ssd_box_encode_decode_utils.py

'''
Includes:
* Function to compute IoU similarity for axis-aligned, rectangular, 2D bounding boxes
* Function to perform greedy non-maximum suppression
* Function to decode raw SSD model output
* Class to encode targets for SSD model training

Copyright (C) 2017 Pierluigi Ferrari

This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.

This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

You should have received a copy of the GNU General Public License
along with this program.  If not, see <http://www.gnu.org/licenses/>.
'''

import numpy as np

def iou(boxes1, boxes2, coords='centroids'):
    '''
    Compute the intersection-over-union similarity (also known as Jaccard similarity)
    of two axis-aligned 2D rectangular boxes or of multiple axis-aligned 2D rectangular
    boxes contained in two arrays with broadcast-compatible shapes.

    Three common use cases would be to compute the similarities for 1 vs. 1, 1 vs. `n`,
    or `n` vs. `n` boxes. The two arguments are symmetric.

    Arguments:
        boxes1 (array): Either a 1D Numpy array of shape `(4, )` containing the coordinates for one box in the
            format specified by `coords` or a 2D Numpy array of shape `(n, 4)` containing the coordinates for `n` boxes.
            Shape must be broadcast-compatible to `boxes2`.
        boxes2 (array): Either a 1D Numpy array of shape `(4, )` containing the coordinates for one box in the
            format specified by `coords` or a 2D Numpy array of shape `(n, 4)` containing the coordinates for `n` boxes.
            Shape must be broadcast-compatible to `boxes1`.
        coords (str, optional): The coordinate format in the input arrays. Can be either 'centroids' for the format
            `(cx, cy, w, h)`, 'minmax' for the format `(xmin, xmax, ymin, ymax)`, or 'corners' for the format
            `(xmin, ymin, xmax, ymax)`.

    Returns:
        A 1D Numpy array of dtype float containing values in [0,1], the Jaccard similarity of the boxes in `boxes1` and `boxes2`.
        0 means there is no overlap between two given boxes, 1 means their coordinates are identical.
    '''

    if len(boxes1.shape) > 2: raise ValueError("boxes1 must have rank either 1 or 2, but has rank {}.".format(len(boxes1.shape)))
    if len(boxes2.shape) > 2: raise ValueError("boxes2 must have rank either 1 or 2, but has rank {}.".format(len(boxes2.shape)))

    if len(boxes1.shape) == 1: boxes1 = np.expand_dims(boxes1, axis=0)
    if len(boxes2.shape) == 1: boxes2 = np.expand_dims(boxes2, axis=0)

    if not (boxes1.shape[1] == boxes2.shape[1] == 4): raise ValueError("It must be boxes1.shape[1] == boxes2.shape[1] == 4, but it is boxes1.shape[1] == {}, boxes2.shape[1] == {}.".format(boxes1.shape[1], boxes2.shape[1]))

    if coords == 'centroids':
        # TODO: Implement a version that uses fewer computation steps (that doesn't need conversion)
        boxes1 = convert_coordinates(boxes1, start_index=0, conversion='centroids2minmax')
        boxes2 = convert_coordinates(boxes2, start_index=0, conversion='centroids2minmax')
    elif not (coords in {'minmax', 'corners'}):
        raise ValueError("Unexpected value for `coords`. Supported values are 'minmax', 'corners' and 'centroids'.")

    if coords in {'minmax', 'centroids'}:
        intersection = np.maximum(0, np.minimum(boxes1[:,1], boxes2[:,1]) - np.maximum(boxes1[:,0], boxes2[:,0])) * np.maximum(0, np.minimum(boxes1[:,3], boxes2[:,3]) - np.maximum(boxes1[:,2], boxes2[:,2]))
        union = (boxes1[:,1] - boxes1[:,0]) * (boxes1[:,3] - boxes1[:,2]) + (boxes2[:,1] - boxes2[:,0]) * (boxes2[:,3] - boxes2[:,2]) - intersection
    elif coords == 'corners':
        intersection = np.maximum(0, np.minimum(boxes1[:,2], boxes2[:,2]) - np.maximum(boxes1[:,0], boxes2[:,0])) * np.maximum(0, np.minimum(boxes1[:,3], boxes2[:,3]) - np.maximum(boxes1[:,1], boxes2[:,1]))
        union = (boxes1[:,2] - boxes1[:,0]) * (boxes1[:,3] - boxes1[:,1]) + (boxes2[:,2] - boxes2[:,0]) * (boxes2[:,3] - boxes2[:,1]) - intersection

    return intersection / union

def convert_coordinates(tensor, start_index, conversion):
    '''
    Convert coordinates for axis-aligned 2D boxes between two coordinate formats.

    Creates a copy of `tensor`, i.e. does not operate in place. Currently there are
    three supported coordinate formats that can be converted from and to each other:
        1) (xmin, xmax, ymin, ymax) - the 'minmax' format
        2) (xmin, ymin, xmax, ymax) - the 'corners' format
        2) (cx, cy, w, h) - the 'centroids' format

    Note that converting from one of the supported formats to another and back is
    an identity operation up to possible rounding errors for integer tensors.

    Arguments:
        tensor (array): A Numpy nD array containing the four consecutive coordinates
            to be converted somewhere in the last axis.
        start_index (int): The index of the first coordinate in the last axis of `tensor`.
        conversion (str, optional): The conversion direction. Can be 'minmax2centroids',
            'centroids2minmax', 'corners2centroids', 'centroids2corners', 'minmax2corners',
            or 'corners2minmax'.

    Returns:
        A Numpy nD array, a copy of the input tensor with the converted coordinates
        in place of the original coordinates and the unaltered elements of the original
        tensor elsewhere.
    '''
    ind = start_index
    tensor1 = np.copy(tensor).astype(np.float)
    if conversion == 'minmax2centroids':
        tensor1[..., ind] = (tensor[..., ind] + tensor[..., ind+1]) / 2.0 # Set cx
        tensor1[..., ind+1] = (tensor[..., ind+2] + tensor[..., ind+3]) / 2.0 # Set cy
        tensor1[..., ind+2] = tensor[..., ind+1] - tensor[..., ind] # Set w
        tensor1[..., ind+3] = tensor[..., ind+3] - tensor[..., ind+2] # Set h
    elif conversion == 'centroids2minmax':
        tensor1[..., ind] = tensor[..., ind] - tensor[..., ind+2] / 2.0 # Set xmin
        tensor1[..., ind+1] = tensor[..., ind] + tensor[..., ind+2] / 2.0 # Set xmax
        tensor1[..., ind+2] = tensor[..., ind+1] - tensor[..., ind+3] / 2.0 # Set ymin
        tensor1[..., ind+3] = tensor[..., ind+1] + tensor[..., ind+3] / 2.0 # Set ymax
    elif conversion == 'corners2centroids':
        tensor1[..., ind] = (tensor[..., ind] + tensor[..., ind+2]) / 2.0 # Set cx
        tensor1[..., ind+1] = (tensor[..., ind+1] + tensor[..., ind+3]) / 2.0 # Set cy
        tensor1[..., ind+2] = tensor[..., ind+2] - tensor[..., ind] # Set w
        tensor1[..., ind+3] = tensor[..., ind+3] - tensor[..., ind+1] # Set h
    elif conversion == 'centroids2corners':
        tensor1[..., ind] = tensor[..., ind] - tensor[..., ind+2] / 2.0 # Set xmin
        tensor1[..., ind+1] = tensor[..., ind+1] - tensor[..., ind+3] / 2.0 # Set ymin
        tensor1[..., ind+2] = tensor[..., ind] + tensor[..., ind+2] / 2.0 # Set xmax
        tensor1[..., ind+3] = tensor[..., ind+1] + tensor[..., ind+3] / 2.0 # Set ymax
    elif (conversion == 'minmax2corners') or (conversion == 'corners2minmax'):
        tensor1[..., ind+1] = tensor[..., ind+2]
        tensor1[..., ind+2] = tensor[..., ind+1]
    else:
        raise ValueError("Unexpected conversion value. Supported values are 'minmax2centroids', 'centroids2minmax', 'corners2centroids', 'centroids2corners', 'minmax2corners', and 'corners2minmax'.")

    return tensor1

def convert_coordinates2(tensor, start_index, conversion):
    '''
    A matrix multiplication implementation of `convert_coordinates()`.
    Supports only conversion between the 'centroids' and 'minmax' formats.

    This function is marginally slower on average than `convert_coordinates()`,
    probably because it involves more (unnecessary) arithmetic operations (unnecessary
    because the two matrices are sparse).

    For details please refer to the documentation of `convert_coordinates()`.
    '''
    ind = start_index
    tensor1 = np.copy(tensor).astype(np.float)
    if conversion == 'minmax2centroids':
        M = np.array([[0.5, 0. , -1.,  0.],
                      [0.5, 0. ,  1.,  0.],
                      [0. , 0.5,  0., -1.],
                      [0. , 0.5,  0.,  1.]])
        tensor1[..., ind:ind+4] = np.dot(tensor1[..., ind:ind+4], M)
    elif conversion == 'centroids2minmax':
        M = np.array([[ 1. , 1. ,  0. , 0. ],
                      [ 0. , 0. ,  1. , 1. ],
                      [-0.5, 0.5,  0. , 0. ],
                      [ 0. , 0. , -0.5, 0.5]]) # The multiplicative inverse of the matrix above
        tensor1[..., ind:ind+4] = np.dot(tensor1[..., ind:ind+4], M)
    else:
        raise ValueError("Unexpected conversion value. Supported values are 'minmax2centroids' and 'centroids2minmax'.")

    return tensor1

def greedy_nms(y_pred_decoded, iou_threshold=0.45, coords='corners'):
    '''
    Perform greedy non-maximum suppression on the input boxes.

    Greedy NMS works by selecting the box with the highest score and
    removing all boxes around it that are too close to it measured by IoU-similarity.
    Out of the boxes that are left over, once again the one with the highest
    score is selected and so on, until no boxes with too much overlap are left.

