Problem Description

You are given a grayscale image that is represented by a 2D grid image where the intensity of each pixel is a value between 0 and 255. The grid is 0-indexed, which means that the row and column indices start from 0. Additionally, you are given a non-negative integer threshold.

The problem involves finding regions within this image grid. A region is defined as a 3x3 subgrid where the absolute difference in intensity between any two adjacent pixels is less than or equal to the threshold. A pixel is adjacent to another if it is directly above, below, to the left, or to the right of the other pixel. It's important to note that a single pixel can be part of multiple regions.

You are required to create a new grid result where each entry result[i][j] represents the average intensity of the region(s) to which the pixel at image[i][j] belongs. If a pixel belongs to multiple regions, you should take the average of the rounded down averages for those regions and then round down this final value. If a pixel does not belong to any region, result[i][j] should simply be the original intensity from image[i][j].

The goal is to calculate this result grid given the image grid and the threshold.

Intuition

The main intuition behind the solution is to iterate over the image grid and identify the 3x3 subgrids that qualify as regions based on the given threshold condition. Once we identify such a region, we compute its average pixel intensity.

We maintain two auxiliary grids: ans, which keeps the sum of average intensities of the regions that each pixel belongs to, and ct, which counts the number of regions each pixel is a part of.

We loop through each possible top-left coordinate (i, j) of a potential 3x3 region in the image. For each position, we check if every pair of adjacent pixels within the 3x3 subgrid satisfies the threshold condition. If they do, we have found a valid region, and we calculate the sum of pixel intensities for this region.

For each pixel within this region, we add the floor of the region's average intensity to the corresponding position in ans, and increment the region count in ct. After all potential regions have been processed, we go through every pixel in the image. If a pixel was part of at least one region (ct[i][j] is not zero), we update result[i][j] to be the floor of the average of the sums from ans divided by the count from ct. If a pixel was not part of any region, we just set result[i][j] to the original pixel intensity image[i][j].

The approach leverages the idea that a pixel can be part of multiple regions and ensures that its final intensity in result is an average of those regions' intensities, rounded down as specified in the problem statement.

Solution Approach

The implementation utilizes nested for-loops to iterate over possible top-left corners (i, j) of potential 3x3 regions within the image. It uses a set of additional nested loops to check the conditions for identifying a region and to compute the total intensity and update the ans and ct auxiliary matrices.

Firstly, the algorithm establishes two matrices, ans and ct, both the size of the original image grid, initialized to zeroes. These matrices serve two purposes:

• ans[i][j] holds the cumulative sum of average intensities for the pixel at (i, j) for all regions it belongs to.
• ct[i][j] holds the count of regions that the pixel at (i, j) is part of.

Next, we begin our core logic, walking the image grid for potential regions:

• Loop through each cell (i, j) of the grid, excluding the last two rows and columns since a 3x3 region cannot start there (would extend beyond grid boundaries).
• For each potential region's top-left corner, we assess if the pixels within this 3x3 subgrid form a valid region by comparing each adjacent pair of pixels and checking whether their intensity difference is within the threshold. This is done using additional loops and the abs() function.
• If we confirm a valid region, we sum the intensities of all nine pixels within this 3x3 subgrid.
• We then distribute this total sum (averaged by dividing by 9 and taking the floor) across the nine corresponding cells in the ans matrix and increment the counts in ct.

After the regions have been processed, we perform a final pass over the entire grid:

• For each pixel (i, j), we check ct[i][j]. If it's greater than zero, it means the pixel is part of one or more regions. We then finalize ans[i][j] by dividing the accumulated sum by the count (the number of regions the pixel belongs to) and rounding down.
• If ct[i][j] is zero, we simply assign the original intensity value from image[i][j] to ans[i][j], as the pixel wasn't part of any region.

The resultant ans matrix now reflects the average intensities as required by the problem, and this matrix is returned as the final answer.

The algorithm is straightforward but computationally intensive given the nested loops, which could lead to a higher time complexity, particularly for larger images.

Example Walkthrough

Let's assume we have the following 5x5 grayscale image image represented by a 2D grid and a threshold of 10:

image = [
    [100, 105, 102, 200, 205],
    [103, 108, 103, 205, 200],
    [101, 106, 102, 202, 207],
    [104, 109, 104, 210, 205],
    [100, 105, 107, 205, 200]
]
threshold = 10

In this example, we will find the regions and calculate the resulting grid result.

