Weibull Distribution Calculator

Model edge magnitudes, texture scales, and sensor reliability. Compute PDF, CDF, interval probabilities & visualize.

Image processing context: Weibull models edge magnitude distributions, SAR image texture, and lifetime of imaging sensors. Shape k < 1 → decreasing failure rate, k = 1 → exponential, k > 1 → increasing.
λ > 0; characteristic life (63.2% fail)
k > 0; determines failure rate behavior
X ≥ 0 (PDF = 0 for negative)
≥0
P(X₁ < X < X₂)
λ=1, k=1.5 λ=1, k=1 (exponential) λ=1, k=3 (sensor) λ=2, k=0.8 (decreasing) λ=1.5, k=2 Interval [0.5,1.5]
Computing Weibull probabilities...

Weibull Distribution in Imaging

The Weibull distribution is a versatile model for positive‑valued phenomena. In image processing, it is used for:

  • Edge magnitude distribution – natural images yield Weibull‑like histograms of gradient magnitudes.
  • Texture analysis – parameters λ and k characterize coarseness and regularity.
  • SAR image statistics – amplitude in some clutter models follows Weibull.
  • Sensor lifetime – reliability of imaging sensors (k > 1 indicates wear‑out).

Weibull PDF & CDF:

f(x; λ,k) = (k/λ) (x/λ)k-1 e-(x/λ)^k, x ≥ 0

F(x) = 1 - e-(x/λ)^k

Special case: k = 1 → exponential(1/λ); k = 2 → Rayleigh (scale = λ√2).

1
Scale λ — characteristic life; 63.2% of distribution lies below λ.
2
Shape k — failure rate: k < 1 decreases, k = 1 constant, k > 1 increases.

Interpreting Results

  • PDF(x): Likelihood of edge magnitude x. High k → peaked, low k → heavy tail.
  • CDF(x): Fraction of pixels with edge strength ≤ x.
  • P(X > x): Tail probability – e.g., probability of extreme sensor stress.
  • Interval probability: e.g., probability edge magnitude between 0.8 and 1.2.
Texture example: For a certain fabric texture, gradient magnitudes follow Weibull with λ = 0.6, k = 1.8. Using this calculator, you can find that P(X > 1.0) = e-(1/0.6)^1.8 ≈ 0.13, indicating a low probability of strong edges.

Frequently Asked Questions

Natural image gradient histograms are heavy‑tailed and well‑approximated by Weibull. The shape parameter k correlates with texture coarseness: smooth regions → k ≈ 2, textured → k ≈ 0.5–1.

Rayleigh distribution is Weibull with shape k = 2 and scale σ = λ/√2. In SAR amplitude imagery, Rayleigh models speckle in homogeneous regions; Weibull extends to heterogeneous terrain.

Yes, X=0 is allowed. For k > 1, PDF(0)=0; for k = 1, PDF(0)=1/λ; for k < 1, PDF(0)=∞. The calculator handles these cases correctly.

Maximum likelihood estimation or linear regression on a Weibull plot (ln(-ln(1-F)) vs ln x). Many image processing libraries provide these estimators.