    This is a basic, straight-forward NMS algorithm that is relatively efficient,
    but it has a number of downsides. One of those downsides is that the box with
    the highest score might not always be the box with the best fit to the object.
    There are more sophisticated NMS techniques like [this one](https://lirias.kuleuven.be/bitstream/123456789/506283/1/3924_postprint.pdf)
    that use a combination of nearby boxes, but in general there will probably
    always be a trade-off between speed and quality for any given NMS technique.

    Arguments:
        y_pred_decoded (list): A batch of decoded predictions. For a given batch size `n` this
            is a list of length `n` where each list element is a 2D Numpy array.
            For a batch item with `k` predicted boxes this 2D Numpy array has
            shape `(k, 6)`, where each row contains the coordinates of the respective
            box in the format `[class_id, score, xmin, xmax, ymin, ymax]`.
            Technically, the number of columns doesn't have to be 6, it can be
            arbitrary as long as the first four elements of each row are
            `xmin`, `xmax`, `ymin`, `ymax` (in this order) and the last element
            is the score assigned to the prediction. Note that this function is
            agnostic to the scale of the score or what it represents.
        iou_threshold (float, optional): All boxes with a Jaccard similarity of
            greater than `iou_threshold` with a locally maximal box will be removed
            from the set of predictions, where 'maximal' refers to the box score.
            Defaults to 0.45 following the paper.
        coords (str, optional): The coordinate format of `y_pred_decoded`.
            Can be one of the formats supported by `iou()`. Defaults to 'corners'.

    Returns:
        The predictions after removing non-maxima. The format is the same as the input format.
    '''
    y_pred_decoded_nms = []
    for batch_item in y_pred_decoded: # For the labels of each batch item...
        boxes_left = np.copy(batch_item)
        maxima = [] # This is where we store the boxes that make it through the non-maximum suppression
        while boxes_left.shape[0] > 0: # While there are still boxes left to compare...
            maximum_index = np.argmax(boxes_left[:,1]) # ...get the index of the next box with the highest confidence...
            maximum_box = np.copy(boxes_left[maximum_index]) # ...copy that box and...
            maxima.append(maximum_box) # ...append it to `maxima` because we'll definitely keep it
            boxes_left = np.delete(boxes_left, maximum_index, axis=0) # Now remove the maximum box from `boxes_left`
            if boxes_left.shape[0] == 0: break # If there are no boxes left after this step, break. Otherwise...
            similarities = iou(boxes_left[:,2:], maximum_box[2:], coords=coords) # ...compare (IoU) the other left over boxes to the maximum box...
            boxes_left = boxes_left[similarities <= iou_threshold] # ...so that we can remove the ones that overlap too much with the maximum box
        y_pred_decoded_nms.append(np.array(maxima))

    return y_pred_decoded_nms

def _greedy_nms(predictions, iou_threshold=0.45, coords='corners'):
    '''
    The same greedy non-maximum suppression algorithm as above, but slightly modified for use as an internal
    function for per-class NMS in `decode_y()`.
    '''
    boxes_left = np.copy(predictions)
    maxima = [] # This is where we store the boxes that make it through the non-maximum suppression
    while boxes_left.shape[0] > 0: # While there are still boxes left to compare...
        maximum_index = np.argmax(boxes_left[:,0]) # ...get the index of the next box with the highest confidence...
        maximum_box = np.copy(boxes_left[maximum_index]) # ...copy that box and...
        maxima.append(maximum_box) # ...append it to `maxima` because we'll definitely keep it
        boxes_left = np.delete(boxes_left, maximum_index, axis=0) # Now remove the maximum box from `boxes_left`
        if boxes_left.shape[0] == 0: break # If there are no boxes left after this step, break. Otherwise...
        similarities = iou(boxes_left[:,1:], maximum_box[1:], coords=coords) # ...compare (IoU) the other left over boxes to the maximum box...
        boxes_left = boxes_left[similarities <= iou_threshold] # ...so that we can remove the ones that overlap too much with the maximum box
    return np.array(maxima)

def _greedy_nms2(predictions, iou_threshold=0.45, coords='corners'):
    '''
    The same greedy non-maximum suppression algorithm as above, but slightly modified for use as an internal
    function in `decode_y2()`.
    '''
    boxes_left = np.copy(predictions)
    maxima = [] # This is where we store the boxes that make it through the non-maximum suppression
    while boxes_left.shape[0] > 0: # While there are still boxes left to compare...
        maximum_index = np.argmax(boxes_left[:,1]) # ...get the index of the next box with the highest confidence...
        maximum_box = np.copy(boxes_left[maximum_index]) # ...copy that box and...
        maxima.append(maximum_box) # ...append it to `maxima` because we'll definitely keep it
        boxes_left = np.delete(boxes_left, maximum_index, axis=0) # Now remove the maximum box from `boxes_left`
        if boxes_left.shape[0] == 0: break # If there are no boxes left after this step, break. Otherwise...
        similarities = iou(boxes_left[:,2:], maximum_box[2:], coords=coords) # ...compare (IoU) the other left over boxes to the maximum box...
        boxes_left = boxes_left[similarities <= iou_threshold] # ...so that we can remove the ones that overlap too much with the maximum box
    return np.array(maxima)

def decode_y(y_pred,
             confidence_thresh=0.01,
             iou_threshold=0.45,
             top_k=200,
             input_coords='centroids',
             normalize_coords=False,
             img_height=None,
             img_width=None):
    '''
    Convert model prediction output back to a format that contains only the positive box predictions
    (i.e. the same format that `enconde_y()` takes as input).

    After the decoding, two stages of prediction filtering are performed for each class individually:
    First confidence thresholding, then greedy non-maximum suppression. The filtering results for all
    classes are concatenated and the `top_k` overall highest confidence results constitute the final
    predictions for a given batch item. This procedure follows the original Caffe implementation.
    For a slightly different and more efficient alternative to decode raw model output that performs
    non-maximum suppresion globally instead of per class, see `decode_y2()` below.

    Arguments:
        y_pred (array): The prediction output of the SSD model, expected to be a Numpy array
            of shape `(batch_size, #boxes, #classes + 4 + 4 + 4)`, where `#boxes` is the total number of
            boxes predicted by the model per image and the last axis contains
            `[one-hot vector for the classes, 4 predicted coordinate offsets, 4 anchor box coordinates, 4 variances]`.
        confidence_thresh (float, optional): A float in [0,1), the minimum classification confidence in a specific
            positive class in order to be considered for the non-maximum suppression stage for the respective class.
            A lower value will result in a larger part of the selection process being done by the non-maximum suppression
            stage, while a larger value will result in a larger part of the selection process happening in the confidence
            thresholding stage. Defaults to 0.01, following the paper.
        iou_threshold (float, optional): A float in [0,1]. All boxes with a Jaccard similarity of greater than `iou_threshold`
            with a locally maximal box will be removed from the set of predictions for a given class, where 'maximal' refers
            to the box score. Defaults to 0.45 following the paper.
        top_k (int, optional): The number of highest scoring predictions to be kept for each batch item after the
            non-maximum suppression stage. Defaults to 200, following the paper.
        input_coords (str, optional): The box coordinate format that the model outputs. Can be either 'centroids'
            for the format `(cx, cy, w, h)` (box center coordinates, width, and height), 'minmax' for the format
            `(xmin, xmax, ymin, ymax)`, or 'corners' for the format `(xmin, ymin, xmax, ymax)`. Defaults to 'centroids'.
        normalize_coords (bool, optional): Set to `True` if the model outputs relative coordinates (i.e. coordinates in [0,1])
            and you wish to transform these relative coordinates back to absolute coordinates. If the model outputs
            relative coordinates, but you do not want to convert them back to absolute coordinates, set this to `False`.
            Do not set this to `True` if the model already outputs absolute coordinates, as that would result in incorrect
            coordinates. Requires `img_height` and `img_width` if set to `True`. Defaults to `False`.
        img_height (int, optional): The height of the input images. Only needed if `normalize_coords` is `True`.
        img_width (int, optional): The width of the input images. Only needed if `normalize_coords` is `True`.

    Returns:
        A python list of length `batch_size` where each list element represents the predicted boxes
        for one image and contains a Numpy array of shape `(boxes, 6)` where each row is a box prediction for
        a non-background class for the respective image in the format `[class_id, confidence, xmin, ymin, xmax, ymax]`.
    '''
    if normalize_coords and ((img_height is None) or (img_width is None)):
        raise ValueError("If relative box coordinates are supposed to be converted to absolute coordinates, the decoder needs the image size in order to decode the predictions, but `img_height == {}` and `img_width == {}`".format(img_height, img_width))

    # 1: Convert the box coordinates from the predicted anchor box offsets to predicted absolute coordinates

    y_pred_decoded_raw = np.copy(y_pred[:,:,:-8]) # Slice out the classes and the four offsets, throw away the anchor coordinates and variances, resulting in a tensor of shape `[batch, n_boxes, n_classes + 4 coordinates]`

