Poisson blending is based on the principle of manipulating the gradient fields of images. Instead of directly transferring pixel values from a source image to a target image, Poisson blending calculates the gradient (the change in intensity) of the source image and applies these changes to the target. This method focuses on blending the images in such a way that the transitions between them are smooth and seamless. The core idea behind Poisson blending is to solve a Poisson equation that aims to preserve the gradient of the source image while conforming to the boundary conditions set by the target image. The result is a harmonious integration of the source into the target, with edges and textures blending smoothly without the abrupt transitions characteristic of simpler methods.

The toy problem in gradient domain processing is a fundamental technique used to understand how to manipulate image gradients for tasks like image blending. It involves calculating the gradients in the x and y directions from a source image and using these gradients to reconstruct an image.

The source image is denoted by s(x,y) and the target image by v(x,y). The task is to solve for v(x,y) such that it maintains the gradient of s while also adhering to certain constraints.

1. Minimize the difference in x-gradients between v and s which can be written as v(x+1, y) - v(x, y) \approx s(x+1, y) - s(x, y) .
2. Minimize the difference in y-gradients between v and s written as v(x, y+1) - v(x, y) \approx s(x, y+1) - s(x, y) .
3. Ensure the top left corners of v and s have the same color, adding the constraint v(1, 1) = s(1, 1).

These constraints are set up as a least squares problem which can be mathematically represented and solved using linear algebra. The least squares approach will find the pixel values for image v that best fit the gradient constraints taken from image s. The implementation involves setting up a system of equations Av = b, where A is a matrix representing the gradient constraints, v is the vectorized form of the image to be solved, and b is a known vector derived from the gradients of s. This system can be solved using numerical methods such as the least squares solver in NumPy's linear algebra package. Once you solve for v, you can reshape this vector back into the form of an image and see if the reconstructed image v matches the original source image s. If the reconstruction is correct, you will have effectively demonstrated how to manipulate and reconstruct images in the gradient domain.

Mixed Gradients Blending is an image processing technique that refines the concept of Poisson blending by selectively using the largest gradient magnitude from either the source or the target image to guide the blending process.

The process aims to minimize the squared differences between the blended image gradients and the larger gradients from the source or target. The mathematical expression is as follows:

v = argmin_v ∑ (|∇v - d_ij|)^2

where d_ij is the gradient chosen from the source or the target image at each pixel.

The selection of gradients is based on their magnitude, favoring the gradient that has the higher magnitude between the source and target images. This decision is made for each pixel, ensuring that the most dominant features are preserved in the blending process.

if abs(∇s_i - ∇s_j) > abs(∇t_i - ∇t_j):
    d_ij = ∇s_i - ∇s_j
else:
    d_ij = ∇t_i - ∇t_j

This technique provides several benefits:

• It preserves sharp features and ensures smooth transitions by selectively choosing gradients.
• The blending often appears more natural by accommodating texture and lighting variations.
• It's particularly effective when blending images with plain backgrounds into more textured scenes.

Mixed Blend result:

Color2Gray

Converting color images to grayscale is a common process, but standard methods like MATLAB's rgb2gray can result in the loss of important contrast information. This makes details harder to distinguish, particularly when the image is used in different contexts, such as printing.

The goal is to create a grayscale image that maintains the intensity levels of a standard rgb2gray output but retains the contrast present in the original color image. This task is akin to a tone-mapping problem, often encountered when converting High Dynamic Range (HDR) images for display on standard RGB screens.

Gradient-domain processing offers a solution. By examining the gradients of the image, particularly in the HSV (Hue, Saturation, Value) color space, we can address the contrast issue. This approach involves:

• Converting the color image to HSV space.
• Analyzing the gradients in the Value channel, which corresponds to the image's intensity.
• Approaching the conversion as a mixed gradients problem to preserve both the grayscale intensity and the contrast of the original image.

To implement this technique:

1. Transform the image into the HSV color space.
2. Examine the gradient in each channel, focusing on the Value channel for intensity information.
3. Utilize the gradient information to guide the conversion, ensuring that the intensity is maintained while enhancing contrast where necessary.

This method ensures that important visual details are retained, making the grayscale image more understandable and closer to the perception of the original color version.

color2gray result: