Rotate image and crop out black borders

Question

My application: I am trying to rotate an image (using OpenCV and Python)

At the moment I have developed the below code which rotates an input image, padding it with black borders, giving me A. What I want is B - the largest possible area crop window within the rotated image. I call this the axis-aligned boundED box.

This is essentially the same as Rotate and crop, however I cannot get the answer on that question to work. Additionally, that answer is apparently only valid for square images. My images are rectangular.

Code to give A:

import cv2 import numpy as np   def getTranslationMatrix2d(dx, dy):     """     Returns a numpy affine transformation matrix for a 2D translation of     (dx, dy)     """     return np.matrix([[1, 0, dx], [0, 1, dy], [0, 0, 1]])   def rotateImage(image, angle):     """     Rotates the given image about it's centre     """      image_size = (image.shape[1], image.shape[0])     image_center = tuple(np.array(image_size) / 2)      rot_mat = np.vstack([cv2.getRotationMatrix2D(image_center, angle, 1.0), [0, 0, 1]])     trans_mat = np.identity(3)      w2 = image_size[0] * 0.5     h2 = image_size[1] * 0.5      rot_mat_notranslate = np.matrix(rot_mat[0:2, 0:2])      tl = (np.array([-w2, h2]) * rot_mat_notranslate).A[0]     tr = (np.array([w2, h2]) * rot_mat_notranslate).A[0]     bl = (np.array([-w2, -h2]) * rot_mat_notranslate).A[0]     br = (np.array([w2, -h2]) * rot_mat_notranslate).A[0]      x_coords = [pt[0] for pt in [tl, tr, bl, br]]     x_pos = [x for x in x_coords if x > 0]     x_neg = [x for x in x_coords if x < 0]      y_coords = [pt[1] for pt in [tl, tr, bl, br]]     y_pos = [y for y in y_coords if y > 0]     y_neg = [y for y in y_coords if y < 0]      right_bound = max(x_pos)     left_bound = min(x_neg)     top_bound = max(y_pos)     bot_bound = min(y_neg)      new_w = int(abs(right_bound - left_bound))     new_h = int(abs(top_bound - bot_bound))     new_image_size = (new_w, new_h)      new_midx = new_w * 0.5     new_midy = new_h * 0.5      dx = int(new_midx - w2)     dy = int(new_midy - h2)      trans_mat = getTranslationMatrix2d(dx, dy)     affine_mat = (np.matrix(trans_mat) * np.matrix(rot_mat))[0:2, :]     result = cv2.warpAffine(image, affine_mat, new_image_size, flags=cv2.INTER_LINEAR)      return result 

Answer 1

The math behind this solution/implementation is equivalent to this solution of an analagous question, but the formulas are simplified and avoid singularities. This is python code with the same interface as largest_rotated_rect from the other solution, but giving a bigger area in almost all cases (always the proven optimum):

def rotatedRectWithMaxArea(w, h, angle):   """   Given a rectangle of size wxh that has been rotated by 'angle' (in   radians), computes the width and height of the largest possible   axis-aligned rectangle (maximal area) within the rotated rectangle.   """   if w <= 0 or h <= 0:     return 0,0    width_is_longer = w >= h   side_long, side_short = (w,h) if width_is_longer else (h,w)    # since the solutions for angle, -angle and 180-angle are all the same,   # if suffices to look at the first quadrant and the absolute values of sin,cos:   sin_a, cos_a = abs(math.sin(angle)), abs(math.cos(angle))   if side_short <= 2.*sin_a*cos_a*side_long or abs(sin_a-cos_a) < 1e-10:     # half constrained case: two crop corners touch the longer side,     #   the other two corners are on the mid-line parallel to the longer line     x = 0.5*side_short     wr,hr = (x/sin_a,x/cos_a) if width_is_longer else (x/cos_a,x/sin_a)   else:     # fully constrained case: crop touches all 4 sides     cos_2a = cos_a*cos_a - sin_a*sin_a     wr,hr = (w*cos_a - h*sin_a)/cos_2a, (h*cos_a - w*sin_a)/cos_2a    return wr,hr 