    if input_coords == 'centroids':
        y_pred_decoded_raw[:,:,[-2,-1]] = np.exp(y_pred_decoded_raw[:,:,[-2,-1]] * y_pred[:,:,[-2,-1]]) # exp(ln(w(pred)/w(anchor)) / w_variance * w_variance) == w(pred) / w(anchor), exp(ln(h(pred)/h(anchor)) / h_variance * h_variance) == h(pred) / h(anchor)
        y_pred_decoded_raw[:,:,[-2,-1]] *= y_pred[:,:,[-6,-5]] # (w(pred) / w(anchor)) * w(anchor) == w(pred), (h(pred) / h(anchor)) * h(anchor) == h(pred)
        y_pred_decoded_raw[:,:,[-4,-3]] *= y_pred[:,:,[-4,-3]] * y_pred[:,:,[-6,-5]] # (delta_cx(pred) / w(anchor) / cx_variance) * cx_variance * w(anchor) == delta_cx(pred), (delta_cy(pred) / h(anchor) / cy_variance) * cy_variance * h(anchor) == delta_cy(pred)
        y_pred_decoded_raw[:,:,[-4,-3]] += y_pred[:,:,[-8,-7]] # delta_cx(pred) + cx(anchor) == cx(pred), delta_cy(pred) + cy(anchor) == cy(pred)
        y_pred_decoded_raw = convert_coordinates(y_pred_decoded_raw, start_index=-4, conversion='centroids2corners')
    elif input_coords == 'minmax':
        y_pred_decoded_raw[:,:,-4:] *= y_pred[:,:,-4:] # delta(pred) / size(anchor) / variance * variance == delta(pred) / size(anchor) for all four coordinates, where 'size' refers to w or h, respectively
        y_pred_decoded_raw[:,:,[-4,-3]] *= np.expand_dims(y_pred[:,:,-7] - y_pred[:,:,-8], axis=-1) # delta_xmin(pred) / w(anchor) * w(anchor) == delta_xmin(pred), delta_xmax(pred) / w(anchor) * w(anchor) == delta_xmax(pred)
        y_pred_decoded_raw[:,:,[-2,-1]] *= np.expand_dims(y_pred[:,:,-5] - y_pred[:,:,-6], axis=-1) # delta_ymin(pred) / h(anchor) * h(anchor) == delta_ymin(pred), delta_ymax(pred) / h(anchor) * h(anchor) == delta_ymax(pred)
        y_pred_decoded_raw[:,:,-4:] += y_pred[:,:,-8:-4] # delta(pred) + anchor == pred for all four coordinates
        y_pred_decoded_raw = convert_coordinates(y_pred_decoded_raw, start_index=-4, conversion='minmax2corners')
    elif input_coords == 'corners':
        y_pred_decoded_raw[:,:,-4:] *= y_pred[:,:,-4:] # delta(pred) / size(anchor) / variance * variance == delta(pred) / size(anchor) for all four coordinates, where 'size' refers to w or h, respectively
        y_pred_decoded_raw[:,:,[-4,-2]] *= np.expand_dims(y_pred[:,:,-6] - y_pred[:,:,-8], axis=-1) # delta_xmin(pred) / w(anchor) * w(anchor) == delta_xmin(pred), delta_xmax(pred) / w(anchor) * w(anchor) == delta_xmax(pred)
        y_pred_decoded_raw[:,:,[-3,-1]] *= np.expand_dims(y_pred[:,:,-5] - y_pred[:,:,-7], axis=-1) # delta_ymin(pred) / h(anchor) * h(anchor) == delta_ymin(pred), delta_ymax(pred) / h(anchor) * h(anchor) == delta_ymax(pred)
        y_pred_decoded_raw[:,:,-4:] += y_pred[:,:,-8:-4] # delta(pred) + anchor == pred for all four coordinates
    else:
        raise ValueError("Unexpected value for `input_coords`. Supported input coordinate formats are 'minmax', 'corners' and 'centroids'.")

    # 2: If the model predicts normalized box coordinates and they are supposed to be converted back to absolute coordinates, do that

    if normalize_coords:
        y_pred_decoded_raw[:,:,[-4,-2]] *= img_width # Convert xmin, xmax back to absolute coordinates
        y_pred_decoded_raw[:,:,[-3,-1]] *= img_height # Convert ymin, ymax back to absolute coordinates

    # 3: Apply confidence thresholding and non-maximum suppression per class

    n_classes = y_pred_decoded_raw.shape[-1] - 4 # The number of classes is the length of the last axis minus the four box coordinates

    y_pred_decoded = [] # Store the final predictions in this list
    for batch_item in y_pred_decoded_raw: # `batch_item` has shape `[n_boxes, n_classes + 4 coords]`
        pred = [] # Store the final predictions for this batch item here
        for class_id in range(1, n_classes): # For each class except the background class (which has class ID 0)...
            single_class = batch_item[:,[class_id, -4, -3, -2, -1]] # ...keep only the confidences for that class, making this an array of shape `[n_boxes, 5]` and...
            threshold_met = single_class[single_class[:,0] > confidence_thresh] # ...keep only those boxes with a confidence above the set threshold.
            if threshold_met.shape[0] > 0: # If any boxes made the threshold...
                maxima = _greedy_nms(threshold_met, iou_threshold=iou_threshold, coords='corners') # ...perform NMS on them.
                maxima_output = np.zeros((maxima.shape[0], maxima.shape[1] + 1)) # Expand the last dimension by one element to have room for the class ID. This is now an arrray of shape `[n_boxes, 6]`
                maxima_output[:,0] = class_id # Write the class ID to the first column...
                maxima_output[:,1:] = maxima # ...and write the maxima to the other columns...
                pred.append(maxima_output) # ...and append the maxima for this class to the list of maxima for this batch item.
        # Once we're through with all classes, keep only the `top_k` maxima with the highest scores
        if pred: # If there are any predictions left after confidence-thresholding...
            pred = np.concatenate(pred, axis=0)
            if pred.shape[0] > top_k: # If we have more than `top_k` results left at this point, otherwise there is nothing to filter,...
                top_k_indices = np.argpartition(pred[:,1], kth=pred.shape[0]-top_k, axis=0)[pred.shape[0]-top_k:] # ...get the indices of the `top_k` highest-score maxima...
                pred = pred[top_k_indices] # ...and keep only those entries of `pred`...
        y_pred_decoded.append(pred) # ...and now that we're done, append the array of final predictions for this batch item to the output list

    return y_pred_decoded

def decode_y2(y_pred,
              confidence_thresh=0.5,
              iou_threshold=0.45,
              top_k='all',
              input_coords='centroids',
              normalize_coords=False,
              img_height=None,
              img_width=None):
    '''
    Convert model prediction output back to a format that contains only the positive box predictions
    (i.e. the same format that `enconde_y()` takes as input).

    Optionally performs confidence thresholding and greedy non-maximum suppression after the decoding stage.

    Note that the decoding procedure used here is not the same as the procedure used in the original Caffe implementation.
    For each box, the procedure used here assigns the box's highest confidence as its predicted class. Then it removes
    all boxes for which the highest confidence is the background class. This results in less work for the subsequent
    non-maximum suppression, because the vast majority of the predictions will be filtered out just by the fact that
    their highest confidence is for the background class. It is much more efficient than the procedure of the original
    implementation, but the results may also differ.

    Arguments:
        y_pred (array): The prediction output of the SSD model, expected to be a Numpy array
            of shape `(batch_size, #boxes, #classes + 4 + 4 + 4)`, where `#boxes` is the total number of
            boxes predicted by the model per image and the last axis contains
            `[one-hot vector for the classes, 4 predicted coordinate offsets, 4 anchor box coordinates, 4 variances]`.
        confidence_thresh (float, optional): A float in [0,1), the minimum classification confidence in any positive
            class required for a given box to be considered a positive prediction. A lower value will result
            in better recall, while a higher value will result in better precision. Do not use this parameter with the
            goal to combat the inevitably many duplicates that an SSD will produce, the subsequent non-maximum suppression
            stage will take care of those. Defaults to 0.5.
        iou_threshold (float, optional): `None` or a float in [0,1]. If `None`, no non-maximum suppression will be
            performed. If not `None`, greedy NMS will be performed after the confidence thresholding stage, meaning
            all boxes with a Jaccard similarity of greater than `iou_threshold` with a locally maximal box will be removed
            from the set of predictions, where 'maximal' refers to the box score. Defaults to 0.45.
        top_k (int, optional): 'all' or an integer with number of highest scoring predictions to be kept for each batch item
            after the non-maximum suppression stage. Defaults to 'all', in which case all predictions left after the NMS stage
            will be kept.
        input_coords (str, optional): The box coordinate format that the model outputs. Can be either 'centroids'
            for the format `(cx, cy, w, h)` (box center coordinates, width, and height), 'minmax' for the format
            `(xmin, xmax, ymin, ymax)`, or 'corners' for the format `(xmin, ymin, xmax, ymax)`. Defaults to 'centroids'.
        normalize_coords (bool, optional): Set to `True` if the model outputs relative coordinates (i.e. coordinates in [0,1])
            and you wish to transform these relative coordinates back to absolute coordinates. If the model outputs
            relative coordinates, but you do not want to convert them back to absolute coordinates, set this to `False`.
            Do not set this to `True` if the model already outputs absolute coordinates, as that would result in incorrect
            coordinates. Requires `img_height` and `img_width` if set to `True`. Defaults to `False`.
        img_height (int, optional): The height of the input images. Only needed if `normalize_coords` is `True`.
        img_width (int, optional): The width of the input images. Only needed if `normalize_coords` is `True`.

    Returns:
        A python list of length `batch_size` where each list element represents the predicted boxes
        for one image and contains a Numpy array of shape `(boxes, 6)` where each row is a box prediction for
        a non-background class for the respective image in the format `[class_id, confidence, xmin, xmax, ymin, ymax]`.
    '''
    if normalize_coords and ((img_height is None) or (img_width is None)):
        raise ValueError("If relative box coordinates are supposed to be converted to absolute coordinates, the decoder needs the image size in order to decode the predictions, but `img_height == {}` and `img_width == {}`".format(img_height, img_width))

    # 1: Convert the classes from one-hot encoding to their class ID
    y_pred_converted = np.copy(y_pred[:,:,-14:-8]) # Slice out the four offset predictions plus two elements whereto we'll write the class IDs and confidences in the next step
    y_pred_converted[:,:,0] = np.argmax(y_pred[:,:,:-12], axis=-1) # The indices of the highest confidence values in the one-hot class vectors are the class ID
    y_pred_converted[:,:,1] = np.amax(y_pred[:,:,:-12], axis=-1) # Store the confidence values themselves, too

