**ANALYSING THE GRADE NODE**

**FROM A MATHEMATICAL PERSPECTIVE**

**0.**

**Introducing Equations of a Straight Line**

**:**

To begin, we must fully comprehend the concepts of the equation of a straight line, as it is key to the formulas in the

*Grade*node.The general equation of a straight line is:

**y = mx + c**

**y**: is the ordinate or vertical plane.

**x**: is the abscissa or horizontal plane.

**m**: is the gradient or

**slope**

**c**: is the intersection of the ordinates or

**y-intercept**(a constant value and the

**y**value when

**x**= 0).

**1. Analysing the Grade node from a mathematical perspective**

**:**

Fortunately,

*The Foundry*has released the source code of some of the*Nuke*plugins (link), including the*Grade*node (link).Concise information about how it works is to be found in the very first lines of the source code:

"Applies a linear ramp followed by a gamma function to each color channel.

A = multiply * (gain-lift)/(whitepoint-blackpoint)

B = offset + lift - A*blackpoint

output = pow(A*input + B, 1/gamma)

The reverse option is also provided [...]. This will do the opposite gamma correction followed by the opposite linear ramp."

The first line:

"Applies a linear ramp followed by a gamma function to each color channel"

Translates as:

The

The

*Grade*node will apply an equation of straight line followed by a gamma function.This is precisely what the last formula in the source code does:

output = pow(A*input + B, 1/gamma).

pow (base, exponent) is the

*C++*format to performing an exponential operation so that formula is in fact:**output = (A*input + B) ^ (1/gamma)**.

From here on, we will call that formula the

**final formula**, as is the one in charge of producing the output (final result) of the node's operation.To recap:

**output**

**=**Applies a

**linear ramp**followed by a

**gamma function**to each color channel.

**output = (A*input + B) ^ (1/gamma)**

Next, we see a rather simplified summary of the execution of the source code in regards to the application of such formulas (lines 79 to 158):

- Runs through the applicable channels → one at a time (e.g. first the red, then green, etc.) → pixel by pixel, one at a time(e.g. red channel, value of pixel at positionx, y)→ andapplies the formulas→ then onto the next pixel until all resolution for that channel is completed → then continues with the next channel until all applicable channels are completed.

**1.1 The final formula without the gamma function:**

**output =**

**(A*input + B)**

^{(1/gamma)}The 'first part' of the

**final formula**(the**final formula**without the**gamma function**) is an equation of a straight line:

*output = A*input + B*

**input**(

**x**, the abscissa) is the input pixel value (before the transformation): e.g. red = 0.212.

**output**

*(*

**y**, the ordinate) is the resulting pixel value (after the transformation): e.g. red = 0.4332.

value

**A**is the**slope**.and value

**B**is the**y-intercept**.In the following example, we will assume certain values for

**A**and**B**to illustrate on the operation of the equation of a straight line:input = 0.212A = 1.1B = 0.2output = (A*input + B) → output = 1.1 * 0.212 + 0.2 → output = 0.4332

This rather simplistic example (the

Let's introduce

*Grade*node is rather simplistic in process, but not in its comprehensibility) shows the transformation of an**input**pixel value of 0.212 into an**output**pixel value of 0.4332 (for a given channel, e.g., red). This example could easily represent an operation in the Grade without gamma (since we have stripped that part of the equation for now).Let's introduce

**B**next.**1.2 B:**

**B = offset + lift - A*blackpoint**The formula that returns

**B**is also an equation of a straight line. However, this equation of a straight line has a negative (or downward')

**slope**. The negative sign in the

**slope**means that any change in the value of

**x**results in the opposite effect to the value of

**y**:

Equation of a straight line with a positive slope: Equation of a straight line with a negative slope:

- increment in

**x**→ increment in**y**. - increment in**x**→ reduction in**y**.- reduction in

**x**→ reduction in**y**. - reduction in**x**→ increment in**y**.This negative

**slope**consolidates the function of the**blackpoint**knob. For in the formula of**B**, the**blackpoint**is**x**and any change in the value of**x**(**blackpoint**value) produces the opposite effect to the value of**y**(the**B**value). Increasing the**blackpoint**value reduces the**output**value to the point that when**blackpoint**value =**input**value*→***output**value = 0.To illustrate, let us assume the following values:

input = 0.2blackpoint = 0.2All the other knobs at their default valueA = multiply * (gain-lift)/(whitepoint-blackpoint)→A = 1 * (1-0)/(1-0.2)→A = 1.25B = offset + lift - A*blackpoint → B = 0 - 1.25 * 0.2 → B = - 0.25output = input*A + B → output = 0.2 *1.25+ -0.25→ output = 0

Hence, because

**blackpoint**drives the

**B**value, and the

**B**value (i.e.

**y-intercept**) is the minimum

**output**value in the

**final formula**, the definition of

**blackpoint**:

- Blackpoint: sets the black point (typically the darkest pixel).

Let's proceed to analyze

**offset**and**lift**,**which in the formula of****B**make the**y-intercept**. As**y-intercept**,**they add a fixed value to****y**, in other words, to the**B**value.- Offset: offsetting is to add a fixed value to the result.
- Lift: any pixels that are black are set to this color.

To illustrate this, let us build up from the previous example by adding values to

**offset**and**lift**:input = 0.2blackpoint = 0.2offset = 0.1lift = 0.5All the other knobs at their default valueA = multiply * (gain-lift)/(whitepoint-blackpoint)→A = 1 * (1-0.5)/(1-0.2)→A = 0.625B = offset + lift - A*blackpoint→ B = 0.1 + 0.5 -0.625* 0.2 → B = 0.475output = input*A + B→ output = 0.2 *0.625+0.475→ output = 0.6

As we can see, a combined fixed value of 0.6 (

**offset**: 0.1 +**lift**: 0.5) has been added to the output result (**output**value), because the value of**B**is the**y-intercept**of the**final formula**.According to the above example, one might think that

**offset**and**lift**serve the same function. However, that is not the case because**lift**is also present in the formula of**A**, where it finally sees its function redefined as the opposite of**blackpoint**.Without further ado, let's analyze

**A**.**1.3 A:**

**A**is the

**slope**in both the formula for

**B**(

**B = offset + lift - A*blackpoint**) and the

**final formula**(

**output = A*input + B**).

*A = multiply * (gain-lift)/(whitepoint-blackpoint)***multiply**multiplies the result of the right-hand side part of the formula (i.e. a multiplication to the

**slope**value). This multiplication is effectively a multiplication to the

**output**value but performed via the value of the

**slope**(

**A**value) so as not to affect the

**y-intercept**value (

**B**value), hence the definition:

- Multiply: multiplies the result of the Grade node by this factor.

The remaining part of the equation is the 'core' of the

**slope**value:

*(gain-lift)/(whitepoint-blackpoint)*

The value for the

**slope**is the result of a division between the two ends of the node's spectrum:**gain**,**whitepoint**; and**lift**,**blackpoint**. The result of a division between two numbers is in a sense a scale factor; in our case, it represents the change to the range of the image, the coefficient of**input**and**output**.-

**Slope**value = 1*→*no change (1:1 of**input****to****output**).-

**Slope**value > 1*→*increase of**output**.-

**Slope**value < 1*→*decrease of**output**.While the above is the impact of the

**slope**value as a whole, the function of each knob is defined by its position in the numerator (the top number of the fraction) or denominator (the bottom number of the fraction). While the commonality between**gain**,**lift**and**whitepoint**,**blackpoint**is defined by their shared position in the numerator or denominator field, their antagonism is defined by the mathematical operation of subtraction among them.Because

**gain**is in the numerator field: any change to its value produces the same effect on the**slope**value (**A**value) which carries the effect onto the**output**value. e.g. increase in**gain**value → increase in**slope**value (**A**value) → increase in**output**value.- Gain: any pixels that are white are set to this color.

To illustrate this, let us assume that:

input = 1gain = 0.7all other knobs are at their default value.A = multiply * (gain-lift)/(whitepoint-blackpoint) → A = 1 * (0.7-0/1-0) → A = 0.7B = offset + lift - A*blackpoint → B = 0 + 0 - 0.7 * 0 → B = 0output = A * input + B → output = 0.7 * 1 + 0 → output = 0.7

As we can see, the reduction of the

**gain**value by 0.3 (from 1 to 0.7) produced (via the**slope**value) a reduction of 0.3 of the**input**value to the**output**value (from 1 to 0.7). That is: what was previously 1 (i.e. white) is now equal to the value of**gain**(0.7).Because

**lift**is also in the numerator field, the same 'dynamic' applies:- Lift: any pixels that are black are set to this color.

To illustrate, let us assume that:

input = 0lift = 0.3all other knobs are at their default value.A = multiply * (gain-lift)/(whitepoint-blackpoint)→ A = 1 * (1-0.3/1-0) → A = 0.7B = offset + lift - A*blackpoint→ B = 0 + 0.3 - 0.7 * 0 → B = 0.3output = A * input + B→ output = 0.7 * 0 + 0.3 → output = 0.3

As we can see, an

**input**value of 0 has become equal to the value of**lift**.On the other hand,

**whitepoint**is in the denominator field, which means that any change to its value produces the opposite effect on the**slope**value (**A**value) and therefore on the**output**value. e.g. decrease in**whitepoint**value*→*increase in**slope**value (**A**value)*→*increase in**output**value.That is, as we decrease the

**whitepoint**value, the**output**value increases to the point that when**whitepoint**value =**input**value →**output**value = 1, which is precisely this knob's definition:- Whitepoint: sets the white point (typically the lightest pixel). Any pixels of this value are set to 1. In other words, this color is turned into pure white.

To illustrate, let us assume that:

input = 0.7whitepoint = 0.7all other knobs are at their default value.A = multiply * (gain-lift)/(whitepoint-blackpoint)→ A = 1 * (1-0/0.7-0) → A = 1,4285B = offset + lift - A*blackpoint→ B = 0 + 0 - 0.7 * 0 → B = 0output = A * input + B→ output =1,4285 * 0.7 + 0 → output = 1

As

**blackpoint**is also in the denominator field, any change to its value produces the opposite effect on the**slop****e**value:- Blackpoint: sets the black point (typically the darkest pixel).

To illustrate this, let us assume that:

input = 0.3blackpoint = 0.3all other knobs are at their default value.A = multiply * (gain-lift)/(whitepoint-blackpoint)→ A = 1 * (1-0/1-0.3) → A = 3,3333B = offset + lift - A*blackpoint→ B = 0 + 0 – 3,3333*0.3 → B = -1output = A * input + B→ output =3,3333 * 0.3 - 1 → output = 0

As we have seen, because of their shared position in the fields of the division and in regards to the

**slope**value:**lift**has the same dynamic as**gain**and**blackpoint**as**whitepoint**, however,**lift**and**blackpoint**are (despite the two other) present in the formula of**B**where they ultimately see their function defined.Contrary to certain misconceptions,

**multiply**and**gain**do not have the same function. However, they are the same if the rest of the other knobs are at their default value (excluding**gamma**, as it being outside the formula of the equation of a straight line does not apply as it is ).**1.4 The gamma function:**

The complete

**final formula**consists of the equation of a straight line followed by a**gamma function.**The**gamma function**is the part in which the result from the equation of a straight line (the**base**) is raised to**1/gamma**(the**exponent**):

**output = (A*input + B)**^{(1/gamma)}The function of the

**gamma**knob is carried out via the**gamma function**as a result of the following intrinsic property of the exponential operations: the strength of the multiplicative effect is maximal when the value of the**base**is 0.5 and decreases as the value of the**base**approaches either 0 or 1.Let us illustrate in the following example:

0.15^{2}= 0.02250.5^{2}= 0.250.85^{2}= 0.7225

The effect has been maximal when the

**base**is 0.5 (0.5 - 0.25 = 0.25*difference*), but as the**base**is close to 0 (0.15) the effect has been reduced (0.15 - 0.0225 = 0.1275*difference*) as well as when the**base**is close to 1 (0.85) (0.85 - 0.7225 = 0.1275*difference*).- Gamma: applies a constant gamma value to the result of the Grade node. This lightens or darkens the midtones.

However, as shown in the example above, raising a

**base**value of less than 1 to a higher**exponent**value produces, rather than an increase, a decrease in the**output**value. Hence, to avoid it exists the division of**1**by the value of**gamma**in the formula, to invert the value of**gamma**(e.g. 1/0.75 = 1,333) so that any change in the**gamma**value produces an effect in the same 'direction' on the**output**value.To illustrate this, the following chart shows a comparison:

Gamma value: 2 | Gamma value with 1/gamma → 1/2 = 0.5 |

0.15 ^{2} = 0.0225 | 0.15 ^{(1/2)} = 0.3872 |

0.5 ^{2} = 0.25 | 0.5 ^{(1/2)} = 0.7071 |

0.85 ^{2} = 0.7225 | 0.85 ^{(1/2)} = 0.9219 |

**1.5 The hidden 'side'**

**of the gamma function:**

So far, until now we have worked with the information easily accessible in the first lines of source code, however, the

**gamma function**has some 'additional' parameters established within the code itself. These parameters (lines: 128 to 153) establish that starting from the result of the equation of a straight line (i.e. the value of the**base**) when performing the**gamma function**:- If the value of the
**base**is less than 0: the**output**will be the value of the**base**, in other words, the**gamma function**is discarded. This is because any value less than 0 is considered to be outside the range of action of**gamma**.

- If the value of the
**base**is greater than 1, the**gamma function**is replaced by:**(base value -1) * (1/gamma value) + 1**, which is in fact another equation of a straight line (i.e.**y**=**m*****x**+**c**). This means that from a value of 1, the**gamma**knob continues the trend of the**gamma function**curve in**a linear progression.**