.Tensor] = None,
) -> torch.Tensor:
"""
Draws Keypoints on given RGB image.
The image values should be uint8 in [0, 255] or float in [0, 1].
Keypoints can be drawn for multiple instances at a time.
This method allows that keypoints and their connectivity are drawn based on the visibility of this keypoint.
Args:
image (Tensor): Tensor of shape (3, H, W) and dtype uint8 or float.
keypoints (Tensor): Tensor of shape (num_instances, K, 2) the K keypoint locations for each of the N instances,
in the format [x, y].
connectivity (List[Tuple[int, int]]]): A List of tuple where each tuple contains a pair of keypoints
to be connected.
If at least one of the two connected keypoints has a ``visibility`` of False,
this specific connection is not drawn.
Exclusions due to invisibility are computed per-instance.
colors (str, Tuple): The color can be represented as
PIL strings e.g. "red" or "#FF00FF", or as RGB tuples e.g. ``(240, 10, 157)``.
radius (int): Integer denoting radius of keypoint.
width (int): Integer denoting width of line connecting keypoints.
visibility (Tensor): Tensor of shape (num_instances, K) specifying the visibility of the K
keypoints for each of the N instances.
True means that the respective keypoint is visible and should be drawn.
False means invisible, so neither the point nor possible connections containing it are drawn.
The input tensor will be cast to bool.
Default ``None`` means that all the keypoints are visible.
For more details, see :ref:`draw_keypoints_with_visibility`.
Returns:
img (Tensor[C, H, W]): Image Tensor with keypoints drawn.
"""
if not torch.jit.is_scripting() and not torch.jit.is_tracing():
_log_api_usage_once(draw_keypoints)
# validate image
if not isinstance(image, torch.Tensor):
raise TypeError(f"The image must be a tensor, got {type(image)}")
elif not (image.dtype == torch.uint8 or image.is_floating_point()):
raise ValueError(f"The image dtype must be uint8 or float, got {image.dtype}")
elif image.dim() != 3:
raise ValueError("Pass individual images, not batches")
elif image.size()[0] != 3:
raise ValueError("Pass an RGB image. Other Image formats are not supported")
# validate keypoints
if keypoints.ndim != 3:
raise ValueError("keypoints must be of shape (num_instances, K, 2)")
# validate visibility
if visibility is None: # set default
visibility = torch.ones(keypoints.shape[:-1], dtype=torch.bool)
# If the last dimension is 1, e.g., after calling split([2, 1], dim=-1) on the output of a keypoint-prediction
# model, make sure visibility has shape (num_instances, K).
# Iff K = 1, this has unwanted behavior, but K=1 does not really make sense in the first place.
visibility = visibility.squeeze(-1)
if visibility.ndim != 2:
raise ValueError(f"visibility must be of shape (num_instances, K). Got ndim={visibility.ndim}")
if visibility.shape != keypoints.shape[:-1]:
raise ValueError(
"keypoints and visibility must have the same dimensionality for num_instances and K. "
f"Got {visibility.shape = } and {keypoints.shape = }"
)
original_dtype = image.dtype
if original_dtype.is_floating_point:
from torchvision.transforms.v2.functional import to_dtype # noqa
image = to_dtype(image, dtype=torch.uint8, scale=True)
ndarr = image.permute(1, 2, 0).cpu().numpy()
img_to_draw = Image.fromarray(ndarr)
draw = ImageDraw.Draw(img_to_draw)
img_kpts = keypoints.to(torch.int64).tolist()
img_vis = visibility.cpu().bool().tolist()
for kpt_inst, vis_inst in zip(img_kpts, img_vis):
for kpt_coord, kp_vis in zip(kpt_inst, vis_inst):
if not kp_vis:
continue
x1 = kpt_coord[0] - radius
x2 = kpt_coord[0] + radius
y1 = kpt_coord[1] - radius
y2 = kpt_coord[1] + radius
draw.ellipse([x1, y1, x2, y2], fill=colors, outline=None, width=0)
if connectivity:
for connection in connectivity:
if (not vis_inst[connection[0]]) or (not vis_inst[connection[1]]):
continue
start_pt_x = kpt_inst[connection[0]][0]
start_pt_y = kpt_inst[connection[0]][1]
end_pt_x = kpt_inst[connection[1]][0]
end_pt_y = kpt_inst[connection[1]][1]
draw.line(
((start_pt_x, start_pt_y), (end_pt_x, end_pt_y)),
width=width,
)
out = torch.from_numpy(np.array(img_to_draw)).permute(2, 0, 1)
if original_dtype.is_floating_point:
out = to_dtype(out, dtype=original_dtype, scale=True)
return out
# Flow visualization code adapted from https://github.com/tomrunia/OpticalFlow_Visualization
@torch.no_grad()
def flow_to_image(flow: torch.Tensor) -> torch.Tensor:
"""
Converts a flow to an RGB image.