    # 2: Convert the box coordinates from the predicted anchor box offsets to predicted absolute coordinates
    if input_coords == 'centroids':
        y_pred_converted[:,:,[4,5]] = np.exp(y_pred_converted[:,:,[4,5]] * y_pred[:,:,[-2,-1]]) # exp(ln(w(pred)/w(anchor)) / w_variance * w_variance) == w(pred) / w(anchor), exp(ln(h(pred)/h(anchor)) / h_variance * h_variance) == h(pred) / h(anchor)
        y_pred_converted[:,:,[4,5]] *= y_pred[:,:,[-6,-5]] # (w(pred) / w(anchor)) * w(anchor) == w(pred), (h(pred) / h(anchor)) * h(anchor) == h(pred)
        y_pred_converted[:,:,[2,3]] *= y_pred[:,:,[-4,-3]] * y_pred[:,:,[-6,-5]] # (delta_cx(pred) / w(anchor) / cx_variance) * cx_variance * w(anchor) == delta_cx(pred), (delta_cy(pred) / h(anchor) / cy_variance) * cy_variance * h(anchor) == delta_cy(pred)
        y_pred_converted[:,:,[2,3]] += y_pred[:,:,[-8,-7]] # delta_cx(pred) + cx(anchor) == cx(pred), delta_cy(pred) + cy(anchor) == cy(pred)
        y_pred_converted = convert_coordinates(y_pred_converted, start_index=-4, conversion='centroids2corners')
    elif input_coords == 'minmax':
        y_pred_converted[:,:,2:] *= y_pred[:,:,-4:] # delta(pred) / size(anchor) / variance * variance == delta(pred) / size(anchor) for all four coordinates, where 'size' refers to w or h, respectively
        y_pred_converted[:,:,[2,3]] *= np.expand_dims(y_pred[:,:,-7] - y_pred[:,:,-8], axis=-1) # delta_xmin(pred) / w(anchor) * w(anchor) == delta_xmin(pred), delta_xmax(pred) / w(anchor) * w(anchor) == delta_xmax(pred)
        y_pred_converted[:,:,[4,5]] *= np.expand_dims(y_pred[:,:,-5] - y_pred[:,:,-6], axis=-1) # delta_ymin(pred) / h(anchor) * h(anchor) == delta_ymin(pred), delta_ymax(pred) / h(anchor) * h(anchor) == delta_ymax(pred)
        y_pred_converted[:,:,2:] += y_pred[:,:,-8:-4] # delta(pred) + anchor == pred for all four coordinates
        y_pred_converted = convert_coordinates(y_pred_converted, start_index=-4, conversion='minmax2corners')
    elif input_coords == 'corners':
        y_pred_converted[:,:,2:] *= y_pred[:,:,-4:] # delta(pred) / size(anchor) / variance * variance == delta(pred) / size(anchor) for all four coordinates, where 'size' refers to w or h, respectively
        y_pred_converted[:,:,[2,4]] *= np.expand_dims(y_pred[:,:,-6] - y_pred[:,:,-8], axis=-1) # delta_xmin(pred) / w(anchor) * w(anchor) == delta_xmin(pred), delta_xmax(pred) / w(anchor) * w(anchor) == delta_xmax(pred)
        y_pred_converted[:,:,[3,5]] *= np.expand_dims(y_pred[:,:,-5] - y_pred[:,:,-7], axis=-1) # delta_ymin(pred) / h(anchor) * h(anchor) == delta_ymin(pred), delta_ymax(pred) / h(anchor) * h(anchor) == delta_ymax(pred)
        y_pred_converted[:,:,2:] += y_pred[:,:,-8:-4] # delta(pred) + anchor == pred for all four coordinates
    else:
        raise ValueError("Unexpected value for `coords`. Supported values are 'minmax', 'corners' and 'centroids'.")

    # 3: If the model predicts normalized box coordinates and they are supposed to be converted back to absolute coordinates, do that
    if normalize_coords:
        y_pred_converted[:,:,[2,4]] *= img_width # Convert xmin, xmax back to absolute coordinates
        y_pred_converted[:,:,[3,5]] *= img_height # Convert ymin, ymax back to absolute coordinates

    # 4: Decode our huge `(batch, #boxes, 6)` tensor into a list of length `batch` where each list entry is an array containing only the positive predictions
    y_pred_decoded = []
    for batch_item in y_pred_converted: # For each image in the batch...
        boxes = batch_item[np.nonzero(batch_item[:,0])] # ...get all boxes that don't belong to the background class,...
        boxes = boxes[boxes[:,1] >= confidence_thresh] # ...then filter out those positive boxes for which the prediction confidence is too low and after that...
        if iou_threshold: # ...if an IoU threshold is set...
            boxes = _greedy_nms2(boxes, iou_threshold=iou_threshold, coords='corners') # ...perform NMS on the remaining boxes.
        if top_k != 'all' and boxes.shape[0] > top_k: # If we have more than `top_k` results left at this point...
            top_k_indices = np.argpartition(boxes[:,1], kth=boxes.shape[0]-top_k, axis=0)[boxes.shape[0]-top_k:] # ...get the indices of the `top_k` highest-scoring boxes...
            boxes = boxes[top_k_indices] # ...and keep only those boxes...
        y_pred_decoded.append(boxes) # ...and now that we're done, append the array of final predictions for this batch item to the output list

    return y_pred_decoded

class SSDBoxEncoder:
    '''
    A class to transform ground truth labels for object detection in images
    (2D bounding box coordinates and class labels) to the format required for
    training an SSD model, and to transform predictions of the SSD model back
    to the original format of the input labels.

    In the process of encoding ground truth labels, a template of anchor boxes
    is being built, which are subsequently matched to the ground truth boxes
    via an intersection-over-union threshold criterion.
    '''

    def __init__(self,
                 img_height,
                 img_width,
                 n_classes,
                 predictor_sizes,
                 min_scale=0.1,
                 max_scale=0.9,
                 scales=None,
                 aspect_ratios_global=[0.5, 1.0, 2.0],
                 aspect_ratios_per_layer=None,
                 two_boxes_for_ar1=True,
                 steps=None,
                 offsets=None,
                 limit_boxes=False,
                 variances=[1.0, 1.0, 1.0, 1.0],
                 pos_iou_threshold=0.5,
                 neg_iou_threshold=0.3,
                 coords='centroids',
                 normalize_coords=False):
        '''
        Arguments:
            img_height (int): The height of the input images.
            img_width (int): The width of the input images.
            n_classes (int): The number of positive classes, e.g. 20 for Pascal VOC, 80 for MS COCO.
            predictor_sizes (list): A list of int-tuples of the format `(height, width)`
                containing the output heights and widths of the convolutional predictor layers.
            min_scale (float, optional): The smallest scaling factor for the size of the anchor boxes as a fraction
                of the shorter side of the input images. Defaults to 0.1. Note that you should set the scaling factors
                such that the resulting anchor box sizes correspond to the sizes of the objects you are trying
                to detect. Must be >0.
            max_scale (float, optional): The largest scaling factor for the size of the anchor boxes as a fraction
                of the shorter side of the input images. All scaling factors between the smallest and the
                largest will be linearly interpolated. Note that the second to last of the linearly interpolated
                scaling factors will actually be the scaling factor for the last predictor layer, while the last
                scaling factor is used for the second box for aspect ratio 1 in the last predictor layer
                if `two_boxes_for_ar1` is `True`. Defaults to 0.9. Note that you should set the scaling factors
                such that the resulting anchor box sizes correspond to the sizes of the objects you are trying
                to detect. Must be greater than or equal to `min_scale`.
            scales (list, optional): A list of floats >0 containing scaling factors per convolutional predictor layer.
                This list must be one element longer than the number of predictor layers. The first `k` elements are the
                scaling factors for the `k` predictor layers, while the last element is used for the second box
                for aspect ratio 1 in the last predictor layer if `two_boxes_for_ar1` is `True`. This additional
                last scaling factor must be passed either way, even if it is not being used.
                Defaults to `None`. If a list is passed, this argument overrides `min_scale` and
                `max_scale`. All scaling factors must be greater than zero. Note that you should set the scaling factors
                such that the resulting anchor box sizes correspond to the sizes of the objects you are trying
                to detect.
            aspect_ratios_global (list, optional): The list of aspect ratios for which anchor boxes are to be
                generated. This list is valid for all prediction layers. Defaults to [0.5, 1.0, 2.0]. Note that you should
                set the aspect ratios such that the resulting anchor box shapes very roughly correspond to the shapes of the
                objects you are trying to detect. For many standard detection tasks, the default values will yield good
                results.
            aspect_ratios_per_layer (list, optional): A list containing one aspect ratio list for each prediction layer.
                If a list is passed, it overrides `aspect_ratios_global`. Defaults to `None`. Note that you should
                set the aspect ratios such that the resulting anchor box shapes very roughly correspond to the shapes of the
                objects you are trying to detect. For many standard detection tasks, the default values will yield good
                results.
            two_boxes_for_ar1 (bool, optional): Only relevant for aspect ratios lists that contain 1. Will be ignored otherwise.
                If `True`, two anchor boxes will be generated for aspect ratio 1. The first will be generated
                using the scaling factor for the respective layer, the second one will be generated using
                geometric mean of said scaling factor and next bigger scaling factor.
            steps (list, optional): `None` or a list with as many elements as there are predictor layers. The elements can be
                either ints/floats or tuples of two ints/floats. These numbers represent for each predictor layer how many
                pixels apart the anchor box center points should be vertically and horizontally along the spatial grid over
                the image. If the list contains ints/floats, then that value will be used for both spatial dimensions.
                If the list contains tuples of two ints/floats, then they represent `(step_height, step_width)`.
                If no steps are provided, then they will be computed such that the anchor box center points will form an
                equidistant grid within the image dimensions. Defaults to `None`.
            offsets (list, optional): `None` or a list with as many elements as there are predictor layers. The elements can be
                either floats or tuples of two floats. These numbers represent for each predictor layer how many
                pixels from the top and left boarders of the image the top-most and left-most anchor box center points should be
                as a fraction of `steps`. The last bit is important: The offsets are not absolute pixel values, but fractions
                of the step size specified in the `steps` argument. If the list contains floats, then that value will
                be used for both spatial dimensions. If the list contains tuples of two floats, then they represent
                `(vertical_offset, horizontal_offset)`. If no offsets are provided, then they will default to 0.5 of the step size.
                Defaults to `None`.
            limit_boxes (bool, optional): If `True`, limits box coordinates to stay within image boundaries.
                Defaults to `True`.
            variances (list, optional): A list of 4 floats >0 with scaling factors (actually it's not factors but divisors
                to be precise) for the encoded ground truth (i.e. target) box coordinates. A variance value of 1.0 would apply
                no scaling at all to the targets, while values in (0,1) upscale the encoded targets and values greater than 1.0
                downscale the encoded targets. If you want to reproduce the configuration of the original SSD,
                set this to `[0.1, 0.1, 0.2, 0.2]`, provided the coordinate format is 'centroids'. Defaults to `[1.0, 1.0, 1.0, 1.0]`.
            pos_iou_threshold (float, optional): The intersection-over-union similarity threshold that must be
                met in order to match a given ground truth box to a given anchor box. Defaults to 0.5.
            neg_iou_threshold (float, optional): The maximum allowed intersection-over-union similarity of an
                anchor box with any ground truth box to be labeled a negative (i.e. background) box. If an
                anchor box is neither a positive, nor a negative box, it will be ignored during training.
            coords (str, optional): The box coordinate format to be used internally in the model (i.e. this is not the input format
                of the ground truth labels). Can be either 'centroids' for the format `(cx, cy, w, h)` (box center coordinates, width,
                and height), 'minmax' for the format `(xmin, xmax, ymin, ymax)`, or 'corners' for the format `(xmin, ymin, xmax, ymax)`.
                Defaults to 'centroids'.
            normalize_coords (bool, optional): If `True`, the encoder uses relative instead of absolute coordinates.
                This means instead of using absolute tartget coordinates, the encoder will scale all coordinates to be within [0,1].
                This way learning becomes independent of the input image size. Defaults to `False`.
        '''
        predictor_sizes = np.array(predictor_sizes)
        if len(predictor_sizes.shape) == 1:
            predictor_sizes = np.expand_dims(predictor_sizes, axis=0)

        if (min_scale is None or max_scale is None) and scales is None:
            raise ValueError("Either `min_scale` and `max_scale` or `scales` need to be specified.")

