Image Compression – rohit apte

I recently finished Andrew Ng’s CS229 course remotely. The course was extremely challenging and covered a wide range of Machine Learning concepts. Even if you have worked with, and used Machine Learning algorithms, he introduced concepts in novel and interesting ways I didn’t expect. One of the things that struct a chord with me was how we worked on K-Means clustering.

The algorithm is quite simple. Courtesy of CS229…

Given a training set, if we want to group it into n clusters

$\{ x^{(1)},\cdots,x^{(m)} \}$

1. Initialize cluster centroids randomly

$\mu_1,\mu_2,\cdots,\mu_k \in \mathbb{R}^n$

2. Repeat until convergence

For every i set

$c^{(i)} := \arg \min_{j} ||x_{(i)}-\mu_j||^2$

For each j set

$\mu_j := \frac{\sum_{i=1}^{m}1 \{c^{(i)}=j \} x^{(i)}}{\sum_{i=1}^{m}1 \{ c^{(i)}=j \}}$

Here an example of running K-means on the Iris dataset.

But an alternate use for K-Means is image compression. Given an image with the standard RGB colors we have 256x256x256=16.77M colors. We can use K-Means to compress these into less colors. The idea is similar to the above, with a few differences.

We take an image of dimension (m,n,3) (3=R,G,B) and resize it to (mxn,3)
For k clusters, we randomly sample k points from this data
We assign colors closest to the k points we’ve selected to that cluster
The new cluster points are the mean R,G,B points for each cluster
We repeat this process until convergence (points don’t change, or we reach a threshold)

This is what the code looks like

def k_means_code(X,num_clusters):
    min_iters=30
    X_reshaped=X.reshape(-1,3)

    #this variable will get resized
    past_min_indices =np.zeros(X_reshaped.shape[0])
    # first set centroids to random points
    initial_indices=np.random.choice(X_reshaped.shape[0],num_clusters, replace=False)
    centroids=X_reshaped[initial_indices]
    #convert to float to prevent rounding on the averages
    centroids=centroids.astype(float)
    i=1
    bContinueRunning=True
    while bContinueRunning:
        distances=np.array((0,0))
        for centroid_index,centroid in enumerate(centroids):
            distance=np.sqrt(np.sum((X_reshaped - centroid) ** 2, axis=1))
            if centroid_index==0:
                distances=distance.copy()
            else:
                distances=np.vstack((distances,distance))
        min_indices=distances.argmin(axis=0)
        if np.sum(min_indices-past_min_indices)==0:
            bContinueRunning=False
        else:
            past_min_indices=min_indices.copy()
            for centroid_index in range(num_clusters):
                x_indices_in_centroid = np.where(min_indices == centroid_index)[0]
                X_in_centroid = X_reshaped[x_indices_in_centroid]
                centroids[centroid_index] = X_in_centroid.mean(axis=0)
        i+=1
        if i>min_iters:
            print("Exceeded 30")
            bContinueRunning=False
    return centroids

def k_means_code(X,num_clusters):

min_iters=30

X_reshaped=X.reshape(-1,3)

#this variable will get resized

past_min_indices =np.zeros(X_reshaped.shape[0])

# first set centroids to random points

initial_indices=np.random.choice(X_reshaped.shape[0],num_clusters, replace=False)

centroids=X_reshaped[initial_indices]

#convert to float to prevent rounding on the averages

centroids=centroids.astype(float)

i=1

bContinueRunning=True

while bContinueRunning:

distances=np.array((0,0))

for centroid_index,centroid in enumerate(centroids):

distance=np.sqrt(np.sum((X_reshaped - centroid) ** 2, axis=1))

if centroid_index==0:

distances=distance.copy()

else:

distances=np.vstack((distances,distance))

min_indices=distances.argmin(axis=0)

if np.sum(min_indices-past_min_indices)==0:

bContinueRunning=False

else:

past_min_indices=min_indices.copy()

for centroid_index in range(num_clusters):

x_indices_in_centroid = np.where(min_indices == centroid_index)[0]

X_in_centroid = X_reshaped[x_indices_in_centroid]

centroids[centroid_index] = X_in_centroid.mean(axis=0)

i+=1

if i>min_iters:

print("Exceeded 30")

bContinueRunning=False

return centroids

We can run this on a few images, reducing them from 16.77m colors to 16 colors to see how good the compression is

M	T	W	T	F	S	S
						1
2	3	4	5	6	7	8
9	10	11	12	13	14	15
16	17	18	19	20	21	22
23	24	25	26	27	28	29
30	31

Tag: Image Compression

K-Means for image compression