SVD

Quantum Image Compression

Experience the future of image compression through Singular Value Decomposition. Discover how mathematics transforms digital images into efficient quantum representations.

Ultra Fast

Lightning speed processing

Linear Algebra

Applied SVD concepts

High Quality

Preserve image fidelity

Precise Control

Fine-tune compression

Start Compressing

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Interactive SVD Compression

Experience the power of Singular Value Decomposition in real-time. Upload an image or use our default to see how mathematical decomposition can achieve impressive compression ratios while preserving visual quality.

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How Computers See: SVD Image Compression

A comprehensive exploration of Singular Value Decomposition for image compression, combining mathematical rigor with practical implementation.

The Problem

Modern images contain millions of pixels, making storage and transmission expensive. However, images are highly structured - local regions are correlated, smooth areas dominate over high-frequency noise, and edges are sparse.

The Challenge

Can we approximate an image with a much smaller representation while keeping it visually faithful?

My Solution

We use a clean linear-algebraic approach: treat the image as a matrix and compress it with low rank. The Singular Value Decomposition (SVD) provides a mathematically optimal way to achieve this.

The Approach

SVD factors any matrix into orthonormal directions and singular values that quantify structure.

Storage Efficiency

Achieve up to 90% compression while preserving visual quality

Mathematical Rigor

Provably optimal in Frobenius norm (Eckart–Young theorem)

Real-time Processing

Interactive compression with instant visual feedback

Image Compression Comparison

See how SVD compression dramatically reduces file size while preserving visual quality

Compression Analysis

File Size

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Original

—

Compressed

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Storage Values

—

Original

—

Compressed

—

Quality Retention

—

Original

100%

Compressed

—

Key Benefits

Faster loading times

Reduced storage costs

Preserved visual quality

Efficient transmission

How It Works

Decompose

Break image into U, Σ, Vᵀ components

Truncate

Keep only the most important singular values

Reconstruct

Rebuild image with reduced data

How to Compute an SVD

A comprehensive guided walkthrough of the step-by-step SVD computation process with mathematical insights

Mathematical Foundation

The SVD Decomposition

A = U Σ Vᵀ

Where U and V are orthonormal matrices, and Σ is diagonal with singular values σ₁ ≥ σ₂ ≥ ... ≥ σᵣ

Key Properties

U and V are orthonormal (UᵀU = I, VᵀV = I)
Σ contains non-negative singular values
Rank(A) = number of non-zero singular values

Computation Steps

Form AᵀA Matrix

Create the symmetric matrix AᵀA (n×n) with non-negative eigenvalues

AᵀA = (A^T) × A

Symmetric matrix

Non-negative eigenvalues

n×n dimensions

AᵀA is always symmetric and positive semi-definite, ensuring real, non-negative eigenvalues

Compute Eigenvalues & Eigenvectors

Find eigenvalues λ and eigenvectors of AᵀA; singular values are σ = √λ

σᵢ = √λᵢ

Eigenvalues sorted descending

Singular values = √(eigenvalues)

Eigenvectors form V

The singular values σᵢ are the square roots of the eigenvalues of AᵀA, sorted in descending order

Build V Matrix

Construct V from normalized eigenvectors; Vᵀ is its transpose

V = [v₁ | v₂ | ... | vₙ]

Normalized eigenvectors

Orthonormal columns

Vᵀ = V transpose

V contains the right singular vectors as columns, forming an orthonormal basis

Create Σ Matrix

Place singular values σ on the diagonal, pad with zeros to match A's size

Σ = diag(σ₁, σ₂, ..., σᵣ, 0, ..., 0)

Diagonal matrix

Singular values on diagonal

Zeros for padding

Σ is a diagonal matrix with singular values on the main diagonal, padded with zeros to match A's dimensions

Compute U Matrix

Calculate columns of U from A vᵢ = σᵢ uᵢ and normalize

uᵢ = (A vᵢ) / σᵢ

Normalize each column

Orthonormal basis

Left singular vectors

U contains the left singular vectors, computed from the relationship A vᵢ = σᵢ uᵢ