        if scales:
            if (len(scales) != len(predictor_sizes)+1): # Must be two nested `if` statements since `list` and `bool` cannot be combined by `&`
                raise ValueError("It must be either scales is None or len(scales) == len(predictor_sizes)+1, but len(scales) == {} and len(predictor_sizes)+1 == {}".format(len(scales), len(predictor_sizes)+1))
            scales = np.array(scales)
            if np.any(scales <= 0):
                raise ValueError("All values in `scales` must be greater than 0, but the passed list of scales is {}".format(scales))
        else: # If no list of scales was passed, we need to make sure that `min_scale` and `max_scale` are valid values.
            if not 0 < min_scale <= max_scale:
                raise ValueError("It must be 0 < min_scale <= max_scale, but it is min_scale = {} and max_scale = {}".format(min_scale, max_scale))

        if not (aspect_ratios_per_layer is None):
            if (len(aspect_ratios_per_layer) != len(predictor_sizes)): # Must be two nested `if` statements since `list` and `bool` cannot be combined by `&`
                raise ValueError("It must be either aspect_ratios_per_layer is None or len(aspect_ratios_per_layer) == len(predictor_sizes), but len(aspect_ratios_per_layer) == {} and len(predictor_sizes) == {}".format(len(aspect_ratios_per_layer), len(predictor_sizes)))
            for aspect_ratios in aspect_ratios_per_layer:
                if np.any(np.array(aspect_ratios) <= 0):
                    raise ValueError("All aspect ratios must be greater than zero.")
        else:
            if (aspect_ratios_global is None):
                raise ValueError("At least one of `aspect_ratios_global` and `aspect_ratios_per_layer` must not be `None`.")
            if np.any(np.array(aspect_ratios_global) <= 0):
                raise ValueError("All aspect ratios must be greater than zero.")

        if len(variances) != 4:
            raise ValueError("4 variance values must be pased, but {} values were received.".format(len(variances)))
        variances = np.array(variances)
        if np.any(variances <= 0):
            raise ValueError("All variances must be >0, but the variances given are {}".format(variances))

        if neg_iou_threshold > pos_iou_threshold:
            raise ValueError("It cannot be `neg_iou_threshold > pos_iou_threshold`.")

        if not (coords == 'minmax' or coords == 'centroids' or coords == 'corners'):
            raise ValueError("Unexpected value for `coords`. Supported values are 'minmax', 'corners' and 'centroids'.")

        if (not (steps is None)) and (len(steps) != len(predictor_sizes)):
            raise ValueError("You must provide at least one step value per predictor layer.")

        if (not (offsets is None)) and (len(offsets) != len(predictor_sizes)):
            raise ValueError("You must provide at least one offset value per predictor layer.")

        self.img_height = img_height
        self.img_width = img_width
        self.n_classes = n_classes + 1
        self.predictor_sizes = predictor_sizes
        self.min_scale = min_scale
        self.max_scale = max_scale
        if (scales is None):
            self.scales = np.linspace(self.min_scale, self.max_scale, len(self.predictor_sizes)+1)
        else:
            # If a list of scales is given explicitly, we'll use that instead of computing it from `min_scale` and `max_scale`.
            self.scales = scales
        if (aspect_ratios_per_layer is None):
            self.aspect_ratios = [aspect_ratios_global] * len(predictor_sizes)
        else:
            # If aspect ratios are given per layer, we'll use those.
            self.aspect_ratios = aspect_ratios_per_layer
        self.two_boxes_for_ar1 = two_boxes_for_ar1
        if (not steps is None):
            self.steps = steps
        else:
            self.steps = [None] * len(predictor_sizes)
        if (not offsets is None):
            self.offsets = offsets
        else:
            self.offsets = [None] * len(predictor_sizes)
        self.limit_boxes = limit_boxes
        self.variances = variances
        self.pos_iou_threshold = pos_iou_threshold
        self.neg_iou_threshold = neg_iou_threshold
        self.coords = coords
        self.normalize_coords = normalize_coords

        # Compute the number of boxes per cell.
        if aspect_ratios_per_layer:
            self.n_boxes = []
            for aspect_ratios in aspect_ratios_per_layer:
                if (1 in aspect_ratios) & two_boxes_for_ar1:
                    self.n_boxes.append(len(aspect_ratios) + 1)
                else:
                    self.n_boxes.append(len(aspect_ratios))
        else:
            if (1 in aspect_ratios_global) & two_boxes_for_ar1:
                self.n_boxes = len(aspect_ratios_global) + 1
            else:
                self.n_boxes = len(aspect_ratios_global)

        # Compute the anchor boxes for all the predictor layers. We only have to do this once
        # as the anchor boxes depend only on the model configuration, not on the input data.
        # For each conv predictor layer (i.e. for each scale factor)the tensors for that lauer's
        # anchor boxes will have the shape `(feature_map_height, feature_map_width, n_boxes, 4)`.
        self.boxes_list = [] # This will contain the anchor boxes for each predicotr layer.

        self.wh_list_diag = [] # Box widths and heights for each predictor layer
        self.steps_diag = [] # Horizontal and vertical distances between any two boxes for each predictor layer
        self.offsets_diag = [] # Offsets for each predictor layer
        self.centers_diag = [] # Anchor box center points as `(cy, cx)` for each predictor layer

        for i in range(len(self.predictor_sizes)):
            boxes, center, wh, step, offset = self.generate_anchor_boxes_for_layer(feature_map_size=self.predictor_sizes[i],
                                                                                   aspect_ratios=self.aspect_ratios[i],
                                                                                   this_scale=self.scales[i],
                                                                                   next_scale=self.scales[i+1],
                                                                                   this_steps=self.steps[i],
                                                                                   this_offsets=self.offsets[i],
                                                                                   diagnostics=True)
            self.boxes_list.append(boxes)
            self.wh_list_diag.append(wh)
            self.steps_diag.append(step)
            self.offsets_diag.append(offset)
            self.centers_diag.append(center)

    def generate_anchor_boxes_for_layer(self,
                                        feature_map_size,
                                        aspect_ratios,
                                        this_scale,
                                        next_scale,
                                        this_steps=None,
                                        this_offsets=None,
                                        diagnostics=False):
        '''
        Compute an array of the spatial positions and sizes of the anchor boxes for one predictor layer
        of size `feature_map_size == [feature_map_height, feature_map_width]`.

        Arguments:
            feature_map_size (tuple): A list or tuple `[feature_map_height, feature_map_width]` with the spatial
                dimensions of the feature map for which to generate the anchor boxes.
            aspect_ratios (list): A list of floats, the aspect ratios for which anchor boxes are to be generated.
                All list elements must be unique.
            this_scale (float): A float in [0, 1], the scaling factor for the size of the generate anchor boxes
                as a fraction of the shorter side of the input image.
            next_scale (float): A float in [0, 1], the next larger scaling factor. Only relevant if
                `self.two_boxes_for_ar1 == True`.
            diagnostics (bool, optional): If true, the following additional outputs will be returned:
                1) A list of the center point `x` and `y` coordinates for each spatial location.
                2) A list containing `(width, height)` for each box aspect ratio.
                3) A tuple containing `(step_height, step_width)`
                4) A tuple containing `(offset_height, offset_width)`
                This information can be useful to understand in just a few numbers what the generated grid of
                anchor boxes actually looks like, i.e. how large the different boxes are and how dense
                their spatial distribution is, in order to determine whether the box grid covers the input images
                appropriately and whether the box sizes are appropriate to fit the sizes of the objects
                to be detected.

        Returns:
            A 4D Numpy tensor of shape `(feature_map_height, feature_map_width, n_boxes_per_cell, 4)` where the
            last dimension contains `(xmin, xmax, ymin, ymax)` for each anchor box in each cell of the feature map.
        '''
        # Compute box width and height for each aspect ratio.