1. Initialize the auxiliary grids ans and ct to zeros:

ans = [
    [0, 0, 0, 0, 0],
    [0, 0, 0, 0, 0],
    [0, 0, 0, 0, 0],
    [0, 0, 0, 0, 0],
    [0, 0, 0, 0, 0]
]
ct = [
    [0, 0, 0, 0, 0],
    [0, 0, 0, 0, 0],
    [0, 0, 0, 0, 0],
    [0, 0, 0, 0, 0],
    [0, 0, 0, 0, 0]
]

2. Loop through the potential top-left corners of the 3x3 regions (excluding the last two rows and columns since a 3x3 region cannot start there):

• Any given cell (i, j) can be part of a region if i is 0, 1, 2, 3 and j is 0, 1, 2, 3. For example, starting at the top-left corner (0, 0), let's check if it can be a region:

Top-left corner (0, 0) region:
[
    [100, 105, 102],
    [103, 108, 103],
    [101, 106, 102]
]

• We compare each pair of adjacent pixels within this 3x3 grid to see if their absolute difference is less than or equal to 10. In this case, all pairs satisfy the condition, so this is a valid region.

• Compute the sum of pixel intensities in the region and divide by 9 (floor if necessary):

Sum = 100 + 105 + 102 + 103 + 108 + 103 + 101 + 106 + 102 = 930
Average = floor(930 / 9) = floor(103.33) = 103

• We distribute this average to all 9 pixels of the ans matrix and increment ct for each:

ans after processing top-left corner (0, 0):
[
    [103, 103, 103,   0,   0],
    [103, 103, 103,   0,   0],
    [103, 103, 103,   0,   0],
    [  0,   0,   0,   0,   0],
    [  0,   0,   0,   0,   0]
]
ct after processing top-left corner (0, 0):
[
    [1, 1, 1, 0, 0],
    [1, 1, 1, 0, 0],
    [1, 1, 1, 0, 0],
    [0, 0, 0, 0, 0],
    [0, 0, 0, 0, 0]
]

• This process continues for other top-left corners, such as (0, 1), (1, 0), etc., until all possible starting points for regions have been checked.

3. After determining all potential regions, process each pixel to compute the final values in result.

• For each pixel (i, j), if ct[i][j] is greater than zero, result[i][j] is set to floor(ans[i][j] / ct[i][j]).
• If ct[i][j] is zero, result[i][j] remains as image[i][j].

The final ans matrix is the output of the solution, showcasing the average intensities of pixels across their respective regions, rounded according to the rules specified in the problem description.

Solution Implementation

Time and Space Complexity

The given Python code defines a method resultGrid that modifies a grid image based on a certain threshold. It also calculates the average value of a region that fits the given condition.

Time Complexity

The time complexity of this code involves several nested loops:

• There are two external loops over the rows and columns (except for the last two), which give us O((n-2) * (m-2)).
• Inside these loops, there are two more nested loops to check the threshold condition for each 3x3 region, which always check 8 differences, resulting in O(8) operations per region.
• Then again, it iterates over the 3x3 region, this time to sum the values and to increment the count; this is O(9) for each region.

The combined time complexity when considering the nested loops for threshold checking and summing over the 3x3 region is O(8 + 9). However, since these are constants, they do not affect the asymptotic behavior compared to the number of times the loops are executed, which depends on the dimensions of the image (n, m).

Therefore, combining all these, the overall time complexity is O((n-2) * (m-2)), which simplifies to O(n * m) as n and m become large.

Lastly, there is one final loop which iterates over all the cells to adjust the final values based on the count ct, which has a time complexity of O(n * m).

When combined, since these operations are sequential, the overall time complexity remains O(n * m).

Space Complexity

The space complexity is determined by the additional space used by the algorithm, which is not inclusive of the input itself:

• The ans array holds the modified image data, which occupies O(n * m) space.
• The ct array holds the count of how many times a cell value has been included in a sum, which also occupies O(n * m) space.

Adding these together, the total space required for the algorithm is 2 * O(n * m), but we drop the constant in Big O notation, rendering the space complexity to O(n * m).

Learn more about how to find time and space complexity quickly using problem constraints.