Assemble & Truncate

Form A = U Σ Vᵀ and optionally truncate to rank k

Aₖ = Uₖ Σₖ Vₖᵀ

Full decomposition

Rank-k truncation

Optimal approximation

The rank-k approximation Aₖ is the optimal rank-k approximation in Frobenius norm (Eckart-Young theorem)

Color Image Processing

Per-Channel Processing

Apply SVD separately to R, G, B channels for independent compression

Independent compression per channel

Channel-specific quality control

RGB space processing

Advantages:

• Simple implementation
• Independent optimization
• Easy to understand

Luminance/Chrominance

Transform to YCbCr space for better energy concentration

Y: luminance channel (most important)

Cb, Cr: chrominance channels

Energy concentration in Y

Advantages:

• Better compression ratios
• Perceptually meaningful
• Industry standard

Error Analysis

Monitor quality vs. compression trade-offs systematically

Rapid improvement for small k

Diminishing returns for larger k

Channel structure importance

Advantages:

• Quantitative quality assessment
• Optimal rank selection
• Performance prediction

Error Analysis & Insights

Error Curves

Typically show rapid improvement for small k, then diminishing returns

Rapid improvement for small k

Diminishing returns for larger k

Quality Insights

Retaining only small σ yields poor quality until many σ are included

Largest σ dominate quality

Green channel often has most structure

Channel Observations

Channels with more structure (often green) can dominate perceived quality

Red

Warm tones

Green

Most structure

Blue

Cool tones

Implementation Notes

Numerical Stability

• Use normalized eigenvectors
• Handle zero singular values
• Consider numerical precision
• Avoid ill-conditioned matrices

Performance Tips

• Parallel processing for RGB
• Caching for repeated computations
• Adaptive rank selection
• Memory-efficient algorithms

Practical Considerations

• Choose appropriate rank k
• Balance quality vs. compression
• Consider application requirements
• Test with various image types

Test Your Knowledge

Answer these questions to test your understanding of SVD and image compression.

Question 1

SVD can compress any matrix, not just images.

Question 2

Higher rank-k approximation always means better image quality.

Question 3

SVD compression is lossless (no quality loss).

Question 4

Color images require three separate SVD operations.

SVD

Quantum Image Compression

Ultra Fast

Linear Algebra

High Quality

Precise Control

Interactive SVD Compression

How Computers See: SVD Image Compression

The Problem

The Challenge

My Solution

The Approach

Storage Efficiency

Mathematical Rigor

Real-time Processing

Image Compression Comparison

Compression Analysis

File Size

Storage Values

Quality Retention

Key Benefits

How It Works

Decompose

Truncate

Reconstruct

How to Compute an SVD

Mathematical Foundation

The SVD Decomposition

Key Properties

Computation Steps

Form AᵀA Matrix

Compute Eigenvalues & Eigenvectors

Build V Matrix

Create Σ Matrix

Compute U Matrix

Assemble & Truncate

Color Image Processing

Per-Channel Processing

Luminance/Chrominance

Error Analysis

Error Analysis & Insights

Error Curves

Quality Insights

Channel Observations

Implementation Notes

Numerical Stability

Performance Tips

Practical Considerations

Test Your Knowledge

Question 1

Question 2

Question 3

Question 4

C. Eckart & G. Young

G. H. Golub & C. F. Van Loan

L. N. Trefethen & D. Bau

Maxine Calle

D. Austin

G. Strang

N. Halko, P.-G. Martinsson, J. A. Tropp

Education

Contact

Course

Email

LinkedIn

GitHub

Portfolio

Twitter

Instagram

How Computers See: SVD Image Compression

The Problem

The Challenge

My Solution

The Approach

Storage Efficiency

Mathematical Rigor

Real-time Processing

Image Compression Comparison

Compression Analysis

File Size