        # The shorter side of the image will be used to compute `w` and `h` using `scale` and `aspect_ratios`.
        size = min(self.img_height, self.img_width)
        # Compute the box widths and and heights for all aspect ratios
        wh_list = []
        for ar in aspect_ratios:
            if (ar == 1):
                # Compute the regular anchor box for aspect ratio 1.
                box_height = box_width = this_scale * size
                wh_list.append((box_width, box_height))
                if self.two_boxes_for_ar1:
                    # Compute one slightly larger version using the geometric mean of this scale value and the next.
                    box_height = box_width = np.sqrt(this_scale * next_scale) * size
                    wh_list.append((box_width, box_height))
            else:
                box_width = this_scale * size * np.sqrt(ar)
                box_height = this_scale * size / np.sqrt(ar)
                wh_list.append((box_width, box_height))
        wh_list = np.array(wh_list)
        n_boxes = len(wh_list)

        # Compute the grid of box center points. They are identical for all aspect ratios.

        # Compute the step sizes, i.e. how far apart the anchor box center points will be vertically and horizontally.
        if (this_steps is None):
            step_height = self.img_height / feature_map_size[0]
            step_width = self.img_width / feature_map_size[1]
        else:
            if isinstance(this_steps, (list, tuple)) and (len(this_steps) == 2):
                step_height = this_steps[0]
                step_width = this_steps[1]
            elif isinstance(this_steps, (int, float)):
                step_height = this_steps
                step_width = this_steps
        # Compute the offsets, i.e. at what pixel values the first anchor box center point will be from the top and from the left of the image.
        if (this_offsets is None):
            offset_height = 0.5
            offset_width = 0.5
        else:
            if isinstance(this_offsets, (list, tuple)) and (len(this_offsets) == 2):
                offset_height = this_offsets[0]
                offset_width = this_offsets[1]
            elif isinstance(this_offsets, (int, float)):
                offset_height = this_offsets
                offset_width = this_offsets
        # Now that we have the offsets and step sizes, compute the grid of anchor box center points.
        cy = np.linspace(offset_height * step_height, (offset_height + feature_map_size[0] - 1) * step_height, feature_map_size[0])
        cx = np.linspace(offset_width * step_width, (offset_width + feature_map_size[1] - 1) * step_width, feature_map_size[1])
        cx_grid, cy_grid = np.meshgrid(cx, cy)
        cx_grid = np.expand_dims(cx_grid, -1) # This is necessary for np.tile() to do what we want further down
        cy_grid = np.expand_dims(cy_grid, -1) # This is necessary for np.tile() to do what we want further down

        # Create a 4D tensor template of shape `(feature_map_height, feature_map_width, n_boxes, 4)`
        # where the last dimension will contain `(cx, cy, w, h)`
        boxes_tensor = np.zeros((feature_map_size[0], feature_map_size[1], n_boxes, 4))

        boxes_tensor[:, :, :, 0] = np.tile(cx_grid, (1, 1, n_boxes)) # Set cx
        boxes_tensor[:, :, :, 1] = np.tile(cy_grid, (1, 1, n_boxes)) # Set cy
        boxes_tensor[:, :, :, 2] = wh_list[:, 0] # Set w
        boxes_tensor[:, :, :, 3] = wh_list[:, 1] # Set h

        # Convert `(cx, cy, w, h)` to `(xmin, ymin, xmax, ymax)`
        boxes_tensor = convert_coordinates(boxes_tensor, start_index=0, conversion='centroids2corners')

        # If `limit_boxes` is enabled, clip the coordinates to lie within the image boundaries
        if self.limit_boxes:
            x_coords = boxes_tensor[:,:,:,[0, 2]]
            x_coords[x_coords >= self.img_width] = self.img_width - 1
            x_coords[x_coords < 0] = 0
            boxes_tensor[:,:,:,[0, 2]] = x_coords
            y_coords = boxes_tensor[:,:,:,[1, 3]]
            y_coords[y_coords >= self.img_height] = self.img_height - 1
            y_coords[y_coords < 0] = 0
            boxes_tensor[:,:,:,[1, 3]] = y_coords

        # `normalize_coords` is enabled, normalize the coordinates to be within [0,1]
        if self.normalize_coords:
            boxes_tensor[:, :, :, [0, 2]] /= self.img_width
            boxes_tensor[:, :, :, [1, 3]] /= self.img_height

        # TODO: Implement box limiting directly for `(cx, cy, w, h)` so that we don't have to unnecessarily convert back and forth.
        if self.coords == 'centroids':
            # Convert `(xmin, ymin, xmax, ymax)` back to `(cx, cy, w, h)`.
            boxes_tensor = convert_coordinates(boxes_tensor, start_index=0, conversion='corners2centroids')
        elif self.coords == 'minmax':
            # Convert `(xmin, ymin, xmax, ymax)` to `(xmin, xmax, ymin, ymax).
            boxes_tensor = convert_coordinates(boxes_tensor, start_index=0, conversion='corners2minmax')

        if diagnostics:
            return boxes_tensor, (cy, cx), wh_list, (step_height, step_width), (offset_height, offset_width)
        else:
            return boxes_tensor

    def generate_encode_template(self, batch_size, diagnostics=False):
        '''
        Produces an encoding template for the ground truth label tensor for a given batch.

        Note that all tensor creation, reshaping and concatenation operations performed in this function
        and the sub-functions it calls are identical to those performed inside the conv net model. This, of course,
        must be the case in order to preserve the spatial meaning of each box prediction, but it's useful to make
        yourself aware of this fact and why it is necessary.

        In other words, the boxes in `y_encoded` must have a specific order in order correspond to the right spatial
        positions and scales of the boxes predicted by the model. The sequence of operations here ensures that `y_encoded`
        has this specific form.

        Arguments:
            batch_size (int): The batch size.
            diagnostics (bool, optional): See the documnentation for `generate_anchor_boxes()`. The diagnostic output
                here is similar, just for all predictor conv layers.

        Returns:
            A Numpy array of shape `(batch_size, #boxes, #classes + 12)`, the template into which to encode
            the ground truth labels for training. The last axis has length `#classes + 12` because the model
            output contains not only the 4 predicted box coordinate offsets, but also the 4 coordinates for
            the anchor boxes and the 4 variance values.
        '''
        # Tile the anchor boxes for each predictor layer across all batch items.
        boxes_batch = []
        for boxes in self.boxes_list:
            # Prepend one dimension to `self.boxes_list` to account for the batch size and tile it along.
            # The result will be a 5D tensor of shape `(batch_size, feature_map_height, feature_map_width, n_boxes, 4)`
            boxes = np.expand_dims(boxes, axis=0)
            boxes = np.tile(boxes, (batch_size, 1, 1, 1, 1))

            # Now reshape the 5D tensor above into a 3D tensor of shape
            # `(batch, feature_map_height * feature_map_width * n_boxes, 4)`. The resulting
            # order of the tensor content will be identical to the order obtained from the reshaping operation
            # in our Keras model (we're using the Tensorflow backend, and tf.reshape() and np.reshape()
            # use the same default index order, which is C-like index ordering)
            boxes = np.reshape(boxes, (batch_size, -1, 4))
            boxes_batch.append(boxes)

        # Concatenate the anchor tensors from the individual layers to one.
        boxes_tensor = np.concatenate(boxes_batch, axis=1)

        # 3: Create a template tensor to hold the one-hot class encodings of shape `(batch, #boxes, #classes)`
        #    It will contain all zeros for now, the classes will be set in the matching process that follows
        classes_tensor = np.zeros((batch_size, boxes_tensor.shape[1], self.n_classes))

        # 4: Create a tensor to contain the variances. This tensor has the same shape as `boxes_tensor` and simply
        #    contains the same 4 variance values for every position in the last axis.
        variances_tensor = np.zeros_like(boxes_tensor)
        variances_tensor += self.variances # Long live broadcasting

        # 4: Concatenate the classes, boxes and variances tensors to get our final template for y_encoded. We also need
        #    another tensor of the shape of `boxes_tensor` as a space filler so that `y_encode_template` has the same
        #    shape as the SSD model output tensor. The content of this tensor is irrelevant, we'll just use
        #    `boxes_tensor` a second time.
        y_encode_template = np.concatenate((classes_tensor, boxes_tensor, boxes_tensor, variances_tensor), axis=2)

        if diagnostics:
            return y_encode_template, self.centers_diag, self.wh_list_diag, self.steps_diag, self.offsets_diag
        else:
            return y_encode_template

    def encode_y(self, ground_truth_labels, diagnostics=False):
        '''
        Convert ground truth bounding box data into a suitable format to train an SSD model.

        For each image in the batch, each ground truth bounding box belonging to that image will be compared against each
        anchor box in a template with respect to their jaccard similarity. If the jaccard similarity is greater than
        or equal to the set threshold, the boxes will be matched, meaning that the ground truth box coordinates and class
        will be written to the the specific position of the matched anchor box in the template.

        The class for all anchor boxes for which there was no match with any ground truth box will be set to the
        background class, except for those anchor boxes whose IoU similarity with any ground truth box is higher than
        the set negative threshold (see the `neg_iou_threshold` argument in `__init__()`).

        Arguments:
            ground_truth_labels (list): A python list of length `batch_size` that contains one 2D Numpy array
                for each batch image. Each such array has `k` rows for the `k` ground truth bounding boxes belonging
                to the respective image, and the data for each ground truth bounding box has the format
                `(class_id, xmin, ymin, xmax, ymax)`, and `class_id` must be an integer greater than 0 for all boxes
                as class_id 0 is reserved for the background class.
            diagnostics (bool, optional): If `True`, not only the encoded ground truth tensor will be returned,
                but also a copy of it with anchor box coordinates in place of the ground truth coordinates.
                This can be very useful if you want to visualize which anchor boxes got matched to which ground truth
                boxes.

