Classical Computer Vision and Perspective Transformation for Sudoku Extraction

of AI hype, it seems like everyone seems to be utilizing Imaginative and prescient-Language Fashions and enormous Imaginative and prescient Transformers for each drawback in Pc Imaginative and prescient. Many individuals see these instruments as one-size-fits-all options and instantly use the most recent, shiniest mannequin as an alternative of understanding the underlying sign they wish to extract. However oftentimes there’s magnificence to simplicity. It’s one of the necessary classes I’ve realized as an engineer: don’t overcomplicate options to easy issues.

Let me present you a sensible utility of some easy classical Pc Imaginative and prescient methods to detect rectangular objects on flat surfaces and apply a perspective transformation to remodel the skewed rectangle. Related strategies are broadly used, for instance, in doc scanning and extraction purposes.

Alongside the way in which you’ll be taught some attention-grabbing ideas from customary classical Pc Imaginative and prescient methods to how one can order polygon factors and why that is associated to a combinatoric task drawback.

Detection

To detect Sudoku grids I thought of many alternative approaches starting from easy thresholding, hough line transformations or some type of edge detection to coaching a deep studying mannequin for segmentation or keypoint detection.

Let’s outline some assumptions to scope the issue:

1. The Sudoku grid is clearly and totally seen within the body with a transparent quadrilateral border, with sturdy distinction from the background.
2. The floor on which the Sudoku grid is printed must be flat, however could be captured from an angle and seem skewed or rotated.

I’ll present you a easy pipeline with some filtering steps to detect the bounds of our Sudoku grid. On a excessive stage, the processing pipeline seems as follows:

Grayscale

On this first step we merely convert the enter picture from its three colour channels to a single channel grayscale picture, as we don’t want any colour data to course of these pictures.

def find_sudoku_grid(
    picture: np.ndarray,
) -> np.ndarray | None:
    """
    Finds the biggest square-like contour in a picture, possible the Sudoku grid.

    Returns:
        The contour of the discovered grid as a numpy array, or None if not discovered.
    """

    grey = cv2.cvtColor(picture, cv2.COLOR_BGR2GRAY)

Edge Detection

After changing the picture to grayscale we will use the Canny edge detection algorithm to extract edges. There are two thresholds to decide on for this algorithm that decide if pixels are accepted as edges:

In our case of detecting Sudoku grids, we assume very sturdy edges on the border traces of our grid. We are able to select a excessive higher threshold to reject noise from showing in our masks, and a decrease threshold not too low to reject small noisy edges linked to the primary border from exhibiting up in our masks.

A blur filter is commonly used earlier than passing pictures to Canny to scale back noise, however on this case the perimeters are very sturdy however slender, therefore the blur is omitted.

def find_sudoku_grid(
    picture: np.ndarray,
    canny_threshold_1: int = 100,
    canny_threshold_2: int = 255,
) -> np.ndarray | None:
    """
    Finds the biggest square-like contour in a picture, possible the Sudoku grid.

    Args:
        picture: The enter picture.
        canny_threshold_1: Decrease threshold for the Canny edge detector.
        canny_threshold_2: Higher threshold for the Canny edge detector.

    Returns:
        The contour of the discovered grid as a numpy array, or None if not discovered.
    """

    ...

    canny = cv2.Canny(grey, threshold1=canny_threshold_1, threshold2=canny_threshold_2)

Dilation

On this subsequent step, we post-process the sting detection masks with a dilation kernel to shut small gaps within the masks.

def find_sudoku_grid(
    picture: np.ndarray,
    canny_threshold_1: int = 100,
    canny_threshold_2: int = 255,
    morph_kernel_size: int = 3,
) -> np.ndarray | None:
    """
    Finds the biggest square-like contour in a picture, possible the Sudoku grid.

    Args:
        picture: The enter picture.
        canny_threshold_1: First threshold for the Canny edge detector.
        canny_threshold_2: Second threshold for the Canny edge detector.
        morph_kernel_size: Dimension of the morphological operation kernel.