Here is a comparison of the function with the other solution:

>>> wl,hl = largest_rotated_rect(1500,500,math.radians(20)) >>> print (wl,hl),', area=',wl*hl (828.2888697391496, 230.61639227890998) , area= 191016.990904 >>> wm,hm = rotatedRectWithMaxArea(1500,500,math.radians(20)) >>> print (wm,hm),', area=',wm*hm (730.9511000407718, 266.044443118978) , area= 194465.478358 

With angle angle in [0,pi/2[ the bounding box of the rotated image (width w, height h) has these dimensions:

• width w_bb = w*cos_a + h*sin_a
• height h_bb = w*sin_a + h*cos_a

If w_r, h_r are the computed optimal width and height of the cropped image, then the insets from the bounding box are:

• in horizontal direction: (w_bb-w_r)/2
• in vertical direction: (h_bb-h_r)/2

Proof:

Looking for the axis aligned rectangle between two parallel lines that has maximal area is an optimization problem with one parameter, e.g. x as in this figure:

Let s denote the distance between the two parallel lines (it will turn out to be the shorter side of the rotated rectangle). Then the sides a, b of the sought-after rectangle have a constant ratio with x, s-x, resp., namely x = a sin α and (s-x) = b cos α:

So maximizing the area a*b means maximizing x*(s-x). Because of "theorem of height" for right-angled triangles we know x*(s-x) = p*q = h*h. Hence the maximal area is reached at x = s-x = s/2, i.e. the two corners E, G between the parallel lines are on the mid-line:

This solution is only valid if this maximal rectangle fits into the rotated rectangle. Therefore the diagonal EG must not be longer than the other side l of the rotated rectangle. Since

EG = AF + DH = s/2*(cot α + tan α) = s/(2sin αcos α) = s/sin 2*α

we have the condition s ≤ lsin 2α, where s and l are the shorter and longer side of the rotated rectangle.

In case of s > lsin 2α the parameter x must be smaller (than s/2) and s.t. all corners of the sought-after rectangle are each on a side of the rotated rectangle. This leads to the equation

x*cot α + (s-x)*tan α = l

giving x = sin α*(lcos α - ssin α)/cos 2*α. From a = x/sin α and b = (s-x)/cos α we get the above used formulas.

Answer 2

So, after investigating many claimed solutions, I have finally found a method that works; The answer by Andri and Magnus Hoff on Calculate largest rectangle in a rotated rectangle.

The below Python code contains the method of interest - largest_rotated_rect - and a short demo.

