CCD Image Processing: CCD Image Processing: [ ] [ ] r x y , - PDF document

Correction of Raw Image Correction of Raw Image with Bias, Dark, Flat Images with Bias, Dark, Flat Images CCD Image Processing: CCD Image Processing: [ ] [ ] − r x y , d x y , Raw File [ ] Issues & Solutions Issues & Solutions r x y , Dark Frame “Raw” − “Dark” “Raw” − “Dark” [ ] d x y , “Flat” − “Bias” [ ] [ ] − [ ] [ ] − f x y , b x y , Flat Field Image r x y , d x y , Output [ ] [ ] [ ] − f x y , f x y , b x y , Image Bias Image “Flat” − “Bias” [ ] b x y , Correction of Raw Image Correction of Raw Image CCDs: Noise Sources CCDs : Noise Sources w/ Flat Image, w/o Dark Image w/ Flat Image, w/o Dark Image • Sky “Background” – Diffuse Light from Sky (Usually Variable) [ ] [ ] Assumes Small Dark Current − r x y , b x y , • Dark Current Raw File (Cooled Camera) [ ] r x y , – Signal from Unexposed CCD – Due to Electronic Amplifiers “Raw” − “Bias” • Photon Counting “Raw” − “Bias” Bias Image [ ] “Flat” − “Bias” – Uncertainty in Number of Incoming Photons b x y , [ ] [ ] − f x y , b x y , [ ] [ ] • Read Noise − Output r x y , b x y , [ ] [ ] Image − – Uncertainty in Number of Electrons at a Pixel f x y , b x y , Flat Field Image [ ] f x y , “Flat” − “Bias” Problem with Sky “ Problem with Sky “Background Background” ” Solution for Sky Background Solution for Sky Background • Uncertainty in Number of Photons from • Measure Sky Signal from Images Source – Taken in (Approximately) Same Direction (Region of Sky) at (Approximately) Same – “How much signal is actually from the Time source object instead of from intervening atmosphere? – Use “Off-Object” Region(s) of Source Image • Subtract Brightness Values from Object Values 1

Problem: Dark Current Solution: Dark Current Problem: Dark Current Solution: Dark Current • Signal in Every Pixel Even if NOT • Subtract Image(s) Obtained Without Exposed to Light Exposing CCD – Strength Proportional to Exposure Time – Leave Shutter Closed to Make a “ Dark Frame” • Signal Varies Over Pixels – Same Exposure Time for Image and Dark Frame – Non-Deterministic Signal = “NOISE” • Measure of “Similar” Noise as in Exposed Image • Actually Average Measurements from Multiple Images – Decreases “Uncertainty” in Dark Current Digression on Digression on “ “Noise Noise” ” Statistical Properties of Noise Statistical Properties of Noise 1. Average Value = “Mean” ≡ µ • What is “Noise”? 2. Variation from Average = “Deviation” ≡ σ • Noise is a “Nondeterministic” Signal – “Random” Signal – Exact Form is not Predictable • Distribution of Likelihood of Noise – “Statistical” Properties ARE (usually) – “Probability Distribution” Predictable More General Description of Noise than µ , σ • – Often Measured from Noise Itself • “Histogram” Histogram of “ Histogram of “Gaussian Gaussian” ” Histogram of “ “Uniform Distribution Uniform Distribution” ” Histogram of Distribution Distribution • Values are “Real Numbers” (e.g., 0.0105) • Values are “Real Numbers” • Noise Values Between 0 and 1 “Equally” Likely • NOT “Equally” Likely • Available in Computer Languages • Describes Many Physical Noise Phenomena Mean µ Histogram Noise Sample Mean µ Variation Mean µ Variation Mean µ Mean µ = 0 Values “Close to” µ “More Likely” Variation Mean µ = 0.5 Variation 2

Histogram of Histogram of “ “Poisson Poisson” ” Distribution Distribution Histogram of Histogram of “ “Poisson Poisson” ” Distribution Distribution • Values are “Integers” (e.g., 4, 76, …) • Describes Distribution of “Infrequent” Events, Mean µ e.g., Photon Arrivals Mean µ Variation Mean µ Variation Mean µ Variation Mean µ = 4 Mean µ = 25 Values “Close to” µ “More Likely” Variation “Variation” is NOT Symmetric How to Describe How to Describe “ “Variation Variation” ”: 1 : 1 Description of “ Description of “Variation Variation” ”: 2 : 2 • Measure of the “Spread” (“Deviation”) of • Sum of Errors over all Measurements: the Measured Values (say “x”) from the “Actual” Value, which we can call “ µ ” ∑ ∑ ( ) ε = − µ x n n n n • The “Error” ε of One Measurement is: Can be Positive or Negative ( ) ε = − µ • Sum of Errors Can Be Small, Even If x Errors are Large (Errors can “Cancel”) (which can be positive or negative) Description of “ Description of “Variation Variation” ”: 3 : 3 Description of Description of “ “Variation Variation” ”: 4 : 4 • Use “Square” of Error Rather Than Error • Sum of Squared Errors over all Itself: Measurements: ∑ ∑ ( ) ( ) ε 2 = − µ 2 ≥ x 0 ( ) ε = − µ 2 ≥ n n 2 x 0 n n • Average of Squared Errors ∑ ( ) Must be Positive − µ 2 x n 1 ∑ ( ) ε = ≥ 2 n 0 n N N n 3

σ Description of “ “Variation Variation” ”: 5 : 5 Effect of Averaging on Deviation σ Description of Effect of Averaging on Deviation • Example: Average of 2 Readings from • Standard Deviation σ = Square Root of Uniform Distribution Average of Squared Errors ∑ ( ) − µ 2 x n σ ≡ ≥ n 0 N Effect of Averaging of 2 Samples: Effect of Averaging of 2 Samples: σ Averaging Reduces σ Averaging Reduces Compare the Histograms Compare the Histograms Mean µ Mean µ • Averaging Does Not Change µ σ ≅ 0.205 σ ≅ 0.289 • “Shape” of Histogram is Changed! σ ≅ 0.289 0 . 289 – More Concentrated Near µ σ is Reduced by Factor: ≅ 1 . 41 0 . 205 – Averaging REDUCES Variation σ Averages of 4 and 9 Samples Averages of 4 and 9 Samples Averaging of Random Noise Averaging of Random Noise REDUCES the Deviation σ σ REDUCES the Deviation Samples Averaged N = 2 N = 4 N = 9 Reduction in 1.41 2.01 3.01 Deviation σ σ σ ≅ 0.096 σ ≅ 0.144 σ = One Sample Observation: Average of N Samples N Reduction Factors 0 . 289 0 . 289 ≅ ≅ 3 . 01 2 . 01 0 . 144 0 . 096 4

Why Does “ “Deviation Deviation” ” Decrease Decrease Averaging Over “ “Time Time” ” vs. vs. Why Does Averaging Over if Images are Averaged? if Images are Averaged? Averaging Over “ Averaging Over “Space Space” ” • “Bright” Noise Pixel in One Image may be “Dark” in Second Image • Examples of Averaging Different Noise • Only Occasionally Will Same Pixel be Samples Collected at Different Times “Brighter” (or “Darker”) than the Average in Both Images • Could Also Average Different Noise Samples Over “Space” (i.e., Coordinate x ) • “Average Value” is Closer to Mean Value – “Spatial Averaging” than Original Values Comparison of Histograms After Effect of Averaging on Dark Comparison of Histograms After Effect of Averaging on Dark Spatial Averaging Spatial Averaging Current Current • Dark Current is NOT a “Deterministic” Number – Each Measurement of Dark Current “Should Be” Different – Values Are Selected from Some Distribution of Likelihood (Probability) Spatial Average Spatial Average Uniform Distribution µ = 0.5 of 9 Samples of 4 Samples µ = 0.5 µ = 0.5 σ ≅ 0.289 σ ≅ 0.096 σ ≅ 0.144 Example of Dark Current Example of Dark Current Example of Dark Current Example of Dark Current Readings Readings Reading of Dark Current vs. Position Reading of Dark Current vs. Position • One-Dimensional Examples (1-D in Simulated Dark Image #1 in Simulated Dark Image #2 Functions) – Noise Measured as Function of One Variation Spatial Coordinate 5

Averages of Independent Dark Infrequent Photon Arrivals Averages of Independent Dark Infrequent Photon Arrivals Current Readings Current Readings • Different Mechanism Average of 2 Readings of Average of 9 Readings of – Number of Photons is an “Integer”! Dark Current vs. Position Dark Current vs. Position • Different Distribution of Values Variation “Variation” in Average of 9 Images ≅ 1/ √ 9 = 1/3 of “Variation” in 1 Image Problem: Photon Counting Problem: Photon Counting Simplest Distribution of Integers Simplest Distribution of Integers Statistics Statistics • Photons from Source Arrive • Only Two Possible Outcomes: “Infrequently” – YES – Few Photons – NO • Measurement of Number of Source • Only One Parameter in Distribution Photons (Also) is NOT Deterministic – “Likelihood” of Outcome YES – Random Numbers – Call it “ p ” – Distribution of Random Numbers of “Rarely – Just like Counting Coin Flips Occurring” Events is Governed by Poisson – Examples with 1024 Flips of a Coin Statistics Example with Example with p = p = 0.5 0.5 Second Example with Second Example with p p = 0.5 = 0.5 “H” “T” String of Outcomes String of Outcomes Histogram Histogram N = 1024 N = 1024 N heads = 522 N heads = 511 µ = 522/1024 > 0.5 p = 511/1024 < 0.5 6