        Returns:
            `y_encoded`, a 3D numpy array of shape `(batch_size, #boxes, #classes + 4 + 4 + 4)` that serves as the
            ground truth label tensor for training, where `#boxes` is the total number of boxes predicted by the
            model per image, and the classes are one-hot-encoded. The four elements after the class vecotrs in
            the last axis are the box coordinates, the next four elements after that are just dummy elements, and
            the last four elements are the variances.
        '''

        # 1: Generate the template for y_encoded
        y_encode_template = self.generate_encode_template(batch_size=len(ground_truth_labels), diagnostics=False)
        y_encoded = np.copy(y_encode_template) # We'll write the ground truth box data to this array

        # 2: Match the boxes from `ground_truth_labels` to the anchor boxes in `y_encode_template`
        #    and for each matched box record the ground truth coordinates in `y_encoded`.
        #    Every time there is no match for a anchor box, record `class_id` 0 in `y_encoded` for that anchor box.

        class_vector = np.eye(self.n_classes) # An identity matrix that we'll use as one-hot class vectors

        for i in range(y_encode_template.shape[0]): # For each batch item...
            available_boxes = np.ones((y_encode_template.shape[1])) # 1 for all anchor boxes that are not yet matched to a ground truth box, 0 otherwise
            negative_boxes = np.ones((y_encode_template.shape[1])) # 1 for all negative boxes, 0 otherwise
            for true_box in ground_truth_labels[i]: # For each ground truth box belonging to the current batch item...
                true_box = true_box.astype(np.float)
                if abs(true_box[3] - true_box[1] < 0.001) or abs(true_box[4] - true_box[2] < 0.001): continue # Protect ourselves against bad ground truth data: boxes with width or height equal to zero
                if self.normalize_coords:
                    true_box[[1,3]] /= self.img_width # Normalize xmin and xmax to be within [0,1]
                    true_box[[2,4]] /= self.img_height # Normalize ymin and ymax to be within [0,1]
                if self.coords == 'centroids':
                    true_box = convert_coordinates(true_box, start_index=1, conversion='corners2centroids')
                elif self.coords == 'minmax':
                    true_box = convert_coordinates(true_box, start_index=1, conversion='corners2minmax')
                similarities = iou(y_encode_template[i,:,-12:-8], true_box[1:], coords=self.coords) # The iou similarities for all anchor boxes
                negative_boxes[similarities >= self.neg_iou_threshold] = 0 # If a negative box gets an IoU match >= `self.neg_iou_threshold`, it's no longer a valid negative box
                similarities *= available_boxes # Filter out anchor boxes which aren't available anymore (i.e. already matched to a different ground truth box)
                available_and_thresh_met = np.copy(similarities)
                available_and_thresh_met[available_and_thresh_met < self.pos_iou_threshold] = 0 # Filter out anchor boxes which don't meet the iou threshold
                assign_indices = np.nonzero(available_and_thresh_met)[0] # Get the indices of the left-over anchor boxes to which we want to assign this ground truth box
                if len(assign_indices) > 0: # If we have any matches
                    y_encoded[i,assign_indices,:-8] = np.concatenate((class_vector[int(true_box[0])], true_box[1:]), axis=0) # Write the ground truth box coordinates and class to all assigned anchor box positions. Remember that the last four elements of `y_encoded` are just dummy entries.
                    available_boxes[assign_indices] = 0 # Make the assigned anchor boxes unavailable for the next ground truth box
                else: # If we don't have any matches
                    best_match_index = np.argmax(similarities) # Get the index of the best iou match out of all available boxes
                    y_encoded[i,best_match_index,:-8] = np.concatenate((class_vector[int(true_box[0])], true_box[1:]), axis=0) # Write the ground truth box coordinates and class to the best match anchor box position
                    available_boxes[best_match_index] = 0 # Make the assigned anchor box unavailable for the next ground truth box
                    negative_boxes[best_match_index] = 0 # The assigned anchor box is no longer a negative box
            # Set the classes of all remaining available anchor boxes to class zero
            background_class_indices = np.nonzero(negative_boxes)[0]
            y_encoded[i,background_class_indices,0] = 1

        # 3: Convert absolute box coordinates to offsets from the anchor boxes and normalize them
        if self.coords == 'centroids':
            y_encoded[:,:,[-12,-11]] -= y_encode_template[:,:,[-12,-11]] # cx(gt) - cx(anchor), cy(gt) - cy(anchor)
            y_encoded[:,:,[-12,-11]] /= y_encode_template[:,:,[-10,-9]] * y_encode_template[:,:,[-4,-3]] # (cx(gt) - cx(anchor)) / w(anchor) / cx_variance, (cy(gt) - cy(anchor)) / h(anchor) / cy_variance
            y_encoded[:,:,[-10,-9]] /= y_encode_template[:,:,[-10,-9]] # w(gt) / w(anchor), h(gt) / h(anchor)
            y_encoded[:,:,[-10,-9]] = np.log(y_encoded[:,:,[-10,-9]]) / y_encode_template[:,:,[-2,-1]] # ln(w(gt) / w(anchor)) / w_variance, ln(h(gt) / h(anchor)) / h_variance (ln == natural logarithm)
        elif self.coords == 'corners':
            y_encoded[:,:,-12:-8] -= y_encode_template[:,:,-12:-8] # (gt - anchor) for all four coordinates
            y_encoded[:,:,[-12,-10]] /= np.expand_dims(y_encode_template[:,:,-10] - y_encode_template[:,:,-12], axis=-1) # (xmin(gt) - xmin(anchor)) / w(anchor), (xmax(gt) - xmax(anchor)) / w(anchor)
            y_encoded[:,:,[-11,-9]] /= np.expand_dims(y_encode_template[:,:,-9] - y_encode_template[:,:,-11], axis=-1) # (ymin(gt) - ymin(anchor)) / h(anchor), (ymax(gt) - ymax(anchor)) / h(anchor)
            y_encoded[:,:,-12:-8] /= y_encode_template[:,:,-4:] # (gt - anchor) / size(anchor) / variance for all four coordinates, where 'size' refers to w and h respectively
        else:
            y_encoded[:,:,-12:-8] -= y_encode_template[:,:,-12:-8] # (gt - anchor) for all four coordinates
            y_encoded[:,:,[-12,-11]] /= np.expand_dims(y_encode_template[:,:,-11] - y_encode_template[:,:,-12], axis=-1) # (xmin(gt) - xmin(anchor)) / w(anchor), (xmax(gt) - xmax(anchor)) / w(anchor)
            y_encoded[:,:,[-10,-9]] /= np.expand_dims(y_encode_template[:,:,-9] - y_encode_template[:,:,-10], axis=-1) # (ymin(gt) - ymin(anchor)) / h(anchor), (ymax(gt) - ymax(anchor)) / h(anchor)
            y_encoded[:,:,-12:-8] /= y_encode_template[:,:,-4:] # (gt - anchor) / size(anchor) / variance for all four coordinates, where 'size' refers to w and h respectively

        if diagnostics:
            # Here we'll save the matched anchor boxes (i.e. anchor boxes that were matched to a ground truth box, but keeping the anchor box coordinates).
            y_matched_anchors = np.copy(y_encoded)
            y_matched_anchors[:,:,-12:-8] = 0 # Keeping the anchor box coordinates means setting the offsets to zero.
            return y_encoded, y_matched_anchors
        else:
            return y_encoded

################################################################################
# Debugging tools, not relevant for normal use
################################################################################

# The functions below are for debugging, so you won't normally need them. That is,
# unless you need to debug your model, of course.

def decode_y_debug(y_pred,
                   confidence_thresh=0.01,
                   iou_threshold=0.45,
                   top_k=200,
                   input_coords='centroids',
                   normalize_coords=False,
                   img_height=None,
                   img_width=None,
                   variance_encoded_in_target=False):
    '''
    This decoder performs the same processing as `decode_y()`, but the output format for each left-over
    predicted box is `[box_id, class_id, confidence, xmin, ymin, xmax, ymax]`.

    That is, in addition to the usual data, each predicted box has the internal index of that box within
    the model (`box_id`) prepended to it. This allows you to know exactly which part of the model made a given
    box prediction; in particular, it allows you to know which predictor layer made a given prediction.
    This can be useful for debugging.

    Arguments:
        y_pred (array): The prediction output of the SSD model, expected to be a Numpy array
            of shape `(batch_size, #boxes, #classes + 4 + 4 + 4)`, where `#boxes` is the total number of
            boxes predicted by the model per image and the last axis contains
            `[one-hot vector for the classes, 4 predicted coordinate offsets, 4 anchor box coordinates, 4 variances]`.
        confidence_thresh (float, optional): A float in [0,1), the minimum classification confidence in a specific
            positive class in order to be considered for the non-maximum suppression stage for the respective class.
            A lower value will result in a larger part of the selection process being done by the non-maximum suppression
            stage, while a larger value will result in a larger part of the selection process happening in the confidence
            thresholding stage. Defaults to 0.01, following the paper.
        iou_threshold (float, optional): A float in [0,1]. All boxes with a Jaccard similarity of greater than `iou_threshold`
            with a locally maximal box will be removed from the set of predictions for a given class, where 'maximal' refers
            to the box score. Defaults to 0.45 following the paper.
        top_k (int, optional): The number of highest scoring predictions to be kept for each batch item after the
            non-maximum suppression stage. Defaults to 200, following the paper.
        input_coords (str, optional): The box coordinate format that the model outputs. Can be either 'centroids'
            for the format `(cx, cy, w, h)` (box center coordinates, width, and height), 'minmax' for the format
            `(xmin, xmax, ymin, ymax)`, or 'corners' for the format `(xmin, ymin, xmax, ymax)`. Defaults to 'centroids'.
        normalize_coords (bool, optional): Set to `True` if the model outputs relative coordinates (i.e. coordinates in [0,1])
            and you wish to transform these relative coordinates back to absolute coordinates. If the model outputs
            relative coordinates, but you do not want to convert them back to absolute coordinates, set this to `False`.
            Do not set this to `True` if the model already outputs absolute coordinates, as that would result in incorrect
            coordinates. Requires `img_height` and `img_width` if set to `True`. Defaults to `False`.
        img_height (int, optional): The height of the input images. Only needed if `normalize_coords` is `True`.
        img_width (int, optional): The width of the input images. Only needed if `normalize_coords` is `True`.