Args:
flow (Tensor): Flow of shape (N, 2, H, W) or (2, H, W) and dtype torch.float.
Returns:
img (Tensor): Image Tensor of dtype uint8 where each color corresponds
to a given flow direction. Shape is (N, 3, H, W) or (3, H, W) depending on the input.
"""
if flow.dtype != torch.float:
raise ValueError(f"Flow should be of dtype torch.float, got {flow.dtype}.")
orig_shape = flow.shape
if flow.ndim == 3:
flow = flow[None] # Add batch dim
if flow.ndim != 4 or flow.shape[1] != 2:
raise ValueError(f"Input flow should have shape (2, H, W) or (N, 2, H, W), got {orig_shape}.")
max_norm = torch.sum(flow**2, dim=1).sqrt().max()
epsilon = torch.finfo((flow).dtype).eps
normalized_flow = flow / (max_norm + epsilon)
img = _normalized_flow_to_image(normalized_flow)
if len(orig_shape) == 3:
img = img[0] # Remove batch dim
return img
@torch.no_grad()
def _normalized_flow_to_image(normalized_flow: torch.Tensor) -> torch.Tensor:
"""
Converts a batch of normalized flow to an RGB image.
Args:
normalized_flow (torch.Tensor): Normalized flow tensor of shape (N, 2, H, W)
Returns:
img (Tensor(N, 3, H, W)): Flow visualization image of dtype uint8.
"""
N, _, H, W = normalized_flow.shape
device = normalized_flow.device
flow_image = torch.zeros((N, 3, H, W), dtype=torch.uint8, device=device)
colorwheel = _make_colorwheel().to(device) # shape [55x3]
num_cols = colorwheel.shape[0]
norm = torch.sum(normalized_flow**2, dim=1).sqrt()
a = torch.atan2(-normalized_flow[:, 1, :, :], -normalized_flow[:, 0, :, :]) / torch.pi
fk = (a + 1) / 2 * (num_cols - 1)
k0 = torch.floor(fk).to(torch.long)
k1 = k0 + 1
k1[k1 == num_cols] = 0
f = fk - k0
for c in range(colorwheel.shape[1]):
tmp = colorwheel[:, c]
col0 = tmp[k0] / 255.0
col1 = tmp[k1] / 255.0
col = (1 - f) * col0 + f * col1
col = 1 - norm * (1 - col)
flow_image[:, c, :, :] = torch.floor(255 * col)
return flow_image
def _make_colorwheel() -> torch.Tensor:
"""
Generates a color wheel for optical flow visualization as presented in:
Baker et al. "A Database and Evaluation Methodology for Optical Flow" (ICCV, 2007)
URL: http://vision.middlebury.edu/flow/flowEval-iccv07.pdf.
Returns:
colorwheel (Tensor[55, 3]): Colorwheel Tensor.