    Returns:
        The contour of the discovered grid as a numpy array, or None if not discovered.
    """

    ...

    kernel = cv2.getStructuringElement(
        form=cv2.MORPH_RECT, ksize=(morph_kernel_size, morph_kernel_size)
    )
    masks = cv2.morphologyEx(canny, op=cv2.MORPH_DILATE, kernel=kernel, iterations=1)

Contour Detection

Now that the binary masks is prepared, we will run a contour detection algorithm to search out coherent blobs and filter all the way down to a single contour with 4 factors.

contours, _ = cv2.findContours(
    masks, mode=cv2.RETR_EXTERNAL, methodology=cv2.CHAIN_APPROX_SIMPLE
)

This preliminary contour detection will return a listing of contours that comprise each single pixel that’s a part of the contour. We are able to use the Douglas–Peucker algorithm to iteratively scale back the variety of factors within the contour and approximate the contour with a easy polygon. We are able to select a minimal distance between factors for the algorithm.

If we assume that even for among the most skewed rectangle, the shortest facet is at the least 10% of the circumference of the form, we will filter the contours all the way down to polygons with precisely 4 factors.

contour_candidates: record[np.ndarray] = []
for cnt in contours:
    # Approximate the contour to a polygon
    epsilon = 0.1 * cv2.arcLength(curve=cnt, closed=True)
    approx = cv2.approxPolyDP(curve=cnt, epsilon=epsilon, closed=True)

    # Maintain solely polygons with 4 vertices
    if len(approx) == 4:
        contour_candidates.append(approx)

Lastly we take the biggest detected contour, presumably the ultimate Sudoku grid. We kind the contours by space in reverse order after which take the primary component, comparable to the biggest contour space.

best_contour = sorted(contour_candidates, key=cv2.contourArea, reverse=True)[0]

Perspective Transformation

Lastly we have to rework the detected grid again to its sq.. To realize this, we will use a perspective transformation. The transformation matrix could be calculated by specifying the place the 4 factors of our Sudoku grid contour should be positioned ultimately: the 4 corners of the picture.

rect_dst = np.array(
    [[0, 0], [width - 1, 0], [width - 1, height - 1], [0, height - 1]],
)

To match the contour factors to the corners, they should be ordered first, to allow them to be assigned appropriately. Let’s outline the next order for our nook factors:

Variant A: Easy kind primarily based on sum/diff

To kind the extracted corners and assign them to those goal factors, a easy algorithm might take a look at the sum and variations of the x and y coordinates for every nook.

p_sum = p_x + p_y
p_diff = p_x - p_y

Primarily based on these values, it’s now attainable to distinguish the corners:

• The highest left nook has each a small x and y worth, it has the smallest sum argmin(p_sum)
• Backside proper nook has the biggest sum argmax(p_sum)
• High proper nook has the biggest diff argmax(p_diff)
• Backside left nook has the smallest distinction argmin(p_diff)

Within the following animation, I attempted to visualise this task of the 4 corners of a rotating sq.. The coloured traces characterize the respective picture nook assigned to every sq. nook.

def order_points(pts: np.ndarray) -> np.ndarray:
    """
    Orders the 4 nook factors of a contour in a constant
    top-left, top-right, bottom-right, bottom-left sequence.

    Args:
        pts: A numpy array of form (4, 2) representing the 4 corners.

    Returns:
        A numpy array of form (4, 2) with the factors ordered.
    """
    # Reshape from (4, 1, 2) to (4, 2) if wanted
    pts = pts.reshape(4, 2)
    rect = np.zeros((4, 2), dtype=np.float32)

    # The highest-left level may have the smallest sum, whereas
    # the bottom-right level may have the biggest sum
    s = pts.sum(axis=1)
    rect[0] = pts[np.argmin(s)]
    rect[2] = pts[np.argmax(s)]

    # The highest-right level may have the smallest distinction,
    # whereas the bottom-left may have the biggest distinction
    diff = np.diff(pts, axis=1)
    rect[1] = pts[np.argmin(diff)]
    rect[3] = pts[np.argmax(diff)]

    return rect

This works effectively except the rectangle is closely skewed, like the next one. On this case, you possibly can clearly see that this methodology is flawed, as there the identical rectangle nook is assigned a number of picture corners.

Variant B: Project Optimization Downside

One other strategy can be to attenuate the distances between every level and its assigned nook. This may be applied utilizing a pairwise_distances calculation between every level and the corners and the linear_sum_assignment perform from scipy, which solves the task drawback whereas minimizing a value perform.

def order_points_simplified(pts: np.ndarray) -> np.ndarray:
    """
    Orders a set of factors to finest match a goal set of nook factors.