    Returns:
        A python list of length `batch_size` where each list element represents the predicted boxes
        for one image and contains a Numpy array of shape `(boxes, 7)` where each row is a box prediction for
        a non-background class for the respective image in the format `[box_id, class_id, confidence, xmin, ymin, xmax, ymax]`.
    '''
    if normalize_coords and ((img_height is None) or (img_width is None)):
        raise ValueError("If relative box coordinates are supposed to be converted to absolute coordinates, the decoder needs the image size in order to decode the predictions, but `img_height == {}` and `img_width == {}`".format(img_height, img_width))

    # 1: Convert the box coordinates from the predicted anchor box offsets to predicted absolute coordinates

    y_pred_decoded_raw = np.copy(y_pred[:,:,:-8]) # Slice out the classes and the four offsets, throw away the anchor coordinates and variances, resulting in a tensor of shape `[batch, n_boxes, n_classes + 4 coordinates]`

    if input_coords == 'centroids':
        if variance_encoded_in_target:
            # Decode the predicted box center x and y coordinates.
            y_pred_decoded_raw[:,:,[-4,-3]] = y_pred_decoded_raw[:,:,[-4,-3]] * y_pred[:,:,[-6,-5]] + y_pred[:,:,[-8,-7]]
            # Decode the predicted box width and heigt.
            y_pred_decoded_raw[:,:,[-2,-1]] = np.exp(y_pred_decoded_raw[:,:,[-2,-1]]) * y_pred[:,:,[-6,-5]]
        else:
            # Decode the predicted box center x and y coordinates.
            y_pred_decoded_raw[:,:,[-4,-3]] = y_pred_decoded_raw[:,:,[-4,-3]] * y_pred[:,:,[-6,-5]] * y_pred[:,:,[-4,-3]] + y_pred[:,:,[-8,-7]]
            # Decode the predicted box width and heigt.
            y_pred_decoded_raw[:,:,[-2,-1]] = np.exp(y_pred_decoded_raw[:,:,[-2,-1]] * y_pred[:,:,[-2,-1]]) * y_pred[:,:,[-6,-5]]
        y_pred_decoded_raw = convert_coordinates(y_pred_decoded_raw, start_index=-4, conversion='centroids2corners')
    elif input_coords == 'minmax':
        y_pred_decoded_raw[:,:,-4:] *= y_pred[:,:,-4:] # delta(pred) / size(anchor) / variance * variance == delta(pred) / size(anchor) for all four coordinates, where 'size' refers to w or h, respectively
        y_pred_decoded_raw[:,:,[-4,-3]] *= np.expand_dims(y_pred[:,:,-7] - y_pred[:,:,-8], axis=-1) # delta_xmin(pred) / w(anchor) * w(anchor) == delta_xmin(pred), delta_xmax(pred) / w(anchor) * w(anchor) == delta_xmax(pred)
        y_pred_decoded_raw[:,:,[-2,-1]] *= np.expand_dims(y_pred[:,:,-5] - y_pred[:,:,-6], axis=-1) # delta_ymin(pred) / h(anchor) * h(anchor) == delta_ymin(pred), delta_ymax(pred) / h(anchor) * h(anchor) == delta_ymax(pred)
        y_pred_decoded_raw[:,:,-4:] += y_pred[:,:,-8:-4] # delta(pred) + anchor == pred for all four coordinates
        y_pred_decoded_raw = convert_coordinates(y_pred_decoded_raw, start_index=-4, conversion='minmax2corners')
    elif input_coords == 'corners':
        y_pred_decoded_raw[:,:,-4:] *= y_pred[:,:,-4:] # delta(pred) / size(anchor) / variance * variance == delta(pred) / size(anchor) for all four coordinates, where 'size' refers to w or h, respectively
        y_pred_decoded_raw[:,:,[-4,-2]] *= np.expand_dims(y_pred[:,:,-6] - y_pred[:,:,-8], axis=-1) # delta_xmin(pred) / w(anchor) * w(anchor) == delta_xmin(pred), delta_xmax(pred) / w(anchor) * w(anchor) == delta_xmax(pred)
        y_pred_decoded_raw[:,:,[-3,-1]] *= np.expand_dims(y_pred[:,:,-5] - y_pred[:,:,-7], axis=-1) # delta_ymin(pred) / h(anchor) * h(anchor) == delta_ymin(pred), delta_ymax(pred) / h(anchor) * h(anchor) == delta_ymax(pred)
        y_pred_decoded_raw[:,:,-4:] += y_pred[:,:,-8:-4] # delta(pred) + anchor == pred for all four coordinates
    else:
        raise ValueError("Unexpected value for `input_coords`. Supported input coordinate formats are 'minmax', 'corners' and 'centroids'.")

    # 2: If the model predicts normalized box coordinates and they are supposed to be converted back to absolute coordinates, do that

    if normalize_coords:
        y_pred_decoded_raw[:,:,[-4,-2]] *= img_width # Convert xmin, xmax back to absolute coordinates
        y_pred_decoded_raw[:,:,[-3,-1]] *= img_height # Convert ymin, ymax back to absolute coordinates

    # 3: For each batch item, prepend each box's internal index to its coordinates.

    y_pred_decoded_raw2 = np.zeros((y_pred_decoded_raw.shape[0], y_pred_decoded_raw.shape[1], y_pred_decoded_raw.shape[2] + 1)) # Expand the last axis by one.
    y_pred_decoded_raw2[:,:,1:] = y_pred_decoded_raw
    y_pred_decoded_raw2[:,:,0] = np.arange(y_pred_decoded_raw.shape[1]) # Put the box indices as the first element for each box via broadcasting.
    y_pred_decoded_raw = y_pred_decoded_raw2

    # 4: Apply confidence thresholding and non-maximum suppression per class

    n_classes = y_pred_decoded_raw.shape[-1] - 5 # The number of classes is the length of the last axis minus the four box coordinates and minus the index

    y_pred_decoded = [] # Store the final predictions in this list
    for batch_item in y_pred_decoded_raw: # `batch_item` has shape `[n_boxes, n_classes + 4 coords]`
        pred = [] # Store the final predictions for this batch item here
        for class_id in range(1, n_classes): # For each class except the background class (which has class ID 0)...
            single_class = batch_item[:,[0, class_id + 1, -4, -3, -2, -1]] # ...keep only the confidences for that class, making this an array of shape `[n_boxes, 6]` and...
            threshold_met = single_class[single_class[:,1] > confidence_thresh] # ...keep only those boxes with a confidence above the set threshold.
            if threshold_met.shape[0] > 0: # If any boxes made the threshold...
                maxima = _greedy_nms_debug(threshold_met, iou_threshold=iou_threshold, coords='corners') # ...perform NMS on them.
                maxima_output = np.zeros((maxima.shape[0], maxima.shape[1] + 1)) # Expand the last dimension by one element to have room for the class ID. This is now an arrray of shape `[n_boxes, 6]`
                maxima_output[:,0] = maxima[:,0] # Write the box index to the first column...
                maxima_output[:,1] = class_id # ...and write the class ID to the second column...
                maxima_output[:,2:] = maxima[:,1:] # ...and write the rest of the maxima data to the other columns...
                pred.append(maxima_output) # ...and append the maxima for this class to the list of maxima for this batch item.
        # Once we're through with all classes, keep only the `top_k` maxima with the highest scores
        pred = np.concatenate(pred, axis=0)
        if pred.shape[0] > top_k: # If we have more than `top_k` results left at this point, otherwise there is nothing to filter,...
            top_k_indices = np.argpartition(pred[:,2], kth=pred.shape[0]-top_k, axis=0)[pred.shape[0]-top_k:] # ...get the indices of the `top_k` highest-score maxima...
            pred = pred[top_k_indices] # ...and keep only those entries of `pred`...
        y_pred_decoded.append(pred) # ...and now that we're done, append the array of final predictions for this batch item to the output list

    return y_pred_decoded

def _greedy_nms_debug(predictions, iou_threshold=0.45, coords='corners'):
    '''
    The same greedy non-maximum suppression algorithm as above, but slightly modified for use as an internal
    function for per-class NMS in `decode_y_debug()`. The difference is that it keeps the indices of all left-over boxes
    for each batch item, which allows you to know which predictor layer predicted a given output box and is thus
    useful for debugging.
    '''
    boxes_left = np.copy(predictions)
    maxima = [] # This is where we store the boxes that make it through the non-maximum suppression
    while boxes_left.shape[0] > 0: # While there are still boxes left to compare...
        maximum_index = np.argmax(boxes_left[:,1]) # ...get the index of the next box with the highest confidence...
        maximum_box = np.copy(boxes_left[maximum_index]) # ...copy that box and...
        maxima.append(maximum_box) # ...append it to `maxima` because we'll definitely keep it
        boxes_left = np.delete(boxes_left, maximum_index, axis=0) # Now remove the maximum box from `boxes_left`
        if boxes_left.shape[0] == 0: break # If there are no boxes left after this step, break. Otherwise...
        similarities = iou(boxes_left[:,2:], maximum_box[2:], coords=coords) # ...compare (IoU) the other left over boxes to the maximum box...
        boxes_left = boxes_left[similarities <= iou_threshold] # ...so that we can remove the ones that overlap too much with the maximum box
    return np.array(maxima)

def get_num_boxes_per_pred_layer(predictor_sizes, aspect_ratios, two_boxes_for_ar1):
    '''
    Returns a list of the number of boxes that each predictor layer predicts.

    `aspect_ratios` must be a nested list, containing a list of aspect ratios
    for each predictor layer.
    '''
    num_boxes_per_pred_layer = []
    for i in range(len(predictor_sizes)):
        if two_boxes_for_ar1:
            num_boxes_per_pred_layer.append(predictor_sizes[i][0] * predictor_sizes[i][1] * (len(aspect_ratios[i]) + 1))
        else:
            num_boxes_per_pred_layer.append(predictor_sizes[i][0] * predictor_sizes[i][1] * len(aspect_ratios[i]))
    return num_boxes_per_pred_layer

def get_pred_layers(y_pred_decoded, num_boxes_per_pred_layer):
    '''
    For a given prediction tensor decoded with `decode_y_debug()`, returns a list
    with the indices of the predictor layers that made each predictions.

    That is, this function lets you know which predictor layer is responsible
    for a given prediction.

    Arguments:
        y_pred_decoded (array): The decoded model output tensor. Must have been
            decoded with `decode_y_debug()` so that it contains the internal box index
            for each predicted box.
        num_boxes_per_pred_layer (list): A list that contains the total number
            of boxes that each predictor layer predicts.
    '''
    pred_layers_all = []
    cum_boxes_per_pred_layer = np.cumsum(num_boxes_per_pred_layer)
    for batch_item in y_pred_decoded:
        pred_layers = []
        for prediction in batch_item:
            if (prediction[0] < 0) or (prediction[0] >= cum_boxes_per_pred_layer[-1]):
                raise ValueError("Box index is out of bounds of the possible indices as given by the values in `num_boxes_per_pred_layer`.")
            for i in range(len(cum_boxes_per_pred_layer)):
                if prediction[0] < cum_boxes_per_pred_layer[i]:
                    pred_layers.append(i)
                    break
        pred_layers_all.append(pred_layers)
    return pred_layers_all