Classical Pc Imaginative and prescient and Perspective Transformation for Sudoku Extraction

of AI hype, it looks like everyone seems to be utilizing Imaginative and prescient-Language Fashions and enormous Imaginative and prescient Transformers for each downside in Pc Imaginative and prescient. Many individuals see these instruments as one-size-fits-all options and instantly use the newest, shiniest mannequin as an alternative of understanding the underlying sign they need to extract. However oftentimes there’s magnificence to simplicity. It’s one of the crucial vital 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 strategies to detect rectangular objects on flat surfaces and apply a perspective transformation to remodel the skewed rectangle. Related strategies are extensively used, for instance, in doc scanning and extraction purposes.

Alongside the best way you’ll study some attention-grabbing ideas from commonplace classical Pc Imaginative and prescient strategies to the way to order polygon factors and why that is associated to a combinatoric task downside.

Detection

To detect Sudoku grids I thought-about many various 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 shade channels to a single channel grayscale picture, as we don’t want any shade info to course of these photographs.

def find_sudoku_grid(
    picture: np.ndarray,
) -> np.ndarray | None:
    """
    Finds the most important square-like contour in a picture, seemingly 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 will 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 related to the primary border from displaying up in our masks.

A blur filter is usually used earlier than passing photographs to Canny to scale back noise, however on this case the sides 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 most important square-like contour in a picture, seemingly 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 most important square-like contour in a picture, seemingly 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: Measurement 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 right 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 an inventory of contours that include each single pixel that’s a part of the contour. We will use the Douglas–Peucker algorithm to iteratively cut back the variety of factors within the contour and approximate the contour with a easy polygon. We will select a minimal distance between factors for the algorithm.

If we assume that even for a number of the most skewed rectangle, the shortest facet is not less than 10% of the circumference of the form, we will filter the contours right down to polygons with precisely 4 factors.

contour_candidates: listing[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 most important detected contour, presumably the ultimate Sudoku grid. We kind the contours by space in reverse order after which take the primary aspect, similar to the most important 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 must 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 must be ordered first, to allow them to be assigned accurately. Let’s outline the next order for our nook factors:

Variant A: Easy kind based mostly 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 doable 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 most important sum argmax(p_sum)
• High proper nook has the most important 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 symbolize 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 could have the smallest sum, whereas
    # the bottom-right level could have the most important sum
    s = pts.sum(axis=1)
    rect[0] = pts[np.argmin(s)]
    rect[2] = pts[np.argmax(s)]

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

    return rect

This works nicely until 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 Drawback

One other method could be to reduce 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 downside 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 space between every level and every goal nook
    D = pairwise_distances(pts, pts_corner)

    # Discover the optimum one-to-one task
    # row_ind[i] ought 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 right slot based mostly 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

Although this answer works, it’s not excellent, 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 must be constructed. After all right here within the case of 4 factors assigned that is negligible, however this answer wouldn’t be nicely fitted 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 based mostly on the centroid place.

For the reason that 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 start out 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)
    heart = pts.imply(axis=0)
    angles = np.arctan2(pts[:, 1] - heart[1], pts[:, 0] - heart[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 will 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 listing 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 the way to 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 venture we walked by way of a easy pipeline utilizing classical Pc Imaginative and prescient strategies to extract Sudoku grids from footage. These strategies present a easy technique to detect the bounds of the Sudoku grids. After all as a result of its simplicity there are some limitations to how nicely this method generalizes to totally different settings and excessive environments reminiscent of low gentle or arduous shadows. Utilizing a deep-learning based mostly method 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, reminiscent of extracting the numbers within the grid and really fixing the Sudoku. In a subsequent article we are going to look additional into these pure subsequent steps on this venture.

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

For extra particulars and the total implementation together with the code for the all of the animations and visualizations, take a look at the supply code of this venture on my GitHub:

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

All visualizations on this put up have been created by the writer.