"""
RY = 15
YG = 6
GC = 4
CB = 11
BM = 13
MR = 6
ncols = RY + YG + GC + CB + BM + MR
colorwheel = torch.zeros((ncols, 3))
col = 0
# RY
colorwheel[0:RY, 0] = 255
colorwheel[0:RY, 1] = torch.floor(255 * torch.arange(0, RY) / RY)
col = col + RY
# YG
colorwheel[col : col + YG, 0] = 255 - torch.floor(255 * torch.arange(0, YG) / YG)
colorwheel[col : col + YG, 1] = 255
col = col + YG
# GC
colorwheel[col : col + GC, 1] = 255
colorwheel[col : col + GC, 2] = torch.floor(255 * torch.arange(0, GC) / GC)
col = col + GC
# CB
colorwheel[col : col + CB, 1] = 255 - torch.floor(255 * torch.arange(CB) / CB)
colorwheel[col : col + CB, 2] = 255
col = col + CB
# BM
colorwheel[col : col + BM, 2] = 255
colorwheel[col : col + BM, 0] = torch.floor(255 * torch.arange(0, BM) / BM)
col = col + BM
# MR
colorwheel[col : col + MR, 2] = 255 - torch.floor(255 * torch.arange(MR) / MR)
colorwheel[col : col + MR, 0] = 255
return colorwheel
def _generate_color_palette(num_objects: int):
palette = torch.tensor([2**25 - 1, 2**15 - 1, 2**21 - 1])
return [tuple((i * palette) % 255) for i in range(num_objects)]
def _parse_colors(
colors: Union[None, str, Tuple[int, int, int], List[Union[str, Tuple[int, int, int]]]],
*,
num_objects: int,
dtype: torch.dtype = torch.uint8,
) -> List[Tuple[int, int, int]]:
"""
Parses a specification of colors for a set of objects.
Args:
colors: A specification of colors for the objects. This can be one of the following:
- None: to generate a color palette automatically.
- A list of colors: where each color is either a string (specifying a named color) or an RGB tuple.
- A string or an RGB tuple: to use the same color for all objects.
If `colors` is a tuple, it should be a 3-tuple specifying the RGB values of the color.
If `colors` is a list, it should have at least as many elements as the number of objects to color.
num_objects (int): The number of objects to color.
Returns:
A list of 3-tuples, specifying the RGB values of the colors.
Raises:
ValueError: If the number of colors in the list is less than the number of objects to color.
If `colors` is not a list, tuple, string or None.
"""
if colors is None:
colors = _generate_color_palette(num_objects)
elif isinstance(colors, list):
if len(colors) < num_objects:
raise ValueError(
f"Number of colors must be equal or larger than the number of objects, but got {len(colors)} < {num_objects}."
)
elif not isinstance(colors, (tuple, str)):
raise ValueError("`colors` must be a tuple or a string, or a list thereof, but got {colors}.")
elif isinstance(colors, tuple) and len(colors) != 3:
raise ValueError("If passed as tuple, colors should be an RGB triplet, but got {colors}.")
else: # colors specifies a single color for all objects
colors = [colors] * num_objects
colors = [ImageColor.getrgb(color) if isinstance(color, str) else color for color in colors]
if dtype.is_floating_point: # [0, 255] -> [0, 1]
colors = [tuple(v / 255 for v in color) for color in colors]
return colors
def _log_api_usage_once(obj: Any) -> None:
"""
Logs API usage(module and name) within an organization.
In a large ecosystem, it's often useful to track the PyTorch and
TorchVision APIs usage. This API provides the similar functionality to the
logging module in the Python stdlib. It can be used for debugging purpose
to log which methods are used and by default it is inactive, unless the user
manually subscribes a logger via the `SetAPIUsageLogger method <https://github.com/pytorch/pytorch/blob/eb3b9fe719b21fae13c7a7cf3253f970290a573e/c10/util/Logging.cpp#L114>`_.
Please note it is triggered only once for the same API call within a process.
It does not collect any data from open-source users since it is no-op by default.
For more information, please refer to
* PyTorch note: https://pytorch.org/docs/stable/notes/large_scale_deployments.html#api-usage-logging;
* Logging policy: https://github.com/pytorch/vision/issues/5052;
Args:
obj (class instance or method): an object to extract info from.
"""
module = obj.__module__
if not module.startswith("torchvision"):
module = f"torchvision.internal.{module}"
name = obj.__class__.__name__
if isinstance(obj, FunctionType):
name = obj.__name__
torch._C._log_api_usage_once(f"{module}.{name}")
def _make_ntuple(x: Any, n: int) -> Tuple[Any, ...]:
"""
Make n-tuple from input x. If x is an iterable, then we just convert it to tuple.
Otherwise, we will make a tuple of length n, all with value of x.
reference: https://github.com/pytorch/pytorch/blob/master/torch/nn/modules/utils.py#L8
Args:
x (Any): input value
n (int): length of the resulting tuple
"""
if isinstance(x, collections.abc.Iterable):
return tuple(x)
return tuple(repeat(x, n))