import math import cv2 import numpy as np   def rotate_image(image, angle):     """     Rotates an OpenCV 2 / NumPy image about it's centre by the given angle     (in degrees). The returned image will be large enough to hold the entire     new image, with a black background     """      # Get the image size     # No that's not an error - NumPy stores image matricies backwards     image_size = (image.shape[1], image.shape[0])     image_center = tuple(np.array(image_size) / 2)      # Convert the OpenCV 3x2 rotation matrix to 3x3     rot_mat = np.vstack(         [cv2.getRotationMatrix2D(image_center, angle, 1.0), [0, 0, 1]]     )      rot_mat_notranslate = np.matrix(rot_mat[0:2, 0:2])      # Shorthand for below calcs     image_w2 = image_size[0] * 0.5     image_h2 = image_size[1] * 0.5      # Obtain the rotated coordinates of the image corners     rotated_coords = [         (np.array([-image_w2,  image_h2]) * rot_mat_notranslate).A[0],         (np.array([ image_w2,  image_h2]) * rot_mat_notranslate).A[0],         (np.array([-image_w2, -image_h2]) * rot_mat_notranslate).A[0],         (np.array([ image_w2, -image_h2]) * rot_mat_notranslate).A[0]     ]      # Find the size of the new image     x_coords = [pt[0] for pt in rotated_coords]     x_pos = [x for x in x_coords if x > 0]     x_neg = [x for x in x_coords if x < 0]      y_coords = [pt[1] for pt in rotated_coords]     y_pos = [y for y in y_coords if y > 0]     y_neg = [y for y in y_coords if y < 0]      right_bound = max(x_pos)     left_bound = min(x_neg)     top_bound = max(y_pos)     bot_bound = min(y_neg)      new_w = int(abs(right_bound - left_bound))     new_h = int(abs(top_bound - bot_bound))      # We require a translation matrix to keep the image centred     trans_mat = np.matrix([         [1, 0, int(new_w * 0.5 - image_w2)],         [0, 1, int(new_h * 0.5 - image_h2)],         [0, 0, 1]     ])      # Compute the tranform for the combined rotation and translation     affine_mat = (np.matrix(trans_mat) * np.matrix(rot_mat))[0:2, :]      # Apply the transform     result = cv2.warpAffine(         image,         affine_mat,         (new_w, new_h),         flags=cv2.INTER_LINEAR     )      return result   def largest_rotated_rect(w, h, angle):     """     Given a rectangle of size wxh that has been rotated by 'angle' (in     radians), computes the width and height of the largest possible     axis-aligned rectangle within the rotated rectangle.      Original JS code by 'Andri' and Magnus Hoff from Stack Overflow      Converted to Python by Aaron Snoswell     """      quadrant = int(math.floor(angle / (math.pi / 2))) & 3     sign_alpha = angle if ((quadrant & 1) == 0) else math.pi - angle     alpha = (sign_alpha % math.pi + math.pi) % math.pi      bb_w = w * math.cos(alpha) + h * math.sin(alpha)     bb_h = w * math.sin(alpha) + h * math.cos(alpha)      gamma = math.atan2(bb_w, bb_w) if (w < h) else math.atan2(bb_w, bb_w)      delta = math.pi - alpha - gamma      length = h if (w < h) else w      d = length * math.cos(alpha)     a = d * math.sin(alpha) / math.sin(delta)      y = a * math.cos(gamma)     x = y * math.tan(gamma)      return (         bb_w - 2 * x,         bb_h - 2 * y     )   def crop_around_center(image, width, height):     """     Given a NumPy / OpenCV 2 image, crops it to the given width and height,     around it's centre point     """      image_size = (image.shape[1], image.shape[0])     image_center = (int(image_size[0] * 0.5), int(image_size[1] * 0.5))      if(width > image_size[0]):         width = image_size[0]      if(height > image_size[1]):         height = image_size[1]      x1 = int(image_center[0] - width * 0.5)     x2 = int(image_center[0] + width * 0.5)     y1 = int(image_center[1] - height * 0.5)     y2 = int(image_center[1] + height * 0.5)      return image[y1:y2, x1:x2]   def demo():     """     Demos the largest_rotated_rect function     """      image = cv2.imread("lenna_rectangle.png")     image_height, image_width = image.shape[0:2]      cv2.imshow("Original Image", image)      print "Press [enter] to begin the demo"     print "Press [q] or Escape to quit"      key = cv2.waitKey(0)     if key == ord("q") or key == 27:         exit()      for i in np.arange(0, 360, 0.5):         image_orig = np.copy(image)         image_rotated = rotate_image(image, i)         image_rotated_cropped = crop_around_center(             image_rotated,             *largest_rotated_rect(                 image_width,                 image_height,                 math.radians(i)             )         )          key = cv2.waitKey(2)         if(key == ord("q") or key == 27):             exit()          cv2.imshow("Original Image", image_orig)         cv2.imshow("Rotated Image", image_rotated)         cv2.imshow("Cropped Image", image_rotated_cropped)      print "Done"   if __name__ == "__main__":     demo() 

Simply place this image (cropped to demonstrate that it works with non-square images) in the same directory as the above file, then run it.