    Args:
        pts: A numpy array of form (N, 2) representing the factors to order.

    Returns:
        A numpy array of form (N, 2) with the factors ordered.
    """
    # Reshape from (N, 1, 2) to (N, 2) if wanted
    pts = pts.reshape(-1, 2)

    # Calculate the gap between every level and every goal nook
    D = pairwise_distances(pts, pts_corner)

    # Discover the optimum one-to-one task
    # row_ind[i] needs to be matched with col_ind[i]
    row_ind, col_ind = linear_sum_assignment(D)

    # Create an empty array to carry the sorted factors
    ordered_pts = np.zeros_like(pts)

    # Place every level within the appropriate slot primarily based on the nook it was matched to.
    # For instance, the purpose matched to target_corners[0] goes into ordered_pts[0].
    ordered_pts[col_ind] = pts[row_ind]

    return ordered_pts

Though this answer works, it isn’t supreme, because it depends on the picture distance between the form factors and the corners and it’s computationally costlier as a result of a distance matrix needs to be constructed. In fact right here within the case of 4 factors assigned that is negligible, however this answer wouldn’t be effectively suited to a polygon with many factors!

Variant C: Cyclic sorting with anchor

This third variant is a really light-weight and environment friendly technique to kind and assign the factors of the form to the picture corners. The thought is to calculate an angle for every level of the form primarily based on the centroid place.

Because the angles are cyclic, we have to select an anchor to ensure absolutely the order of the factors. We merely choose the purpose with the bottom sum of x and y.

def order_points(self, pts: np.ndarray) -> np.ndarray:
    """
    Orders factors by angle across the centroid, then rotates to begin from top-left.

    Args:
        pts: A numpy array of form (4, 2).

    Returns:
        A numpy array of form (4, 2) with factors ordered."""
    pts = pts.reshape(4, 2)
    middle = pts.imply(axis=0)
    angles = np.arctan2(pts[:, 1] - middle[1], pts[:, 0] - middle[0])
    pts_cyclic = pts[np.argsort(angles)]
    sum_of_coords = pts_cyclic.sum(axis=1)
    top_left_idx = np.argmin(sum_of_coords)
    return np.roll(pts_cyclic, -top_left_idx, axis=0)

We are able to now use this perform to kind our contour factors:

rect_src = order_points(grid_contour)

Making use of the Perspective Transformation

Now that we all know which factors must go the place, we will lastly transfer on to probably the most attention-grabbing half: creating and really making use of the attitude transformation to the picture.

Since we have already got our record of factors for the detected quadrilateral sorted in rect_src, and now we have our goal nook factors in rect_dst, we will use the OpenCV methodology for calculating the transformation matrix:

warp_mat = cv2.getPerspectiveTransform(rect_src, rect_dst)

The result’s a 3×3 warp matrix, defining how one can rework from a skewed 3D perspective view to a 2D flat top-down view. To get this flat top-down view of our Sudoku grid, we will apply this attitude transformation to our authentic picture:

warped = cv2.warpPerspective(img, warp_mat, (side_len, side_len))

And voilà, now we have our completely sq. Sudoku grid!

Conclusion

On this challenge we walked by means of a easy pipeline utilizing classical Pc Imaginative and prescient methods to extract Sudoku grids from photos. These strategies present a easy technique to detect the bounds of the Sudoku grids. In fact on account of its simplicity there are some limitations to how effectively this strategy generalizes to completely different settings and excessive environments comparable to low gentle or arduous shadows. Utilizing a deep-learning primarily based strategy might make sense if the detection must generalize to an enormous quantity of various settings.

Subsequent, a perspective transformation is used to get a flat top-down view of the grid. This picture can now be utilized in additional processing, comparable to extracting the numbers within the grid and really fixing the Sudoku. In a subsequent article we’ll look additional into these pure subsequent steps on this challenge.

Take a look at the supply code of the challenge beneath and let me know if in case you have any questions or ideas on this challenge. Till then, joyful coding!

For extra particulars and the complete implementation together with the code for the all of the animations and visualizations, try the supply code of this challenge on my GitHub:

https://github.com/trflorian/sudoku-extraction

All visualizations on this submit had been created by the writer.